MULTIPLIER CONVERGENT SERIES
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MULTIPLIER CONVERGENT SERIES Charles Swartz New Mex...

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Charles Swartz

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MULTIPLIER CONVERGENT SERIES

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MULTIPLIER CONVERGENT SERIES Charles Swartz New Mexico State University, USA

World Scientific NEW JERSEY

•

LONDON

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SINGAPORE

•

BEIJING

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SHANGHAI

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HONG KONG

•

TA I P E I

•

CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

MULTIPLIER CONVERGENT SERIES Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-283-387-7 ISBN-10 981-283-387-0

Printed in Singapore.

ZhangJi - Multiplier Convergent Series.pmd

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To the Memory of My Mother

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Preface

This monograph contains an exposition of the properties and applications of multiplier convergent series with values in a topological vector space. P is a (formal) series If λ is a space of scalar valued sequences and j xjP with values in a topological vector space X, the series j xj is λ multiplier P∞ convergent if the series j=1 tj xj converge in X for every {tj } ∈ λ. For example, if M0 = {χσ : σ ⊂ N}, where χσ is the characteristic function of σ, then M0 multiplier convergence is just subseries convergence. Basic properties of multiplier convergent series are developed in Chapter 2 and applications of multiplier convergent series to topics in topological vector spaces and vector valued measures are given in Chapter 3. A classical result of Orlicz and Pettis states that if a series in a normed linear space is subseries convergent (M0 multiplier convergent) in the weak topology of the space, then the series is actually subseries convergent (M0 multiplier convergent) in the norm topology of the space. Generalizations of this theorem to λ multiplier convergent series with values in a locally convex space are given in Chapters 4, 5 and 6. Another classical theorem of Hahn P and Schur asserts that if j tij is absolutely convergent for every i ∈ N P and if limi j∈σ tij exists for every σ ⊂ N with tj = limi tij , then the series P j tj is absolutely convergent and lim i

∞ X j=1

|tij − tj | = 0.

In Chapter 7 we establish generalizations of the Hahn-Schur Theorem to λ multiplier convergent series with values in a topological vector space. Chapters 8, 9 and 10 contain applications of the Hahn-Schur Theorems to spaces of multiplier convergent series, double series and automatic continuity of matrix mappings between sequence spaces. vii

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Chapter 11 extends the notion of multiplier convergent series to series with operator values and multiplier sequences with values in the domains of the operators. Chapters 12 and 13 extend the Orlicz-Pettis Theorem and Hahn-Schur Theorem to operator valued series and vector valued multipliers. Chapter 13 also contains applications to measures with values in a space of continuous linear operators. Chapter 14 considers automatic continuity results for operator valued matrices acting on vector valued sequence spaces.

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Contents

Preface

vii

1. Introduction

1

2. Basic Properties of Multiplier Convergent Series

5

3. Applications of Multiplier Convergent Series

25

4. The Orlicz-Pettis Theorem

49

5. Orlicz-Pettis Theorems for the Strong Topology

83

6. Orlicz-Pettis Theorems for Linear Operators

89

7. The Hahn-Schur Theorem

101

8. Spaces of Multiplier Convergent Series and Multipliers

133

9. The Antosik Interchange Theorem

145

10. Automatic Continuity of Matrix Mappings

157

11. Operator Valued Series and Vector Valued Multipliers

169

12. Orlicz-Pettis Theorems for Operator Valued Series

187

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13. Hahn-Schur Theorems for Operator Valued Series

191

14. Automatic Continuity for Operator Valued Matrices

201

Appendix A. Topological Vector Spaces

207

Appendix B. Scalar Sequence Spaces

213

Appendix C. Vector Valued Sequence Spaces

229

Appendix D. The Antosik-Mikusinski Matrix Theorems

239

Appendix E. Drewnowski’s Lemma

243

References

245

Index

251

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Chapter 1

Introduction

One of the most interesting and useful theorems in the early history of functional analysis is a result now known as the Orlicz-Pettis Theorem. The result was originally proven by Orlicz for weakly sequentially complete normed spaces although the result in full generality for normed spaces was known by the Polish mathematicians and appears in Banach’s book ([Or], [Ba]). The first version of the theorem available in English was proven by Pettis and was used to treat topics in vector valued measures and vector valued integrals ([Pe]; see [Ka3] and [FL] for discussions of the history of P the theorem). If X is a topological vector space (TVS), a series j xj in P∞ X is subseries convergent in X if the subseries j=1 xnj converges in X for every subsequence {nj }. The Orlicz-Pettis Theorem for normed spaces P states that if the series j xj is subseries convergent in the weak topology of the space, then the series is actually subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was extended to locally convex spaces by McArthur ([Mc]). If σ is any subset of N and χσ is the P characteristic function of σ, then a series j xj in a TVS X is subseries P∞ P convergent iff the series j=1 χσ (j)xj = j∈σ xj converges in X for every σ ⊂ N. Thus, if m0 = span{χσ : σ ⊂ N}, the sequence space of real P valued sequences with finite range, a series j xj in a TVS X is subseries P∞ convergent iff the series j=1 tj xj converges for every t = {tj } ∈ m0 . To obtain a generalization of the notion of subseries convergence, we may replace the space m0 by a general vector space λ of real valued sequences. If λ is a vector space of real valued sequences and {xj } is a sequence in the P TVS X, the (formal) series j xj is said to be λ multiplier convergent if P∞ the series j=1 tj xj converges in X for every t = {tj } ∈ λ; the elements t ∈ λ are called multipliers. This suggests that generalizations of the Orlicz1

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Pettis Theorem might be obtained by replacing subseries convergent series by λ multiplier convergent series for certain sequence spaces λ. We show that such generalizations are possible in Chapters 4, 5, and 6. Another classical result which involves subseries convergent series is a result which is often referred to as the Hahn-Schur Theorem. One verP sion of the Hahn-Schur Theorem states that if j xij is subseries conP∞ vergent for every i ∈ N, limi j=1 xinj exists for every subsequence {nj } P and if xj = limi xij , then the series j xj is subseries convergent and P P limi j∈σ xij = j∈σ xj uniformly for σ ⊂ N ([Ha], [Sc], [Sw1]; this version of the theorem actually holds for series with values in an Abelian P topological group). A series j xj in a TVS is said to be bounded mulP tiplier convergent if the series j xj is l∞ multiplier convergent ([Day]). There is a version of the Hahn-Schur Theorem for bounded multiplier conP vergent series which states that if j xij is bounded multiplier converP∞ gent for every i ∈ N, limi j=1 tj xij exists for every t = {tj } ∈ l∞ and P if xj = limi xij , then the series j xj is bounded multiplier convergent P∞ P∞ and limi j=1 tj xij = j=1 tj xj uniformly for t ∈ l∞ , ktk∞ ≤ 1 ([Sw2]). Again this suggest that one might obtain generalizations of both versions of the Hahn-Schur Theorem by replacing subseries and bounded multiplier convergent series by λ multiplier convergent series for certain sequence spaces λ. We show in Chapter 7 that versions of the Hahn-Schur Theorem are obtainable for λ multiplier convergent series if the sequence space λ satisfies sufficient conditions. There are further applications of λ multiplier convergent series to topics in Banach space theory, sequence spaces and matrix mappings. For example, a result of Bessaga and Pelczynski states that a Banach space X contains no subspace isomorphic to c0 iff every c0 multiplier convergent series in X is subseries convergent (or bounded multiplier convergent) ([BP]). A generalization of this result to sequentially complete locally convex topological vector spaces (LCTVS) is given in Chapter 3.15. A characterization of dual spaces not containing c0 is given in terms of subseries convergent series in 3.20, a characterization of locally complete LCTVS in terms of c0 multiplier convergent series is given in 3.10 and a characterization of Banach-Mackey spaces in terms of l 1 multiplier convergent series is given in 3.23. In Chapter 3, we also give applications of multiplier convergent series to vector valued measures. We give a characterization of bounded vector measures in terms of c0 multiplier convergent series in 3.33 and a characterization of strongly bounded (strongly additive) vector measures in terms of subseries convergence in 3.43.

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Introduction

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3

Further applications to various topics in sequence spaces and matrix mappings between sequence spaces are given in later chapters. Multiplier convergent series are interesting in their own right and we develop their basic properties in Chapter 2. In the last four chapters we consider operator valued series and vector valued spaces of multipliers. Let X, Y be TVS, L (X, Y ) the space of all continuous linear operators from X into Y and E be a vector space of X P valued sequences. A series j Tj in L(X, Y, ) is E multiplier convergent P∞ if the series j=1 Tj xj converges in Y for every sequence {xj } ∈ E. The basic properties of operator valued series with vector valued multipliers sometimes closely parallels the properties of series with scalar multipliers but sometimes require additional assumptions. We present these properties in Chapter 11. Versions of the Orlicz-Pettis Theorem and the Hahn-Schur Theorem for operator valued series and vector valued multipliers are presented in Chapters 12 and 13. The basic notations, definitions and terminology are presented in Appendices A, B and C. Appendices D and E contain material not easily accessible and which is used in the text.

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Chapter 2

Basic Properties of Multiplier Convergent Series

In this chapter we will develop the basic properties of multiplier convergent series. In what follows λ will denote a scalar sequence space which contains the subspace c00 of sequences which are eventually 0 and Λ ⊂ λ will denote a subset while X will denote a Hausdorff topological vector space (TVS). P Definition 2.1. A (formal) series j xj in X is Λ multiplier convergent P∞ in X if the series j=1 tj xj converges in X for every t = {tj } ∈ Λ . The P∞ series is Λ multiplier Cauchy in X if the series j=1 tj xj satisfies a Cauchy condition for every t = {tj } ∈ Λ. The elements t = {tj } ∈ Λ are called multipliers. P subseries A series j xj which is m0 multiplier convergent is said to be P convergent; thus, a series is subseries convergent iff the subseries j xnj is P convergent for every subsequence {nj }. A series j xj which is l∞ multiplier convergent is said to be bounded multiplier convergent. We now establish the basic properties of Λ multiplier convergent series. We begin by considering boundedness properties of Λ multiplier convergent series. P convergent series. The summing operator Let j xj be a λ multiplier P S (with respect to λ and j xj ) is the linear map S : λ → X defined by P∞ St = S({tj }) = j=1 tj xj for t = {tj } ∈ λ. Recall that the β-dual of λ, λβ , is defined to be ∞ X λβ = {sj } : sj tj = s · t converges for every t ∈ λ j=1

β

and λ and λ form a dual pair under the bilinear pairing s · t (Appendix A). 5

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If Y, Y 0 are a dual pair, a Hellinger-Toeplitz topology defined for dual pairs is a topology w(Y, Y 0 ) with the property that whenever a linear map T : Y → Z is σ(Y, Y 0 ) − σ(Z, Z 0 ) continuous, then T is w(Y, Y 0 ) − w(Z, Z 0 ) continuous (Appendix A.1). Theorem 2.2. Let X be a Hausdorff locally convex TVS (LCTVS) and P The summing operator j xj a λ multiplier convergent series in X. S : λ → X is σ(λ, λβ ) − σ(X, X 0 ) continuous and, therefore, w(λ, λβ ) − w(X, X 0 ) continuous with respect to any Hellinger-Toeplitz topology w. Proof: Let x0 ∈ X 0 , t ∈ λ. Then hx0 , Sti =

∞ X j=1

tj hx0 , xj i = {hx0 , xj i} · t

since {hx0 , xj i} ∈ λβ by the convergence of the series. This implies that S is σ(λ, λβ ) − σ(X, X 0 ) continuous. The last statement follows from the definition of Hellinger-Toeplitz topologies. Theorem 2.2 gives a boundedness result for multiplier convergent series Corollary 2.3. If B is σ(λ, λβ ) bounded, then SB = { is bounded in X.

P∞

j=1 tj xj

: t ∈ B}

For topological sequence spaces we also have a boundedness result for the sums of multiplier convergent series. Corollary 2.4. Let λ be a K-space. If λβ ⊂ λ0 and B ⊂ λ is bounded, then P SB = { ∞ j=1 tj xj : t ∈ B} is bounded in X.

Proof: Since B is bounded in X and λβ ⊂ λ0 , B is σ(λ, λβ ) bounded so the result follows from Corollary 2.3. For a general condition which guarantees that λβ ⊂ λ0 , we have Proposition 2.5. (i) If λ is a barrelled K-space, then λβ ⊂ λ0 . (ii) If λ is an AK-space, then λ0 ⊂ λβ . (iii) If λ is a barrelled AK-space, then λ0 = λβ .

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Proof: (i): Let s ∈ λβ . For each n define fn : λ → R by fn (t) = Pn j=1 sj tj . Since λ is a K-space, each fn is continuous. Now fn (t) → P∞ j=1 sj tj = s · t so the linear functional t → s · t is continuous since λ is barrelled. Therefore, s ∈ λ0 .

P∞ j (ii): Let f ∈ λ0 . Set sj = f, e j . If t ∈ λ, then t = j=1 tj e P P∞ ∞ j (convergence in λ) so hf, ti = j=1 tj f, e = j=1 tj sj = s · t so f = s ∈ λβ . (iii) follows from (i) and (ii). Proposition 2.5 is applicable, in particular, if λ is a Banach or Frechet space. Concerning the strong boundedness of partial sums of multiplier convergent series, we have Corollary 2.6. Suppose that λ is a barrelled AB space (Appendix B.3) and Pn P j=1 tj xj : j xj is λ multiplier convergent. If B ⊂ λ is bounded, then { n ∈ N, t ∈ B} is β(X, X 0 ) bounded. Pn Proof: Let Pn : λ → λ be the sectional operator Pn (t) = j=1 tj ej . By the AB assumption {Pn : n} is pointwise bounded on λ and, therefore, {Pn : n} is equicontinuous by the barrelledness assumption. Since λ is barrelled, 0 0 λ has the strong topology β(λ, λ ) so {Pn B : n} is β(λ, λ ) bounded. By Proposition 2.5, λβ ⊂ λ0 so {Pn B : n} is β(λ, λβ ) bounded. The result now follows from Theorem 2.2 since the strong topology is a Hellinger-Toeplitz topology. In particular, Corollary 2.6 is applicable to bounded multiplier convergent series. P Corollary 2.7. Let j xj be bounded multiplier convergent ( l ∞ multiplier P∞ convergent). Then { j=1 tj xj : k{tj }k∞ ≤ 1} is β(X, X 0 ) bounded. Since m0 is barrelled (Theorem 7.59 or [Sw1] 4.7.9), we also have P Corollary 2.8. Let j xj be subseries convergent (m0 multiplier converP gent). Then { j∈σ xj : σ ⊂ N} is β(X, X 0 ) bounded. The condition that λβ ⊂ λ0 in Corollaries 2.6, 2.7 and 2.8 is important.

Example 2.9. Let λ = c00 with the sup-norm. For any {xj } ⊂ X the P series j xj is c00 multiplier convergent. Take any unbounded sequence

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{xj } in X. Then {ek : k ∈ N} is bounded in c00 , but { 0

1

k ∈ N} is not bounded in X. Note that c00 = l ⊂

cβ00

Pk

j=1

ekj xj = xk :

= s.

We have another result for boundedness in topological sequence spaces. This result requires a gliding hump assumption. An interval in N is a set of the form I = {k ∈ N : m ≤ k ≤ n}, where m ≤ n. A sequence of intervals {Ij } is increasing if max Ik < min Ik+1 for all k. If x = {xk } is any sequence (scalar or vector) and σ ⊂ N, χσ x will denote the coordinatewise product of χσ and x. Definition 2.10. Let λ be a K-space. Then λ has the zero gliding hump property (0-GHP) if whenever {Ij } is an increasing sequence of intervals and {tj } ⊂ λ converges to 0 in λ, there is a subsequence {nj } such that the P∞ coordinatewise sum of the series j=1 χInj tnj belongs to λ. For examples of spaces with 0-GHP, see Appendix B. P Theorem 2.11. Let λ be a K-space with 0-GHP. If j xj is λ multiplier convergent in X, then the summing operator S : λ → X is sequentially continuous and, therefore, bounded.

Proof: Suppose the conclusion fails. Then there exist a closed neighborP∞ /U hood of 0, U , in X and a null sequence {ti } in λ such that j=1 tij xj ∈ P n 1 m1 for every i. Set m1 = 1. There exists n1 > m1 such that j=1 t j xj ∈ / U. Pick a closed symmetric neighborhood of 0, V , such that V + V ⊂ U . Since P 1 m2 t xj ∈ V . limi tij = 0 for every j, there exists m2 > m1 such that nj=1 P 2 j m P n 2 m2 / t 2 xj ∈ / U . Hence, nj=n There exists n2 > n1 such that j=1 tj xj ∈ 1 +1 j V . Continuing this construction produces increasing sequences {mj }, {nj } such that (∗)

X

j∈Ik

k tm / V, where Ik = [nk + 1, nk+1 ]. j xj ∈

P∞ By 0-GHP, there is a subsequence {pk } such that t = k=1 χIpk tpk ∈ P∞ P P∞ mp k xj should converge. But, this λ. Hence, j=1 tj xj = k=1 j∈Ipk tj contradicts (∗). Corollary 2.12. Let the assumptions be as in Theorem 2.11. If B ⊂ λ is bounded, the SB ⊂ X is bounded. Remark 2.13. Note that λ = c00 does not have 0-GHP so this assumption in Theorem 2.11 cannot be dropped. See Example 2.9.

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We next consider uniform convergence for multiplier convergent series when the multipliers range over certain bounded subsets of the multiplier space. These results also require gliding hump assumptions. In the following definition a sign is a variable which assumes the values {±1}. Definition 2.14. Let λ be a K-space and Λ ⊂ λ. The set Λ has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {tk } in Λ and every increasing sequence of intervals {Ik }, there is a subsequence {nk } and a sequence of signs {sk } such that the coordinatewise P∞ sum k=1 sk χInk tnk belongs to Λ. The set has the strong gliding hump property (SGHP) if the signs in the definition above can all be chosen to be equal to 1. See Appendix B for examples. For example, l ∞ has SGHP and bs has signed-SGHP but not SGHP. Λ = {χσ : σ ⊂ N} = M0 has SGHP whereas m0 = spanM0 does not. We first establish a lemma. P Lemma 2.15. Let j xj be Λ multiplier convergent where Λ ⊂ λ. If the P series ∞ j=1 tj xj do not converge uniformly for t ∈ B ⊂ Λ, then there exist a symmetric neighborhood of 0,V , in X , tk ∈ B and an increasing sequence P / V. of intervals {Ik } such that j∈Ik tkj xj ∈ P∞ Proof: If the series j=1 tj xj do not converge uniformly for t ∈ B, there exists a symmetric neighborhood, U , of 0 such that for every k there P∞ / U . For k = 1, let m1 tk x ∈ exist tk ∈ B, mk ≥ k such that j=m P∞k j j1 1 / U . Pick a symmetric and t ∈ B satisfy this condition so j=m1 tj xj ∈ neighborhood of 0, V , such that V + V ⊂ U . There exists n1 > m1 such P∞ that j=n1 +1 t1j xj ∈ V . Then n1 X

j=m1

t1j xj

=

∞ X

j=m1

t1j xj

−

∞ X

j=n1 +1

t1j xj ∈ / V.

Put I1 = [m1 , n1 ] and continue the construction. Theorem 2.16. Let λ be a K-space and Λ ⊂ λ have signed-SGHP. If the P P series j xj is Λ multiplier convergent, then the series ∞ j=1 tj xj converge uniformly for t belonging to bounded subsets of Λ. P∞ Proof: Suppose that B ⊂ Λ is bounded but the series j=1 tj xj do not converge uniformly for t ∈ B. Let the notation be as in Lemma 2.15. Let nk and sk be as in the definition of signed-SGHP above and

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let t = P

P∞

In k t k=1 sk χP

j∈Ink tj xj

= sk Cauchy condition.

nk

j∈Ink

P∞ ∈ Λ. Then tj xj does not converge since j=1 P∞ tnj k xj ∈ / V , i.e., j=1 tj xj does not satisfy the

Theorem 2.16 implies two well known results for bounded multiplier and subseries convergent series which we now state. P Corollary 2.17. Let j xj be l∞ multiplier convergent. Then the series P∞ j=1 tj xj converge uniformly for k{tj }k∞ ≤ 1. Proof: l∞ has SGHP.

P Corollary 2.18. Let Then the series j xj be subseries convergent. P∞ t x converge uniformly for t ∈ M = {χ : σ ⊂ N}. j j 0 σ j=1 Proof: M0 has SGHP and is a bounded subset of m0 .

A sequence space λ is normal (solid ) if t ∈ λ and |sj | ≤ |tj | for all j implies that s = {sj } ∈ λ. For example, c0 and lp , 0 < p ≤ ∞, are normal whereas c and m0 are not normal. From Corollary 2.17, we have Corollary 2.19. Let λ be a normal K-space with signed-SGHP and with P the property that {s ∈ λ : |s| ≤ |t|} is bounded for every t ∈ λ. If j xj P∞ is λ multiplier convergent and t ∈ λ, then the series s x converge j=1 j j uniformly for |sj | ≤ |tj |. Without some assumptions on the multiplier space, the conclusion of Theorem 2.16 may fail even when the multiplier space satisfies WGHP or 0-GHP (see Appendix B). P Example 2.20. The series j ej is lp multiplier convergent in (l p , k·kp ) P∞ for any 1 ≤ p < ∞, but the series j=1 tj ej do not converge uniformly P k j k p for k{tj }kp ≤ 1 [ take tk = ek so ∞ j=1 tj e = e ]. Note that l has both WGHP and 0-GHP (Appendix B). The SGHP assumption in Theorem 2.16 is only a sufficient condition for the uniform convergence conclusion. P Example 2.21. Let x = {xj } ∈ l2 . Then the series j xj ej is l2 multiplier convergent in l 1 . If B ⊂ l2 is bounded, M = sup{ktk2 : t ∈ B} and t ∈ B, then

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2

∞

∞ ∞ ∞ ∞ X X X X

X

2 2 2 j 2

t x e = ( |t x |) ≤ |t | |x | ≤ M |xj | → 0 j j j j j j

j=n

j=n j=n j=n j=n 1

so the conclusion of Theorem 2.16 holds but l 2 does not have SGHP. Another uniform convergence result holds for multiplier spaces satisfying 0-GHP. P Theorem 2.22. Let λ be a K-space with 0-GHP and let j xj be λ mulP ∞ i tiplier convergent. If ti → 0 in λ, then the series j=1 tj xj converge uniformly for i ∈ N. Proof: Suppose the conclusion fails to hold. Then there exists a closed neighborhood, U, of 0 such that for every k there exist pk , mk > k such P pk that ∞ / U . For k = 1, let p1 , m1 > 1 satisfy this condition so j=mk tj xj ∈ ∞ X

j=m1

tpj 1 xj ∈ / U.

Pick a symmetric neighborhood of 0,V, such that V + V ⊂ U . There exists P∞ n1 > m1 such that j=n1 +1 tpj 1 xj ∈ V . Then n1 X

tpj 1 xj =

∞ X

j=m1

j=m1

tpj 1 xj −

Pn

i j=m tj xj

∞ X

j=n1 +1

tpj 1 xj ∈ / V.

∈ V for 1 ≤ i ≤ p1 , n ≥ m ≥ N1 . Let There exists N1 such that p2 , m2 > N1 , n2 satisfy the conditions above for N1 . Note that we must have p2 > p1 . Continuing this construction produces increasing sequences {pj }, {mj }, {nj } such that (#)

nk X

j=mk

tpj k xj ∈ / V.

Set Ik = [mk , nk ]. By 0-GHP, since {tpk } → 0, there is a subsequence {qk } P∞ P∞ of {pk } such that t = k=1 χIqk tqk ∈ λ. Since j=1 tj xj converges, we P should have that j∈Iq tqj k xj → 0 contradicting the condition (#). k

Without the 0-GHP assumption the conclusion of Theorem 2.22 may fail. P∞ Example 2.23. Let λ = c00 = X. Then j=1 jej is c00 multiplier converP∞ gent in c00 . Now ti = ei /i → 0 but the series j=1 tij jej do not converge uniformly.

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We can obtain another uniform convergence result from the continuity of the summing operator. Theorem 2.24. Let λ be a complete, metrizable AK-space and let X be a P Mackey space (i.e., X has the Mackey topology). If j xj is λ multiplier P∞ convergent in X and K ⊂ λ is compact, then the series j=1 tj xj converge uniformly for t ∈ K. Proof: Since λ is metrizable, λ carries the Mackey toplology ([Sw2] 18.8) so the summing operator S : λ → X is continuous (Theorem 2.2). Since K P∞ is compact, limn j=n tj ej = 0 uniformly for t ∈ K ([Sw2] 10.15). Thus, P∞ P∞ limn j=n tj Sej = limn j=n tj xj = 0 uniformly for t ∈ K.

We next consider uniform convergence for families of multiplier convergent series. The β-dual of Λ ⊂ λ with respect to X is defined to be X xj is Λ multiplier convergent}. ΛβX = {{xj } : j

P∞ If t ∈ λ and x ∈ λβX , we write t · x = j=1 tj xj [see Appendix B]. We define the topology ω(λβX , λ) on λβX to be the weakest topology on λβX such that the mappings x = {xj } →

∞ X j=1

t j xj = t · x

from λβX into X are continuous for every t ∈ λ. Thus, if X is the scalar field, then λβX = λβ and ω(λβX , λ) = σ(λβ , λ). We now give a definition for another gliding hump property which will be used. Definition 2.25. Let Λ ⊂ λ. The space Λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ Λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } and a sequence of signs P∞ {sj } such that the coordinatewise sum of the series j=1 sj χInj t ∈ Λ. The space Λ has the weak gliding hump property (WGHP) if the signs above can all be chosen equal to 1 for every t ∈ λ. Examples of spaces with signed-WGHP and WGHP are given in Appendix B. For example, any monotone space such as c00 , c0 , lp (0 < p ≤ ∞), and m0 has WGHP while the space bs has signed-WGHP but not WGHP.

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Theorem 2.26. Let Λ ⊂ λ have signed-WGHP. If {xk } ⊂ ΛβX is such that limk t · xk exists for every t ∈ Λ and limk xkj exists for every j, then P∞ for every t ∈ Λ the series j=1 tj xkj converge uniformly for k ∈ N. Proof: If the conclusion fails, then

(∗) there exists a neighborhood of 0, U , in X such that f or every n P nn there exist kn , nn > mn > n such that j=m tj xkj n ∈ / U. n

P 1 tj xkj 1 ∈ / U. By (∗) for n = 1, there exist k1 , n1 > m1 such that nj=m 1 P n 0 k 0 There exists m > n1 such that t x ∈ U for n > m > m , 1 ≤ j j j=m Pn2 k2 0 k ≤ k1 . By (∗) there exist k2 , n2 > m2 > m such that j=m2 tj xj ∈ / U. Hence, k2 > k1 . Continuing this construction produces increasing sequences {ki }, {mi }, {ni } with mi < ni < mi+1 and (#)

x k i · χ Ii t ∈ / U, where Ii = [mi , ni ].

Define the matrix M by M = [mij ] = [xki · χIj t]. We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Second, given any increasing sequence of integers, there is a subsequence {pk } and a sequence of signs {sk } such P∞ that z = {zj } = j=1 sj χIpj t ∈ Λ. Then ∞ X j=1

sj mipj =

∞ X j=1

s j x k i · χ Ip j t = x k i · z

and lim xki · z exists. Hence, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (#). From Theorem 2.26, we can obtain an important weak sequential completeness result due to Stuart ([St1], [St2], [Sw1]). First, we establish a lemma. Lemma 2.27. Let Λ ⊂ λ. If {xk } ⊂ ΛβX is such that limk t · xk exists for every t ∈ Λ, limk xkj = xj exists for each j and for each t ∈ Λ the series P∞ k βX such that j=1 tj xj converge uniformly for k ∈ N, there exists x ∈ Λ k t · x → t · x for every t ∈ Λ.

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Proof: Set x = {xj }. We claim that x ∈ ΛβX and t · xk → t · x for every P∞ t ∈ Λ. Put u = lim t · xk . It suffices to show that u = j=1 tj xj . Let U be a balanced neighborhood of 0 in X and pick a balanced neighborhood P∞ V such that V + V + V ⊂ U . There exists p such that j=n tj xkj ∈ V for P∞ n ≥ p, k ∈ N. Fix n ≥ p. Pick k = kn such that j=1 tj xkj − u ∈ V and Pn k j=1 tj (xj − xj ) ∈ V . Then

n X j=1

t j xj − u = (

∞ X j=1

tj xkj − u) +

n X j=1

tj (xj − xkj ) −

∞ X

j=n+1

tj xkj ∈ V + V + V ⊂ U

and the result follows. Stuart’s result now follows from Lemma 2.27 and Theorem 2.26. Corollary 2.28. (Stuart) Let λ have signed-WGHP and let X be sequentially complete. Then (λβX , ω(λβX , λ)) is sequentially complete. Proof: If {xk } is ω(λβX , λ) Cauchy and X is sequentially complete, then limk t · xk exists for every t ∈ λ so Theorem 2.26 and Lemma 2.27 apply. Since any monotone space has WGHP, Corollary 2.28 applies to monotone spaces, in particular to c0 , lp (0 < p ≤ ∞) and m0 . Corollary 2.28 also applies to the space of bounded series bs which has signed-WGHP but not WGHP as originally noted by Stuart (Appendix B). A subset F of λβX is said to be conditionally ω(λβX , λ) sequentially compact if every sequence {xk } ⊂ F has a subsequence which is such that lim t · xk exists for every t ∈ λ ([Din]). From Theorem 2.26 we have Corollary 2.29. Let λ have signed-WGHP. If F ⊂ λβX is conditionally P∞ ω(λβX , λ) sequentially compact and t ∈ λ, then the series j=1 tj xj converge uniformly for x ∈ F . Without the gliding hump assumptions, the conclusions in Theorem 2.26 and Corollary 2.28 may fail. Example 2.30. Let λ = c so λβ = l1 . Then {ek } is ω(l1 , c) = σ(l1 , c) P∞ Cauchy, but if e is the constant sequence {1}, the series j=1 ekj ej do not converge uniformly and the sequence {ek } is not σ(l1 , c) convergent. We next consider another uniform convergence result with another gliding hump assumption.

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Definition 2.31. The space λ has the infinite gliding hump property (∞GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate P∞ sum j=1 apj χIpj t ∈ λ. The term ”infinite gliding hump” is used to suggest that the ”humps”, χIpj t, are multiplied by a sequence of scalars which tend to ∞; there are other gliding hump properties where the humps are multiplied by elements of classical sequence spaces (Appendix B). Examples of spaces with ∞-GHP are given in Appendix B. For example, λ = lp (0 < p < ∞) and λ = cs have ∞-GHP. The spaces l ∞ , m0 , bs and bv do not have ∞-GHP. Theorem 2.32. Assume that λ has ∞-GHP. If B ⊂ λβX is pointwise P∞ bounded on λ , then for every t ∈ λ the series j=1 tj xj converge uniformly for x ∈ B. Proof: If the conclusion fails, there exist > 0, a continuous semi-norm p on X, {xk } ⊂ B and subsequences {mk }, {nk } with m1 < n1 < m2 < ... and nk X (∗) p( tl xkl ) > . l=mk

Put Ik = [mk , nk ]. By ∞-GHP there exist {pk }, apk > 0, apk → ∞ such that any subsequence of {pk } has a further subsequence {qk } such that ∞ X

k=1

Define a matrix

aqk χIqk t ∈ λ.

M = [mij ] by mij =

X

apj tl xpl i /api .

l∈Ipj

We claim that M is a K matrix (Appendix D.2). First, the columns of M converge to 0 since B is pointwise bounded on λ and 1/api → 0. Next, given any subsequence of {pj } there is a further subsequence {qj } such that P∞ u = k=1 aqk χIqk t ∈ λ . Then ∞ X j=1

miqj = (1/api )

∞ X l=1

ul xpl i = (1/api )xpi · u → 0.

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Hence, M is a K matrix so the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). From Lemma 2.27 and Theorem 2.32 we obtain another weak sequential completeness result. Corollary 2.33. Assume that λ has ∞-GHP and X is sequentially complete. Then (λβX , ω(λβX , λ)) is sequentially complete. Remark 2.34. The signed-WGHP and ∞-GHP are independent so Corollaries 2.28 and 2.33 cover different spaces. For example, the space bs has signed-WGHP but not ∞-GHP while the space bv0 has ∞-GHP but not signed-WGHP. We next consider uniform convergence results when the elements range over both subsets of λβX and λ. These results have stronger conclusions but require stronger assumptions. First, we consider an improvement of Theorem 2.26. Theorem 2.35. Assume that Λ ⊂ λ has signed-SGHP and limk xkj = xj exists for each j. If {xk } ⊂ λβX is such that lim t · xk exists for every t ∈ Λ P∞ and B ⊂ Λ is bounded, then the series j=1 tj xkj converge uniformly for k ∈ N, t ∈ B. Proof: If the conclusion fails, (∗) there is a neighborhood U of 0 such that for every n there exist P nn tn xkj n ∈ / U. kn , tn ∈ B, nn > mn > n such that j=m n j 1 By (∗) for n = 1 there exist k1 , t ∈ B, n1 > m1 > 1 such that P n1 / U. By Theorem 2.16 there exists m0 > n1 such that t 1 xk 1 ∈ 1 j j Pj=m q k for 1 ≤ k ≤ k1 , t ∈ B, q ≥ p ≥ m0 . By (∗) there j=p tj xj ∈ U Pn2 2 k2 exist k2 , t2 ∈ B, n2 > m2 > m0 such that ∈ / U . Hence, j=m2 tj xj k2 > k1 . Continuing this construction produces increasing sequences {ki }, {mi }, {ni }, mi+1 > ni > mi , ti ∈ B such that (∗∗)

ni X

j=mi

/ U. tij xkj i ∈

Set Ii = [mi , ni ] and define a matrix M = [mij ] = [xki · χIj tj ].

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We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Next, for any increasing sequence of positive integers, there is a subsequence {pj } and a sequence of signs {sj } such that P∞ z = j=1 sj χIpj tpj ∈ Λ. Then ∞ X j=1

ki

sj mipj =

∞ X j=1

xki · sj χIpj tpj = xki · z

and lim x ·z exists. Thus, M is a signed K-matrix so by the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗∗). Since M0 ⊂ m0 has SGHP and l∞ has SGHP, we have the following corollaries. P x be subseries convergent for every i ∈ N Corollary 2.36. Let P j ij and suppose that limi j∈σ xij exists for every σ ⊂ N. Then the series P converge uniformly for i ∈ N, σ ⊂ N. j∈σ xij P Corollary 2.37. Let j xij be bounded multiplier convergent for every P ∞ i ∈ N and suppose limi j=1 tj xij exists for every {tj } ∈ l∞ . Then the P∞ series j=1 tj xij converge uniformly for i∈ N, k{tj }k∞ ≤ 1. Corollary 2.38. Assume that λ has signed-SGHP. If F ⊂βX is conditionally ω(λβX ,λ) sequentially compact and B ⊂ λ is bounded, then the series P∞ j=1 tj xj converge uniformly for x ∈ F, t ∈ B.

Example 2.30 shows that the gliding hump property in Theorem 2.35 is important. We have another uniform convergence result generalizing Theorem 2.22.

Theorem 2.39. Assume that λ has 0-GHP. If {xk } ⊂ λβX is such that P∞ lim t · xk exists for every t ∈ λ and tk → 0 in λ, then the series j=1 tlj xkj converge uniformly for k, l ∈ N. Proof: If the conclusion fails, (∗) there is a neighborhood U of 0 such that for every n there exist kn , ln , nn > mn > n such that nn X / U. tljn xkj n ∈ j=mn

P n1 / U. tl1 xkj 1 ∈ By (∗) for n = 1, there exist k1 , l1 , m1 < n1 such that j=m 1 j P q 0 i k By Theorem 2.22 there exists m > n1 such that t x ∈ U for j=p j j

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i ∈ N, 1 ≤ k ≤ k1 , q > p > m0 . By (∗) there exist k2 , l2 , n2 > m2 > m0 P n2 l 2 k2 such that / U . Hence, k2 > k1 and l2 > l1 . Continuing j=m2 tj xj ∈ this construction produces increasing sequences {ki }, {li }, {mi }, {ni } with mi+1 > ni > mi and

(∗∗)

ni X

j=mi

tlji xkj i ∈ / U.

Set Ii = [mi , ni ]. Define a matrix M = [mij ] = [xki · χIj tlj ]. We claim that M is a K-matrix (Appendix D.2). First the columns of M converge by hypothesis. Next, given any increasing sequence of positive P pj ∈ λ. Then integers there is a subsequence {pj } such that z = ∞ j=1 χIpj t P∞ ki ki · z and limi x · z exists. By the Antosik-Mikusinski j=1 mipj = x Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗∗). The 0-GHP hypothesis in Theorem 2.39 is important. Pk j Example 2.40. Let λ = c00 so λβ = s. Let xk = j=1 e ∈ s and P l tl = j=1 ej /l so {xk } is σ(s, c00 ) Cauchy and tl → 0 in (c00 , k·k∞ ). Then P∞ l k j=N tj xj = (k − l)/l if k ≥ l ≥ N so the series do not converge uniformly for k, l ∈ N although the series do converge uniformly for fixed k or l. This shows that the 0-GHP cannot be dropped in Theorem 2.39. Corollary 2.41. Assume that λ has 0-GHP. If F ⊂ λβX is conditionally P∞ ω(λβX ,λ) sequentially compact and tl → 0 in λ, then the series j=1 tlj xj converge uniformly for l ∈ N, x ∈ F . We next consider compactness in the range of the summing operator associated with a multiplier convergent series. We first establish a basic lemma. P Lemma 2.42. Let Λ ⊂ λ. If j xj is Λ multiplier convergent and the P∞ series j=1 tj xj converge uniformly for t ∈ Λ, then the summing operator P∞ S : Λ → X, St = j=1 tj xj , is continuous with respect to the topology p of coordinatewise convergence on Λ and the topology of X. Proof: Let tδ = {tδj } be a net in Λ which converges to t ∈ Λ with respect to p. Let U be a neighborhood of 0 in X and pick a symmetric

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neighborhood V such that V + V + V ⊂ U . There exists n such that P∞ t x ∈ V for every t ∈ Λ. There exists δ such that α ≥ δ implies Pj=n jα j (t j 0 for every P P j j. Equip m0 with the norm ktks = ∞ e is j=1 sj |tj |. Then the series subseries convergent in (m0 , k·ks ) but is not bounded multiplier convergent P since, for example, the series j ej /j does not converge to an element of m0 . From Theorem 2.54 and Corollary 2.46, we have P Corollary 2.56. Let X be a sequentially complete LCTVS. If j xj is P∞ subseries convergent, then { j=1 tj xj : ktk∞ ≤ 1} is compact in X.

We now define two additional notions of convergence for series in TVS. P Definition 2.57. The series j xjPis unconditionally convergent (rear∞ rangement convergent) if the series j=1 xπ(j) converges for every permuP tation π : N → N. The series j xj is unconditionally Cauchy if the series P∞ j=1 xπ(j) is Cauchy for every permutation. P Definition 2.58. The series j xj is unordered convergent if the net P { j∈σ xj : σ ∈ F} converges, where F is the family of all finite subP sets of N ordered by inclusion. We write limF j∈σ xj for the limit of this net when the net converges. We have the following relationships. Theorem 2.59. Let X be a TVS and {xj } ⊂ X. Consider the following conditions: P (i) the series j xj is unconditionally convergent, P (ii) the series j xj is unordered convergent, P (iii) the series j xj is subseries Cauchy,

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P (iv) the series j xj is unconditionally Cauchy. Then (i)⇒(ii)⇒(iii)⇒(iv). P∞ P Proof: Assume (i). Let x = j=1 xj . Assume the net { j∈σ xj : σ ∈ F} does not converge to x. Then there exists a symmetric neighborhood, U , of 0 in X such that for every σ ∈ F there exists σ 0 ∈ F, P σ 0 ⊃ σ with x − j∈σ0 xj ∈ / U . Pick a symmetric neighborhood of 0, V , Pn such that V + V ⊂ U . There exists N such that x − j=1 xj ∈ V for n ≥ N . Let d1 = {1, ..., N } and let d01 be as above. Set d2 = {1, ..., max d01 } and let d02 be as above. Continue in this way to obtain a sequence d1 , d01 , d2 , d02 , ... . Define a permutation π of N by enumerating the elements P of d1 , d01 \d2 , d2 \d01 , d02 \d2 , ... . The series ∞ j=1 xπ(j) is not convergent since X X X xj ) ∈ / V. xj = ( xj − x) + (x − j∈d0n \dn

j∈d0n

j∈dn

Hence, (i) implies (ii). P Assume (ii). Note that since limF j∈σ xj is unique, every rearrangeP P ment of j xj converges to the same limit, namely, limF j∈σ xj . Let U be a symmetric neighborhood of 0 in X. There exists σ0 ∈ F such that P P P xnj be a subseries of j xj . j∈σ xj − x ∈ U for every σ ⊃ σ0 . Let Pk j P Pick N > max σ0 . If k > j ≥ N , then i=j xi ∈ U so xnj is Cauchy and (iii) holds. Assume (iii). If (iv) fails, there exist a symmetric neighborhood of 0, U , in X and a permutation π of N and an increasing sequence {mn } such Pmn+1 xπ(i) ∈ / U . Choose a subsequence {mnj } of {mn } such that that i=m n +1 min{π(i) : mnj + 1 ≤ i ≤ mnj+1 +1 } > max{π(i) : mnj ≤ i ≤ mnj +1 }.

Arrange the integers π(i), mnj + 1 ≤ i ≤ mnj+1 , j ∈ N into an increasing P sequence {ij }. Then ∞ j=1 xij does not satisfy the Cauchy condition and (iii) fails. Hence, (iii) implies (iv). Corollary 2.60. Let X be a sequentially complete TVS. Then (i) and (ii) of Theorem 2.59 are equivalent to: P (iii)’ the series j xj is subseries convergent.

Finally, for LCTVS there is the notion of absolute convergence. P Definition 2.61. Let X be an LCTVS. The series j xj is absolutely P∞ convergent if j=1 p(xj ) < ∞ for every continuous semi-norm p on X.

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The notion of absolute convergence is a very strong condition. For example, in a Banach space every convergent series is absolutely convergent iff the space is finite dimensional (Dvoretsky-Rogers Theorem ([Sw2] 30.1)); in a Frechet space every convergent series is absolutely convergent iff the space is nuclear ([Sch] 10.7.2)). However, we do have P Proposition 2.62. If the series j xj is absolutely convergent in the LCTVS X, then the series is subseries Cauchy; if X is sequentially complete, then the series is subseries convergent. The sequential completeness statement in the last part of Theorem 2.62 is important. P Example 2.63. Let X = c00 . Set xj = ej /j 2 . Then the series j xj is absolutely convergent in X but not convergent.

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Chapter 3

Applications of Multiplier Convergent Series

In this chapter we will give several applications of multiplier convergent series to various topics in locally convex spaces and vector valued measures. As before, throughout this chapter λ will denote a sequence space containing c00 , the space of sequences which are eventually 0 and X will denote a (Hausdorff) LCTVS. We begin by establishing a generalization of a result of G. Bennett ([Be]). Proposition 3.1. If λβ for every x0 ∈ X 0 .

P

j

xj is λ multiplier Cauchy in X, then {hx0 , xj i} ∈

Proof: Let t ∈ λ, x0 ∈ X 0 . Then

converges so {hx0 , xj i} ∈ λβ .

E D P P∞ ∞ 0 x0 , j=1 tj xj = j=1 tj hx , xj i

We consider the converse of Proposition 3.1 under additional assumptions. Theorem 3.2. Let (λ, τ ) be a metrizable AK-space such that λ0 = λβ . P Then j xj is λ multiplier Cauchy in (X, τ (X, X 0 )) iff {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Proof: Suppose that {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Define a linear P 0 0 0 map T : c00 → X by T t = ∞ j=1 tj xj . If x ∈ X , t ∈ c00 , then hx , T ti = P∞ 0 0 j=1 tj hx , xj i = t · {hx , xj i} which implies by hypothesis that T is σ(c00 , λβ )−σ(X, X 0 ) continuous and ,therefore, τ (c00 , λβ )−τ (X, X 0 ) continuous. Since λ0 = (c00 , τ |c00 )0 and τ |c00 = τ (λ, λ0 ) |c00 (any metrizable space carries the Mackey topology ([Sw2] 18.8)), τ |c00 = τ (c00 , λ0 ) = τ (c00 , λβ ). P∞ Now, if s ∈ λ, s = j=1 sj ej , where the convergence is in τ = τ (λ, λβ ) 25

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Pn by the AK assumption, so { j=1 sj ej }n is τ (c00 , λβ ) Cauchy. Therefore, Pn Pn Pn { j=1 T (sj ej )}n = { j=1 sj T ej }n = { j=1 sj xj }n is τ (X, X 0 ) Cauchy. The converse is given in Proposition 3.1. Corollary 3.3. If λ satisfies the assumptions of Theorem 3.2 and P (X, τ (X, X 0 )) is sequentially complete, then j xj is λ multiplier convergent in (X, τ (X, X 0 )) iff {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Corollary 3.4. Let λ be as in Theorem 3.2 and let X be an FK-space. Then X contains λ iff c00 ⊂ X and {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Remark 3.5. Bennett’s result corresponds to the case where λ = l p , 1 ≤ p < ∞ or λ = c0 in Corollaries 3.3 and 3.4. Note that Corollaries 3.3 and 3.4 also apply to the spaces cs and bv0 . For conditions which guarantee that λ0 = λβ , see Proposition 2.5. Without some additional assumptions on the multiplier space λ, the converse of Proposition 3.1 may fail. Example 3.6. Let λ = l ∞ so λβ = l1 . Let X = c0 and consider the series P j P j e is not e in c0 . If s ∈ X 0 = l1 , then { s, ej } = {sj } ∈ l1 = λβ , but ∞ l multiplier convergent in c0 . We next consider results which involve series which are c0 multiplier Cauchy and c0 multiplier convergent. These series are often described in a different way which we now consider. P Definition 3.7. A series j xj in X is said to be weakly unconditionally P∞ Cauchy (wuc) if j=1 |hx0 , xj i| < ∞ for every x0 ∈ X 0 . P P Note that a series j xj is wuc iff the series j xj is subseries Cauchy in the weak topology σ(X, X 0 ). A series which is subseries convergent in the weak topology σ(X, X 0 ) is wuc, but a wuc series may not be subseries P convergent in the weak topology (consider the series j ej in c0 ). We give several characterizations of wuc series. Proposition 3.8. Let {xj } ⊂ X. The following are equivalent: P (i) The series j xj is wuc. (ii) {hx0 , xj i} ∈ l1 for every x0 ∈ X 0 . P (iii) The series j xj is c0 multiplier Cauchy. P (iv) { j∈σ xj : σ f inite} is bounded in X.

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(v) For every continuous semi-norm p on X, there exists M > 0 such that P p( j∈σ tj xj ) ≤ M ktk∞ for every t ∈ l ∞ and σ finite. P∞ (vi) The map T : c00 → X, T t = j=1 tj xj , is linear and continuous. P (vii) The series j xj is c0 multiplier Cauchy in σ(X, X 0 ).

Proof: Clearly (i) and (ii) are equivalent, and (ii) and (iii) are equivalent by Bennett’s result in Theorem 3.2. Assume that (i) holds. If x0 ∈ X 0 and σ is finite, then * X + X ∞ 0 x, x |hx0 , xj i| < ∞ ≤ j j=1 j∈σ P so { j∈σ xj : σ finite} is σ(X, X 0 ) bounded and, therefore, bounded in X so (iv) holds. Assume that (iv) holds. Let p be a contimuous semi-norm on X. Set P M = 2 sup{p( j∈σ xj ) : σ finite}. By the McArthur/Rutherford inequality P (Lemma 2.53), p( j∈σ tj xj ) ≤ M ktk∞ for every t ∈ l ∞ so (v) holds. That (v) implies (vi) is immediate. Suppose that (vi) holds. Then the adjoint operator T 0 : X 0 → c000 = l1 P 0 so T 0 x0 = {hx0 , xj i} ∈ l1 . Therefore, ∞ j hx , xj i converges for every j=1 s P∞ P 0 s ∈ c0 and j=1 sj xj is σ(X, X ) Cauchy or j xj is c0 multiplier Cauchy in σ(X, X 0 ). Thus, (vii) holds. P∞ Assume that (vii) holds. Then j=1 sj hx0 , xj i converges for every x0 ∈ P ∞ X 0 and for every s ∈ c0 . Hence, j=1 |hx0 , xj i| < ∞ for every x0 ∈ X 0 and (i) holds.

Note that it follows from Proposition 3.8 that a continuous linear operator between LCTVS carries wuc series into wuc series (condition (iv)). P Corollary 3.9. Let j xj be c0 multiplier convergent in X. Then P (i) j xj is wuc, (ii) for every continuous semi-norm p on X there exists M > 0 such that P p( ∞ t∈c , j=1 tj xj ) ≤ M ktk∞ for every P∞ 0 (iii) the linear map T : c0 → X, T t = j=1 tj xj , is continuous.

Proof: (i) follows from Proposition 3.8 (iii); (ii) follows from Proposition 3.8 (v); (iii) follows directly from (ii). We can now use the notions of wuc series and c0 multiplier convergent series to give a characterization of a locally complete LCTVS due to Madrigal and Arrese ([MA]). Recall that an LCTVS X is locally complete if for every

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closed, bounded, absolutely convex set B ⊂ X, the space XB = spanB equipped with the Minkowski functional pB of B in XB is complete ([K2]). Theorem 3.10. The LCTVS X is locally complete iff every wuc series in X is c0 multiplier convergent. P Proof: Suppose that X is locally complete and let j xj be a wuc series P in X. Then S = { j∈σ xj : σ finite} is bounded in X by Proposition 3.8. Let B be the closed, absolutely convex hull of S so (XB , pB ) is complete. P Since S is bounded in (XB , pB ), j xj is wuc in (XB , pB ) by Proposition 3.8. By the completeness of (XB , pB ) and condition (iii) of Proposition P 3.8, j xj is c0 multiplier convergent in (XB , pB ). Since the inclusion of P (XB , pB ) into X is continuous, j xj is c0 multiplier convergent in X. Let B be a closed, bounded, absolutely convex subset of X and suppose that {xj } is Cauchy in (XB , pB ). Pick an increasing sequence {nj } such that pB xnj+1 − xnj < 1/j2j P∞ for every j and set yj = xnj+1 − xnj . Then j=1 jyj is pB absolutely P∞ P∞ j convergent ( p (jy ) ≤ 1/2 < ∞ ) so by Proposition 3.8 j j=1 B j=1 P∞ P∞ By wuc in (XB , pB ) and, therefore, j=1 jyj is wuc in X. j=1 jyj isP P hypothesis ∞ jy is c multiplier convergent in X so the series j 0 j=1 j yj Pk is convergent to, say, y ∈ X. Thus, j=1 yj = xnk+1 − xn1 → y or xnj+1 → y + xn1 = z in X. Now, {xnj } is Cauchy in (XB , pB ), {xnj } converges in X to z and the topology pB is linked to the relative topology of XB from X so {xnj } converges to z in XB (Appendix A.4). Thus, XB is complete with respect to pB . Theorem 3.10 has an interesting corollary due to Madrigal and Arrese ([MA]). Corollary 3.11. Let X be a locally complete LCTVS. The following are equivalent: (i) every wuc series in X is subseries convergent, (ii) every wuc series in X is l ∞ multiplier convergent, (iii) every continuous linear operator T : c0 → X has a compact extension T : l∞ → X. P Proof: Suppose that (i) holds. Let j xj be wuc and let t ∈ l∞ . By P Theorem 3.10, j xj is c0 multiplier convergent. By Proposition 3.8 and

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P P Corollary 3.9, the series j tj xj is wuc. Hence, j tj xj converges by (i) and (ii) holds. Suppose that (ii) holds. Let T : c0 → X be linear and continuous. P P j P Since e is wuc in c0 , j T ej is wuc in X. By (ii), j T ej is l∞ multiP∞ plier convergent. By Corollary 2.46, { j=1 tj T ej : ktk∞ ≤ 1} is compact. P j Therefore, by Theorem 2.2, T t = j tj T e , defines a compact operator ∞ from l into X which extends T . Hence, (iii) holds. P Suppose that (iii) holds. Let j xj be wuc in X. By Theorem 3.10, P∞ P j xj is c0 multiplier convergent so T t = j=1 tj xj defines a continuous linear operator from c0 into X by Corollary 3.9. By (iii) T is compact so P P S = { j∈σ xj : σ finite} is relatively compact. By Theorem 2.48, j xj is subseries convergent. Bessaga and Pelczynski have shown that a Banach space X contains no subspace isomorphic to c0 iff every wuc series in X is subseries convergent ([BP]). We now extend this characterization to LCTVS. For this we require several preliminary lemmas. Lemma 3.12. Let xij ∈ R, εij > 0 for every i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i, then there exists an increasing sequence {mj } such that xmi mj ≤ εij for i 6= j.

Proof: Set m1 = 1. There exists m2 > m1 such that |xm1 j | < ε12 and |xim1 | < ε21 for all i, j ≥ m2 . There exists m3 > m2 such that |xm1 j | < ε13 , |xm2 j | < ε23 , |xim1 | < ε31 , |xim2 | < ε32 for all i, j ≥ m3 . Now just continue.

Lemma 3.13. Let X be a semi-normed space and xij ∈ X for i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i, then given > 0 there exists a subsequence {mj } such that ∞ X X

xmi mj < . i=1 j6=i

P∞ P∞ Proof: Pick ij > 0 such that i=1 j=1 ij < . Let {mj } be the subsequence from Lemma 3.12 applied to the double sequence kxij k. Then

xmi mj ≤ ij for i 6= j so the result follows.

Lemma 3.14. Let X be a semi-normed space that contains a c0 multiplier P convergent series j xj with kxj k ≥ δ > 0 for every j. Then there exists a subsequence {mj } such that for any subsequence {nj } of {mj }, T {tj } = P∞ T t = j=1 tj xnj defines a topological isomorphism of c0 into X.

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Proof: By replacing X by the linear subspace spanned by {xj }, we may

0 0 0 assume that X is separable. For each j pick x ∈ X , x j j ≤ 1, such that

0 xj , xj = kxj k. By the Banach-Alaoglu Theorem, {x0j } has a subsequence which is weak* convergent to an element x0 ∈ X 0 ; to avoid cumbersome notation later, assume that {x0j } is weak* convergent to x0 . Then 0 x − x0 , xj ≥ δ − |hx0 , xj i| > δ/2 j 0 for large hx 0 j since , xj i → 0; again to avoid cumbersome notation assume 0 that xj − x , xj ≥ δ/2 for all j. The matrix

M = [hx0i − x0 , xj i]

satisfies the assumption of Lemma 3.13 so let {mj } be the subsequence from Lemma 3.13 with = δ/4. Now define a continuous linear operator T : c0 → X by T t = P∞ 0 0 0 j=1 tj xmj (Corollary 3.9). If zi = xmi − x , then by the conclusion of Lemma 3.13, we have 2 kT {tj }k ≥ |hzi0 , T {tj }i| ≥ |ti hzi0 , xmi i| − ≥ |ti | δ/2 − k{tj }k∞ δ/4.

X

tj zi0 , xmj j6=i

Taking the supremum over all i in the inequality above gives kT {tj }k ≥ (δ/8) k{tj }k∞ so T has a bounded inverse. The same computation applies to any subsequence {nj } of {mj } so the result follows. We now give a characterization of sequentially complete LCTVS which have the property that any wuc series is subseries convergent. In the statement below, if X is a semi-normed space, B(X) denotes the closed unit ball of X. Theorem 3.15. Let X be a sequentially complete LCTVS. The following are equivalent: (i) X contains no subspace (topologically) isomorphic to c0 . P (ii) If j xj is c0 multiplier convergent in X, then xj → 0. P P (iii) If j xj is c0 multiplier convergent in X, then j xj is subseries convergent in X.

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P P (iv) If j xj is c0 multiplier convergent in X, then j xj is bounded multiplier convergent in X. P P∞ (v) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges ∞ uniformly for {tj } ∈ B(l ). P P∞ (vi) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges uniformly for {tj } ∈ B(c0 ). P P∞ (vii) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges 1 uniformly for {tj } ∈ B(l ). (viii) Every continuous linear operator T : c0 → X is compact and has a compact extension to l ∞ . Proof: (i) implies (ii): Suppose there exists a c0 multiplier convergent P series j xj with xj 9 0. Then we may assume there exists a continuous semi-norm p on X and δ > 0 such that p(xj ) ≥ δ for all j. By Lemma P∞ 3.14 there is a subsequence {mi } such that H{tj } = j=1 tj xmj defines a topological isomorphism from c0 onto (Hc0 , p). Let I be the continuous inP∞ clusion operator from X onto (X, p). By Corollary 3.9, T {tj } = j=1 tj xmj defines a continuous linear operator from c0 into X, and T −1 = H −1 I is continuous so T defines a linear homeomorphism from c0 into X. (ii) implies (iii): Suppose there exists a c0 multiplier convergent series P P X such that j xj diverges. Since X is sequentially complete, j xj in P n {sn } = { j=1 xj } is not Cauchy. Hence, there exist a neighborhood of 0, V , in X and an increasing sequence {nj } such that yj = snj+1 − snj ∈ /V P∞ P x is c multiplier convergent, the series for all j. Since 0 j j j=1 tj yj converges for every {tj } ∈ c0 . By (ii), yj → 0. This contradiction shows that (ii) implies (iii). That (iii) implies (iv) is given in Theorem 2.54. That (iv) implies (v) is given in Theorem 2.54. That (v) implies (vi) and (vi) implies (vii) is clear. P (vii) implies (ii): Suppose there is a c0 multiplier convergent series j xj P∞ 1 in X such that the series j=1 tj xj converges uniformly for {tj } ∈ B(l ) but xj 9 0. There exists a neighborhood of 0,V , and a subsequence {xnj } such that xnj ∈ / V for every j. Let tk = {tkj } = enk ∈ B(l1 ). Then P∞ k P / V so the series ∞ j=1 tj xj fail to converge uniformly for j=1 tj xj = xnk ∈ {tj } ∈ B(l1 ). (viii) implies (i) since no continuous, linear, 1-1 map from c0 into X can have a continuous inverse by the compactness of the map. Finally, (iv) implies (viii): Let T : c0 → X be linear and continuous and P set T ej = xj . Then j xj is c0 multiplier convergent and, hence, bounded

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multiplier convergent by (iv). By Corollary 2.46, {T {tj } : k{tj }k∞ ≤ 1} = {

∞ X j=1

tj xj : k{tj }k∞ ≤ 1}

is compact so (viii) holds. Remark 3.16. The equivalence of (i) and (iii) for the case when X is a Banach space is a well known result of Bessaga and Pelczynski ([BP]). Bessaga and Pelczynski derive their result from results on basic sequences in B-spaces; Diestel and Uhl give a proof based on Rosenthal’s Lemma ([DU] I.4.5). The equivalence of (i) and (viii) was noted by Li. The conditions (v), (vi) and (vii) are contained in [LB]. Without the sequential completeness assumption, the conclusions in Theorem 3.15 may fail. P j Example 3.17. The series e is wuc in c00 with the sup-norm but is not subseries convergent. However, c00 being of countable algebraic dimension does not contain a subspace isomorphic to c0 . We next derive a result of Pelczynski on unconditionally converging operators. A continuous linear operator T from a Banach space X into a Banach space Y is said to be unconditionally converging if T carries wuc series into subseries convergent series ([Pl]). A weakly compact operator is unconditionally converging [we give a proof of this fact in Chapter 4 after we establish the Orlicz-Pettis Theorem; recall an operator is weakly compact if it carries bounded sets into relatively weakly compact sets]. The identity on l1 gives a example of an unconditionally converging operator which is not weakly compact [recall that a sequence in l 1 is weakly convergent iff the sequence is norm convergent; this result will be established in Chapter 7 when Hahn-Schur Theorems are derived; see also, [Sw2] 16.14]. Theorem 3.18. Let X, Y be Banach spaces and T : X → Y a continuous linear operator which is not unconditionally converging. Then there exist topological isomorphisms I1 : c0 → X and I2 : c0 → Y such that T I1 = I2 [i.e., T has a bounded inverse on a subspace isomorphic to c0 ]. P Proof: By hypothesis there exists a wuc series j xj in X such that P P j T xj is not subseries convergent. Since j T xj contains a subseries P which is not convergent, we may as well assume that the series j T xj diverges. Thus, there exist δ > 0 and a subsequence {nj } such that kzj k ≥

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Pnj+1 δ, where zj = T uj and uj = i=n xi . By Proposition 3.8, the series j +1 P P T u are both wuc. Since kxk ≥ kT xk / kxk for x ∈ X, kuj k ≥ u and j j j j P P δ/ kT k. Applying Lemma 3.14 to the series j uj and j T uj , there is a P∞ P subsequence {mj } such that I1 {tj } = j=1 tj umj and I2 = ∞ j=1 tj T umj define isomorphisms from c0 into X and Y , respectively. Obviously, T I1 = I2 . Remark 3.19. The converse of Theorem 3.18 holds and gives an interesting characterization of unconditionally converging operators (see [Ho]). We next consider wuc series in the strong dual of an LCTVS. Theorem 3.20. Let X be a barrelled LCTVS. The following are equivalent: (i) (X 0 , β(X 0 , X)) contains no subspace isomorphic to c0 , P (ii) every wuc series j x0j in X 0 is β(X 0 , X) subseries 0 convergent, P P < ∞ for every (iii) every series j x0j in X 0 which satisfies ∞ j=1 xj , x 0 x ∈ X is β(X , X) subseries convergent, (iv) every continuous linear operator T : X → l 1 is compact [an operator T is compact if T carries bounded sets into relatively compact sets]. Proof: Conditions (i) and (ii) are equivalent by Theorem 3.15 since β(X 0 , X) is sequentially complete by the barrelledness of X ([Wi] 6.1.16 and 9.3.8). 0 P P x ,x < ∞ Assume that (ii) holds. Let j x0j be such that ∞ j j=1 P for every x ∈ X. Then { j∈σ x0j : σ finite} is weak* bounded and, thereP fore, β(X 0 , X) bounded since X is barrelled. Therefore, j x0j is wuc in P (X 0 , β(X 0 , X)) by Proposition 3.8. Hence, j x0j is β(X 0 , X) subseries convergent by (ii) and (iii) holds. Assume that (iii) holds. Let T : X → l 1 be linear and continuous. Set x0j = T 0 ej . Now T 0 is β(l∞ , l1 ) − β(X 0 , X) continuous so {x0j } is β(X 0 , X) bounded. For x ∈ X, T x ∈ l1 we have ∞ ∞ ∞ X j 0 j X 0 X e , T x < ∞. T e ,x = x ,x = j j=1 0 j xj

j=1

j=1

is β(X , X) subseries convergent and, therefore, l ∞ mulBy (iii), tiplier convergent since β(X 0 , X) is sequentially complete as noted above (Theorem 2.54). Therefore, if B ⊂ X is bounded, then ∞ ∞ X X 0 j e , T x = 0. lim sup xj , x = lim sup P

0

n x∈B j=n

n x∈B j=n

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Hence, T B is relatively compact in l 1 ([Sw2] 10.15) and (iv) holds. P 0 0 0 Assume that (iv) holds. Let j xj be wuc in (X , β(X , X)). Define T :

X → l1 by T x = { x0j , x }. T is obviously linear and is σ(X, X 0 ) − σ(l1 , l∞ ) continuous since if t ∈ l ∞ , x ∈ X, *∞ + ∞ X X

0 0 t · Tx = t j xj , x = t j xj , x j=1

j=1

[ the series j tj x0j is σ(X 0 , X) Cauchy and ,therefore, σ(X 0 , X) convergent since X is barrelled ([Wi] 9.3.8)]. Thus, T is β(X, X 0 )−β(l1 , l∞ ) continuous. By (iv), T is compact. If B ⊂ X is bounded, T B is relatively compact in l1 so ∞ ∞ X X j 0 xj , x = 0 lim sup e , T x = lim sup P

n x∈B j=n

n x∈B j=n

P ([Sw2] 10.15) and j x0j is β(X 0 , X) convergent. The same argument can P be applied to every subseries of j x0j so (ii) holds.

Remark 3.21. If X is barrelled, then X 0 is weak* sequentially complete P so condition (iii) is equivalent to the statement that every series j x0j in X 0 which is σ(X 0 , X) subseries convergent is β(X 0 , X) subseries convergent. This is the statement of an Orlicz-Pettis type Theorem which we will consider in Chapter 4. Without the barrelledness assumption, the conclusion of Theorem 3.20 may fail. P j Example 3.22. Let X = c00 with the sup-norm. The series j e in 1 0 l = X satisfies the condition (iii) in Theorem 3.20 but is not strongly subseries convergent in l 1 and l1 contains no subspace isomorphic to c0 . We next give a characterization of Banach-Mackey spaces in terms of multiplier convergent series. Recall that an LCTVS X is a Banach-Mackey space if every σ(X, X 0 ) bounded subset of X is β(X, X 0 ) bounded; i.e., if B ⊂ X is pointwise bounded on X 0 , then B is uniformly bounded on σ(X 0 , X) bounded subsets of X 0 ([Wi] 10.4.3). The Banach-Mackey Theorem states that any sequentially complete LCTVS is a Banach-Mackey space ([Wi] 10.4.8). Let X be an LCTVS. Let X b (X s ) be the space of all bounded (sequentially continuous) linear functionals on X. Since X 0 ⊂ X s ⊂ X b ,

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(X, X s ) and (X, X b ) both form dual pairs. We now give a characterization of Banach-Mackey spaces in terms of l 1 multiplier convergent series and the spaces X s and X b . Theorem 3.23. Let X be an LCTVS. The following are equivalent: (i) (ii) (iii) (iv)

X is a Banach-Mackey space. P∞ If {x0j } is σ(X 0 , X) bounded and {tj } ∈ l1 , then j=1 tj x0j ∈ X s . P∞ 0 b If {x0j } is σ(X 0 , X) bounded and {tj } ∈ l1 , then j=1

t j xj ∈ X . 0 0 0 0 If {xj } is σ(X , X) Cauchy and hx , xi = lim xj , x for x ∈ X, then x0 ∈ X b .

Proof: Suppose that (i) holds. Let xj → 0 in X. Then {xj } is bounded in X and, therefore, β(X, X 0 ) bounded by (i). Hence, M = sup{|hx0i , xj i| : i, j ∈ N} < ∞

and

X ∞ X ∞ 0 ≤M t x , x |tj | j i j j=n j=n

0 P for {tj } ∈ l1 . Therefore, the series ∞ j=1 tj xj , xi converge uniformly for i ∈ N. Hence, ∞ ∞ X

X

lim tj lim x0j , xi = 0 tj x0j , xi = i

j=1

j=1

i

P∞

0 j=1 tj xj

∈ X s and (ii) holds. That (ii) implies (iii) is immediate. Assume that (iii) holds. We show that (i) holds. Let A ⊂ X be σ(X, X 0 ) bounded and B ⊂ X 0 be σ(X 0 , X) bounded. We show that sup{|hx0 , xi| : x0 ∈ B, x ∈ A} < ∞. If this fails to hold, there exist {x0j } ⊂ B and {xj } ⊂ A such that so

(#)

Consider the matrix

|hx0i , xi i| > i2 for every i.

M = [mij ] = [(1/j) x0j , (1/i)xi ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since {xi } is σ(X, X 0 ) bounded. Given any subsequence {mj } P∞ pick a further subsequence {nj } such that j=1 1/nj < ∞. By (iii) + *∞ ∞ E D X X 0 (1/nj ) x0nj , (1/i)xi → 0. (1/nj )xnj , (1/i)xi = j=1

j=1

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Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (#). We next show that (i) implies (iv). Let A ⊂ X be bounded.

Then A is σ(X, X 0 ) bounded by (i). Since {x0j } is β(X 0 , X) bounded, { x0j , x : x ∈ A, j ∈ N} is bounded. Therefore, {hx0 , xi : x ∈ A} is bounded. Therefore, x0 ∈ X b and (iv) holds. Suppose that (iv) holds. We show that (iii) holds and this will com- P plete the proof. X and {tj } ∈ l1 , then limn nj=1 tj x0j , x

0 If x ∈ P P∞ ∞ = j=1 tj xj , x . By (iv), j=1 tj x0j ∈ X b and (iii) holds.

Theorem 3.23 is contained in [LS], Theorem 7, where other characterizations of Banach-Mackey spaces are given. We make an interesting observation concerning Banach spaces with an unconditional Schauder basis. Let X be a Banach space. A sequence {bj } ⊂ X is a Schauder basis for X if every x ∈ X has a unique series P∞ representation x = j=1 tj bj ; the linear functionals fj : X → R defined by hfj , xi = tj are called the coordinate functionals associated with the basis {bj }. It is known that the coordinate functionals are equicontinuous P∞ ([Sw2] 10.10). If the series x = j=1 hfj , xi bj is unconditionally convergent (subseries convergent) for every x, the basis {bj } is said to be unconditional. Theorem 3.24. Let {bj } be an unconditional basis for the Banach space P∞ X. If x ∈ X, the series j=1 tj bj converge uniformly for |tj | ≤ |hfj , xi|. P∞ Proof: Since X is complete, the series j=1 hfj , xi bj is also bounded multiplier convergent (Theorem 2.54) so the result follows from Theorem 2.54. In the last part of this chapter we present several applications of convergent series to topics in vector valued measures. Let A (Σ) be an algebra (σ-algebra) of subsets of a set S and let X be a TVS. A set function µ : A →X is finitely additive (countably additive) if µ(∅) = 0 and P µ(A∪B) = µ(A)+µ(B) when A, B ∈ A with A∩B = ∅ (µ(A) = ∞ j=1 µ(Aj ) A ∈ A). Note that if when {Aj } ⊂ A, is pairwise disjoint and A = ∪∞ j P∞ j=1 µ is countably additive, then the series µ(A ) is unconditionally j j=1 convergent since the union ∪∞ j=1 Aj is independent of the ordering of the {Aj }. Finitely additive set functions, even scalar valued functions, defined on algebras or σ-algebras are not necessarily bounded as the following examples

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show. First, we give a simple example of an unbounded, finitely additive set function defined on an algebra. Example 3.25. Let A be the algebra of finite/co-finite subsets of N; i.e., A ∈ A iff either A or the complement of A, Ac , is finite. Define µ : A → R by µ(A) equals the number of elements in A when A is finite and µ(A) equals minus the number of elements in Ac when Ac is finite. Then µ is finitely additive but not bounded. To present an example of a finitely additive set function defined on a σ-algebra which is unbounded is more complicated. We present an example due to Giesy ([Gi]). Lemma 3.26. Let A, B be algebras of subsets of a set S with A ⊂ B and let α : A → R be finitely additive. If B ∈ BA and b ∈ R, there exists β : B → R finitely additive such that β is an extension of α with β(B) = b. Proof: Let S(A) (S(B)) be the vector space of all A (B) simple functions. Then α Rinduces a linear functional α0 : S(A) → R via integration, i.e., α0 (f ) = f dα. The linear functional α0 has a linear extension, β 0 , to S(B) such that β 0 (χB ) = b. Then β(E) = β 0 (χE ) defines the desired finitely additive extension of α. We now give an example of a real valued, finitely additive set function defined on the σ-algebra of Lebesgue measurable subsets of R which is not bounded. Example 3.27. Let {Ej }∞ j=0 be a pairwise disjoint sequence of bounded intervals whose union is R. Let Ak be the algebra generated by {R, E0 , E1 , ..., Ek } so A0 ⊂ A1 ⊂ ... ⊂ M, where M is the σ-algebra of Lebesgue measurable subsets of R. Set α0 = 0 on A0 ; let α1 be a finitely additive extension of α0 to A1 such that α1 (E1 ) = 1 (Lemma 3.26). Inductively, there is a sequence {αk } of finitely additive set functions such that αk : Ak → R, αk+1 extends αk and αk (Ek ) = k. Now A = ∪∞ k=0 Ak is an α is finitely additive on A. By Lemma 3.26 there is algebra and α = ∪∞ k k=0 a finitely additive extension of α, µ, to M and we have that µ(Ek ) = k for every k so µ is not bounded. We now give several conditions which characterize bounded, finitely additive set functions with values in LCTVS.

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Theorem 3.28. Let X be an LCTVS and let µ : A → R be finitely additive. The following are equivalent: (i) µ is bounded, (ii) for every pairwise disjoint sequence {Aj } from A, {µ(Aj )} is bounded, P (iii) for every pairwise disjoint sequence {Aj } from A, the series j µ(Aj ) is c0 multiplier Cauchy. Proof: Clearly, (i) implies (ii). Suppose that (ii) holds and µ is not bounded. If E ∈ A, set AE = {A ∩ E: A ∈ A}. Suppose that µ(AE ) is not absorbed by the absolutely convex neighborhood of 0, U , in X. Pick an absolutely convex neighborhood of 0, V , such that V + V ⊂ U . We claim that for every k there exist nk > k and a partition (Ak , Bk ) of E with Ak , Bk ∈ A and µ(Ak ) ∈ / nk V, µ(Bk ) ∈ / nk V . For, there exists nk > k such that µ(E) ∈ nk V . But, µ(AE ) * nk (V + V ) since V + V ⊂ U . Therefore, there exists Ak ∈ AE such that µ(Ak ) ∈ / nk (V + V ). Note that µ(Ak ) ∈ / nk V . Put Bk = EAk . Then µ(Bk ) ∈ / nk V since otherwise µ(Ak ) = µ(E) − µ(Bk ) ∈ nk (V + V ). Since µ(A) is assumed to be unbounded, there exists an absolutely convex neighborhood of 0, U , in X such that µ(A) is not absorbed by U . Pick V as above. By the observation above there exist n1 > 1 and a partition (A1 , B1 ) of S such that µ(A1 ) ∈ / n1 V and µ(B1 ) ∈ / n1 V . Either µ(AA1 ) or µ(AB1 ) is not absorbed by U since otherwise there exists m such that µ(AA1 ) ⊂ mU and µ(AB1 ) ⊂ mU and µ(AS ) = µ(A) ⊂m(U +U ) ⊂m(2U ) since U is convex. Pick whichever of A1 or B1 satisfies this condition, label it F1 and set E1 = SF1 . Now treat F1 as above to obtain a partition (E2 , F2 ) of F1 and n2 > n1 such that µ(E2 ) ∈ / n2 V, µ(F2 ) ∈ / n2 V and µ(AF2 ) is not absorbed by U . Continuing this construction produces a pairwise disjoint sequence {Ek } such that {µ(Ek )} is not absorbed by U . Thus, (ii) fails to hold so (ii) implies (i). Suppose that (iii) holds, {Aj } ⊂ A is pairwise disjoint and t ∈ c0 . Then P j tj µ(Aj ) is Cauchy so lim tj µ(Aj ) = 0. Since t ∈ c0 is arbitrary, {µ(Aj )} is bounded. Thus, (ii) holds. Suppose that (i) holds and let {Aj } ⊂ A be pairwise disjoint. Then X µ(Aj ) : σ finite} = {µ(∪j∈σ Aj ) : σ finite j∈σ

is bounded. Therefore, 3.8.

P

j

µ(Aj ) is c0 multiplier Cauchy by Proposition

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If the LCTVS X is sequentially complete, the condition (iii) in Theorem 3.28 can be strengthened. To establish this we employ a very intersting and simple result of Li Ronglu ([LS] Corollary 10) which will be used several times later. Lemma 3.29. Let {Ej } be a sequence of sets. Let G be an Abelian (HausP∞ dorff ) topological group and fj : Ej → G. If the series fj (tj ) conj=1P ∞ verges for every sequence {tj } with tj ∈ Ej , then the series j=1 fj (tj ) converge uniformly for all sequences {tj } with tj ∈ Ej . Proof: If the conclusion fails to hold, there exist a neighborhood, U , of 0 in G and sequences {tij }j , tij ∈ Ej , and an increasing sequence {ni } P∞ such that j=ni fj (tij ) ∈ / U . Pick a symmetric neighborhood of 0,V , such P∞ P∞ 1 that V + V ⊂ U . Since limk j=k fj (t1j ) = 0 and / U, j=n1 fj( tj ) ∈ Pm1 1 there exists m1 > n1 such that j=n1 f (tj ) ∈ / V . Put N1 = 1 and pick P∞ i2 ni2 = N2 > m1 such that / U . As before pick m2 > j=N2 fj (tj ) ∈ Pm2 i2 / V . Continuing this construction produces N2 such that j=N2 fj (tj ) ∈ increasing sequences {Nk }, {mk } and {ik } such that Nk < mk < Nk+1 P mk and j=N fj (tijk ) ∈ / V . Pick an arbitrary sequence {uj } with uj ∈ Ej for k every j. Define a sequence {sj } with sj ∈ Ej by sj = tijk if Nk ≤ j ≤ mk P∞ and sj = uj otherwise. If the series j=1 fj (sj ) converges, there exists Pmk Pn N such that j=m fj (sj ) ∈ V for n > m ≥ N . But, j=Nk fj (sj ) = P∞ P mk ik / V for large k so the series j=1 fj (sj ) does not satisfy j=Nk fj (tj ) ∈ the Cauchy condition and, therefore, does not converge. This contradicts the hypothesis. To illustate the utility of Lemma 3.29, we derive a couple of previous results for series which were established by other means. First, we consider a version of Corollary 2.18. P Corollary 3.30. Let X be a TVS and j xj a series in X which is subP series convergent. Then the series j∈σ xj converge uniformly for σ ⊂ N. Proof: Let Ej = {0, 1} for every j and define fj : Ej → X by fj (0) = 0 and fj (1) = xj . Then the conclusion follows directly from Lemma 3.29. Next, we consider an improvement of Corollary 2.19. P Corollary 3.31. Let X be a TVS and let λ be a normal space and let j xj P∞ be λ multipier convergent. If t ∈ λ, then the series j=1 sj xj converge uniformly for |sj | ≤ |tj |.

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Proof: Let Ej = {t ∈ R : |t| ≤ |tj |} and define fj : Ej → X by fj (t) = txj . Then the conclusion follows directly from Lemma 3.29. Note that the space λ in Corollary 3.31 is not assume to be a K-space with signed-SGHP as in Corollary 2.19. From Corollary 3.31 we can obtain immediately Corollary 2.17. P Corollary 3.32. Let X be a TVS and let j xj be l∞ multiplier converP∞ gent. Then the series j=1 tj xj converge uniformly for k{tj }k∞ ≤ 1. Proof: Let t be the constant sequence with 1 in each coordinate. Then the result follows immediately from Corollary 3.31.

Notice that Corollaries 3.31 and 3.32 were proven in reverse order in Chapter 2. We now show that Lemma 3.29 can be used to obtain an improvement in Theorem 3.28 when the space X is a sequentially complete LCTVS. Corollary 3.33. Let X be a sequentially complete LCTVS and µ : A →X finitely additive. The following are equivalent: (i) µ is bounded, (ii) for every pairwise disjoint sequence {Aj } from A, {µ(Aj )} is bounded, P (iii)’ for every pairwise disjoint sequence {Aj } from A, the series j µ(Aj ) is c0 multiplier convergent, (iv) for every pairwise disjoint sequence {Aj } from A and t ∈ c0 , the series P∞ j=1 sj µ(Bj ) converge uniformly for Bj ⊂ Aj , Bj ∈ A and |sj | ≤ |tj |. Proof: (i), (ii) and (iii)’ are equivalent by Theorem 3.28. Obviously, (iv) implies (iii)’. Assume (iii)’. We apply Lemma 3.29. Set Ej = {(B, s) : B ∈ A, B ⊂ Aj , |s| ≤ |tj |} and define fj : Ej → X result.

by fj (B, s) = sµ(B). Lemma 3.29 now gives the

We also have the following boundedness result for countably additive set functions defined on σ-algebras. Corollary 3.34. Let X be an LCTVS and Σ a σ-algebra. If µ : Σ → X is countably additive, then µ is bounded.

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Proof: µ satisfies condition (ii) of Theorem 3.28. Remark 3.35. The locally convex assumption in Corollary 3.34 is important. Turpin has given an example of a countably additive set function defined on a σ-algebra with values in a (non-locally convex) TVS which is unbounded ([Rol] 3.6.4). We next consider an important property for vector valued set functions which was introduced by Rickart which lies between finite additivity and countable additivity. Definition 3.36. Let X be a TVS and µ : A →X be finitely additive. Then µ is strongly bounded (strongly additive, exhaustive) if µ(Aj ) → 0 for every pairwise disjoint sequence {Aj } from A. A countably additive set function defined on a σ-algebra is obviously strongly bounded. We show below that bounded, finitely additive scalar valued set functions are strongly bounded and give an example of a bounded, finitely additive set function defined on a σ-algebra with values in a Banach space which is not strongly bounded. Lemma there exists M ≥ 0 such that P 3.37. Let {tj } ⊂ R and assume thatP ∞ j∈σ tj ≤ M for every finite σ ⊂ N. Then j=1 |tj | ≤ 2M .

Proof: Let σ be finite. Set σ+ = {j ∈ σ : tj ≥ 0} and σ− = {j ∈ σ : tj < 0}. Then X X tj ≤ M |tj | = j∈σ+

j∈σ+

and X

j∈σ−

so

P

j∈σ

|tj | = −

X

j∈σ−

|tj | ≤ 2M . Since σ is arbitrary,

Corollary 3.38. Let µ : A → R iff µ is strongly bounded.

tj ≤ M

P∞

j=1

|tj | < 2M .

be finitely additive. Then µ is bounded

Proof: If µ is strongly bounded, then µ is bounded since condition (ii) of Theorem 3.28 is satisfied.

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Suppose that µ is bounded with sup{|µ(A)| : A ∈ A } = M < ∞. Let {Aj } ⊂ A be pairwise disjoint. If σ ⊂ N is finite, then X µ(Aj ) = |µ(∪j∈σ Aj )| ≤ M j∈σ P so by Lemma 3.37, ∞ j=1 |µ(Aj )| ≤ 2M . In particular, µ(Aj ) → 0 so µ is strongly bounded. Remark 3.39. The proof of Corollary 3.38 shows that if µ : A → R is bounded and finitely additive and {Aj } is pairwise disjoint, then the P absolutely convergent. Thus, if µ fails to be countably series ∞ j=1 µ(Aj ) is P additive, the series ∞ j=1 µ(Aj ) converges but may fail to converge to the ”proper value”, namely, µ(∪∞ j=1 Aj ). For vector valued set functions we have the following boundedness result. Corollary 3.40. Let X be an LCTVS. If µ : A →X is strongly bounded, then µ is bounded. Proof: For each x0 ∈ X 0 , x0 ◦ µ = x0 µ : A → R is strongly bounded so {hx , µ(A)i : A ∈ A} is bounded by Corollary 3.38. Thus, {µ(A) : A ∈ A} is weakly bounded in X and, therefore, bounded in X. 0

The example indicated in Remark 3.35 shows that the local convex assumption in Corollary 3.40 is important. The converse of Corollary 3.38 is false, in general. Example 3.41. Let M be the σ-algebra of Lebesgue measurable subsets of [0, 1]. Define µ : M → L∞ [0, 1] by µ(E) = χE . Then µ is bounded, finitely additive but not strongly bounded [take any pairwise disjoint sequence from M with positive Lebesgue measure]. We have a series characterization of strongly additive set functions. Theorem 3.42. Let X be a TVS and µ : A →X be finitely additive. The following are equivalent: (i) µ is strongly bounded, P (ii) for every pairwise disjoint sequence {Aj } ⊂ A , the series j µ(Aj ) is Cauchy.

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Proof: That (ii) implies (i) is clear. Assume that (ii) fails to hold. Then there exist a neighborhood U of 0 in P X and an increasing sequence of intervals {Ij } such that j∈Ik µ(Aj ) ∈ /U for all k. If Bk = ∪j∈Ik Aj , then {Bk } is pairwise disjoint and µ(Bk ) 9 0 so (i) fails. If X is sequentially complete, using Li’s Lemma 3.29 we can strengthen condition (ii). Corollary 3.43. Let X be a sequentially complete TVS and µ : A →X be finitely additive. The following are equivalent: (i) µ is strongly bounded, P (ii)’ for every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) converges, P (iii) for every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Bj ) converge uniformly for Bj ⊂ Aj , Bj ∈ A. Proof: That (i) and (ii)’ are equivalent follows from Theorem 3.42. Obviously (iii) implies (ii)’. Assume that (ii)’ holds. We establish (iii) by using Lemma 3.29. Set Ej = {B ∈ A : B ⊂ Aj } and define fj : Ej → X by fj (B) = µ(B). Lemma 3.29 now gives the result. We consider the semi-variation of set functions µ : A →X with values in a normed space X. The semi-variation is useful in discussing topics in vector measures and vector integration ([DS], [DU] I.1). Definition 3.44. For A ∈ A the semi-variation of µ on A is defined to be

n

X

: {Aj }nj=1 a partition of A and |tj | ≤ 1 . kµk (A) = sup t µ(A ) j j

j=1

We have the following properties of the semi-variation. In the proposition below, the variation of a real valued set function ν is denoted by |ν| ([Sw3] 2.2.1.7). Proposition 3.45. Let µ : A →X.

(i) kµk (A) = sup{|x0 µ| (A) : kx0 k ≤ 1}, (ii) sup{kµ(B)k : B ⊂ A, B ∈ A} ≤ kµk (A) ≤ 2 sup{kµ(B)k : B ⊂ A, B ∈ A}.

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Proof: (i): Let {A1 , ..., An } be a partition of A and let |tj | ≤ 1 for j = 1, ..., n. Then D P E

Pn

n

j=1 tj µ(Aj ) = sup{ x0 , j=1 tj µ(Aj ) : kx0 k ≤ 1} Pn ≤ sup{ j=1 |hx0 , tj µ(Aj )i| : kx0 k ≤ 1} Pn ≤ sup{ j=1 |hx0 , µ(Aj )i| : kx0 k ≤ 1} Pn ≤ sup{ j=1 |x0 µ| (Aj ) : kx0 k ≤ 1} = sup{|x0 µ| (A) : kx0 k ≤ 1}. Therefore, kµk (A) ≤ sup{|x0 µ| (A) : kx0 k ≤ 1}. For the reverse inequality, let x0 ∈ X 0 ,kx0 k ≤ 1 and {A1 , ..., An } be a partition of A. Then Pn P |hx0 , µ(Aj )i| = nj=1 (signx0 µ(Aj ))x0 µ(Aj ) j=1 D P E = x0 , nj=1 (signx0 µ(Aj ))µ(Aj )

P

n ≤ j=1 (signx0 µ(Aj ))µ(Aj ) ≤ kµk (A). Therefore, |x0 µ| (A) ≤ kµk (A) and kµk (A) ≥ sup{|x0 µ(A)| : kx0 k ≤ 1}. Thus, (i) holds. For (ii), recall that for scalar set functions ν we have that sup{|ν(B)| : B ⊂ A, B ∈ A} ≤ |ν| (A) ≤ 2 sup{|ν(B)| : B ⊂ A, B ∈ A} ([Sw3] 2.2.7). Let kx0 k ≤ 1. Then sup{kµ(B)k : B ⊂ A, B ∈ A} = sup{|x0 µ(B)| : B ⊂ A, B ∈ A, kx0 k ≤ 1} ≤ sup{|x0 µ| (B) : B ⊂ A, B ∈ A, kx0 k ≤ 1} = sup{|x0 µ| (A) : kx0 k ≤ 1} ≤ 2 sup{|x0 µ(B)| : B ⊂ A, B ∈ A, kx0 k ≤ 1} = 2 sup{kµ(B)k : B ⊂ A, B ∈ A}. Thus. (ii) follows from (i).

Thus, from Proposition 3.45 µ has finite semi-variation iff µ is bounded. Conditions for µ to be bounded are given in Theorem 3.28 and Corollary 3.33. We next show that the stronger conclusion of Corollary 3.43 can be used to establish a strong boundedness property for strongly bounded set functions. Proposition 3.46. Let X be a Banach space and µ : A →X be strongly bounded. If {Aj } ⊂ A is pairwise disjoint, then for every ε > 0 there exists an N such that kµk (∪nj=m Aj ) < ε for n > m ≥ N . In particular, the semi-variation is strongly bounded in the sense that kµk (Aj ) → 0.

P

Proof: By Corollary 3.43 there exists N such that ∞ j=m µ(Bj ) < ε

P

n for Bj ⊂ Aj , Bj ∈ A, m ≥ N . Therefore, j=m µ(Bj ) ≤ 2ε for Bj ⊂

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Aj , Bj ∈ A, n > m ≥ N . Suppose n > m ≥ N

Pand B ∈ A with

B ⊂

n

n n ∪j=m Aj . Then B = ∪j=m B ∩ Aj so kµ(B)k = j=m µ(B ∩ Aj ) ≤ 2ε. By Proposition 3.45, kµk (∪nj=m Aj ) ≤ 4ε. Using Theorem 3.15 we can derive a result of Diestel connecting bounded, finitely additive set functions and strongly bounded set functions ([DU] I.4.2). For this we first present an example. Example 3.47. Let A be the algebra of finite/co-finite subsets of N; A = {A : either A or Ac is finite}. Define µ : A →c0 by µ(A) = χA if A is finite and µ(A) = −χAc if Ac is finite. Then µ is bounded and finitely additive but not strongly bounded since µ({j}) = ej 9 0. Theorem 3.48. Let X be a sequentially complete LCTVS. Then X contains no subspace isomorphic to c0 iff every bounded, finitely additive X valued set function defined on an algebra of sets is strongly bounded. Proof: Example 3.47 shows that if X contains a subspace isomorphic to c0 , then there is a bounded, finitely additive X valued set function defined on an algebra which is not strongly bounded. Suppose that X contains a subspace isomorphic to c0 and µ : A →X is a bounded, finitely additive set function defined on an algebra A. Let {Aj } P be a pairwise disjoint sequence from A. By Corollary 3.33 j µ(Aj ) is c0 P multiplier convergent and by Theorem 3.15 the series j µ(Aj ) is subseries convergent. Hence, µ(Aj ) → 0 and µ is strongly bounded. Finally, in this section we consider the class of vector valued measures of bounded variation. For simplicities sake, we consider only the case of set functions with values in a normed space. P Recall that a series j xj in a normed space is absolutely convergent P∞ iff j=1 kxj k < ∞. An absolutely convergent series is obviously subseries Cauchy so if X is a Banach space an absolutely convergent series is subseries convergent. The converse holds in a finite dimensional space but P not in infinite dimensional spaces [consider j (1/j)ej in c0 or recall the Dvoretsky-Rogers Theorem ([Day], [Sw2] 30.1.1)]. Definition 3.49. Let X be a normed space and µ: A →X be finitely additive. If E ∈ A, the variation of µ on E is defined to be n X |µ| (E) = sup kµ(Aj )k : {Aj }nj=1 is a partition of E with Aj ∈ A . j=1

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If |µ| (S) < ∞, µ is said to have bounded variation. It is routine to show that the variation |µ| : A → [0, ∞] is finitely additive. Since kµ(A)k ≤ |µ| (A) for A ∈ A , if µ has bounded variation, then µ is bounded. If µ : Σ → R is countably additive, then µ has bounded variation ([Sw3] 2.2.1). This statement is false for vector valued set functions as the following example shows. Example 3.50. Let P be power set of N. Let X be a normed space and P P j∈σ xj . j xj subseries convergent in X. Define µ : P →X by µ(σ) = Then it is easily seen that µ is countably additive and bounded [indeed {µ(σ) : σ ⊂ N} is relatively compact by Theorem 2.47]. However, µ has P∞ P∞ P bounded variation iff j=1 kµ({j})k = j=1 kxj k < ∞, i.e., iff j xj is absolutely convergent. Thus, if X is infinite dimensional, by the DvoretskyRogers Theorem ([Day], [Sw2] 30.1.1), there is a countably additive X valued set function defined on a σ-algebra which is of infinite variation. We have a characterization of set functions having bounded variation in terms of absolutely converging series. Theorem 3.51. Let µ : A →X equivalent:

be finitely additive. The following are

(i) µ has bounded variation, P (ii) for every pairwise disjoint sequence {Aj } ⊂ A , j µ(Aj ) is absolutely convergent, P∞ (iii) for every pairwise disjoint sequence {Aj } ⊂ A, the series j=1 kµ(Bj )k converge uniformly for Bj ⊂ Aj with Bj ∈ A. Proof: That (i) implies (ii) is clear. Suppose that (ii) holds. We establish (iii) by using Li’s Lemma 3.29. Set Ej = {B ∈ A : B ⊂ Aj } and define fj : Ej → R by fj (B) = kµ(B)k. Then Lemma 3.29 gives (iii) immediately. Clearly (iii) implies (ii). Suppose that (ii) holds but (i) fails. Note that µ is bounded by Theorem 3.28. Set M = sup{kµ(A)k : A ∈ A }. There exists a partition {A11 , ..., A1n , A1n+1 } of S such that n+1 X j=1

µ(A1j ) > M + 1,

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where some {A1j }, say, A1n+1 satisfies |µ| (A1n+1 ) = ∞ since |µ| is finitely additive. Then n X

µ(A1j ) ≥ 1 + M − µ(A1n+1 ) ≥ 1. j=1

Now treat A1n+1 as S above to obtain a partition of A1n+1 , Pm 2 2 2 2

≥ 2 and |µ| (A2m+1 ) = {A1 , ..., Am , Am+1 } with j=1 µ(Aj ) ∞. Continuing this construction produces a pairwise disjoint sequence {A11 , ..., A1n , A21 , ..., A2m , ...} which violates condition (ii). The equivalence of (i) and (ii) was established by Thorpe ([Thr]).

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Chapter 4

The Orlicz-Pettis Theorem

As noted earlier the classical version of the Orlicz-Pettis Theorem for normed spaces asserts that a series in a normed space which is subseries convergent in the weak topology of the space is subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was originally established by Orlicz for weakly sequentially complete spaces but was evidently known in full generality by the Polish mathematicians as it appears as a statement in Banach’s book ([Ba]). The first version available in English was established by Pettis in [Pe] where it was used to treat topics in vector valued integration — the Pettis integral. The theorem was extended to locally convex spaces by McArthur ([Mc]). For historical discussions of the theorem see [Ka3], [DU], or [FL]. Since a series is subseries convergent iff the series is m0 multiplier convergent, it is natural to ask what sequence spaces λ have the property that series which are λ multiplier convergent in the weak topology are λ multiplier convergent in some stronger topology such as the Mackey topology. We will refer to such results as Orlicz-Pettis Theorems. The locally convex topologies which we utilize will all be polar topologies which are described briefly in Appendix A. We record the polar topologies which we will encounter. Let X, X 0 be a pair of spaces in duality with the duality pairing h, i. The weak topology σ(X, X 0 ) (strong topology β(X, X 0 )) is the polar topology generated by the finite subsets (σ(X 0 , X) bounded subsets) of X 0 . The Mackey topology is the polar topology τ (X, X 0 ) generated by the absolutely convex, σ(X 0 , X) compact subsets of X 0 . We will also use two other polar topologies. The polar topology λ(X, X 0 )(γ(X, X 0 )) is the polar topology on X generated by the family of all σ(X 0 , X) compact subsets of X 0 [conditionally σ(X 0 , X) sequentially 49

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compact subsets of X 0 ; a subset B ⊂ X 0 is conditionally σ(X 0 , X) sequentially compact if every sequence {x0j } ⊂ B has a subsequence {x0nj } such that limhx0nj , xi exists for every x ∈ X ([Din])]. Obviously, λ(X, X 0 ) is stronger than the Mackey topology τ (X, X 0 ) and can be strictly stronger ([K1] 21.4). The topologies λ(X, X 0 ) and γ(X, X 0 ) are not comparable. We recall some basic results from Appendix A.3-6. Definition 4.1. Let X be a vector space and σ and τ two vector topologies on X. We say that τ is linked to σ if τ has a neighborhood base at 0 consisting of σ closed sets. [The terminology is that of Wilansky ([Wi] 6.1.9).] For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are linked to the weak topology σ(X, X 0 ). Lemma 4.2. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. (i) If {xj } ⊂ X is τ Cauchy and if σ − lim xj = x, then τ − lim xj = x. (ii) If (X, σ) is sequentially complete and σ ⊂ τ, then (X, τ ) is sequentially complete. Remark 4.3. It is important that the topologies σ and τ are linked in Lemma 4.2. For example, consider the space c with its weak topology P σ(c, l1 ) and the topology of pointwise convergence p. The series j ej is p convergent, the partial sums of the series are σ(c, l 1 ) Cauchy, but the series is not σ(c, l1 ) convergent. Lemma 4.4. Let X be a vector space and σ and τ two vector topologies P on X such that τ is linked to σ. If every series j xj which is σ subseries convergent satisfies τ − lim xj = 0, then every series in X which is σ subseries convergent is τ subseries convergent. The proofs of the lemmas can be found in Appendix A.3-6. Throughout this chapter λ will denote a scalar sequence space which contains c00 , the space of sequences which are eventually 0. If ∞ X λβ = {sj } : sj tj converges for every {tj } ∈ λ j=1 P∞ is the β-dual of λ, we write s · t = j=1 sj tj for {sj } ∈ λβ and {tj } ∈ λ. Note that λ and λβ are in duality with respect to the bilinear pairing s · t.

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Recall that if λ has a vector topology τ , then (λ, τ ) is an AK-space if Pn P∞ t = τ − lim j=1 tj ej = j=1 tj ej for each t ∈ λ [Appendix B.2]. We show that the conclusion of any Orlicz-Pettis Theorem for a Hellinger-Toeplitz topology is characterized by the AK-property. Recall that a locally convex topology w(X, X 0 ) defined for dual pairs X, X 0 is said to be a HellingerToeplitz topology if whenever a linear map T : X → Y is σ(X, X 0 )−σ(Y, Y 0 ) continuous, then T is also w(X, X 0 )−w(Y, Y 0 ) continuous ([Wi] 11.1.5 or see Appendix A.1; note that Hellinger-Toeplitz topologies must be defined for dual pairs). For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are Hellinger-Toeplitz topologies [Appendix A.2]. Theorem 4.5. Let w be a Hellinger-Toeplitz topology for dual pairs. The following are equivalent: (i) For every dual pair X, X 0 a series which is λ multiplier convergent for the weak topology σ(X, X 0 ) is λ multiplier convergent with respect to w(X, X 0 ). (ii) (λ, w(λ, λβ )) is an AK-space. P j Proof: Assume (i). Then j e is λ multiplier convergent with reP spect to σ(λ, λβ ) so by (i), j ej is λ multiplier convergent with respect P∞ to w(λ, λβ ). But, this means that if t ∈ λ, then t = j=1 tj ej , where the series is w(λ, λβ ) convergent so (ii) holds. P Assume (ii). Let to j xj be λ multiplier convergent with respect P σ(X, X 0 ). Consider the summing operator S : λ → X , St = ∞ t x j j j=1 [σ(X, X 0 ) limit ]. By Theorem 2.2, S is σ(λ, λβ ) − σ(X, X 0 ) continuous and, therefore, w(λ, λβ ) − w(X, X 0 ) continuous. If t = {tj } ∈ λ, then t = Pn Pn P∞ w(λ, λβ ) − lim j=1 tj ej so T t = w(X, X 0 ) − lim j=1 tj xj = j=1 tj xj . Hence, (i) holds. Condition (i) is, of course, just the conclusion of the Orlicz-Pettis Theorem for the Hellinger Toeplitz topology w(X, X 0 ). Thus, in order to establish an Orlicz-Pettis Theorem for a Hellinger-Toeplitz topology, it suffices to check the AK-property for the topology w(λ, λβ ). We now give several examples where this is the case. P Corollary 4.6. Let λ be a barrelled AK-space. If j xj is λPmultiplier convergent with respect to the weak topology σ(X, X 0 ), then j xj is λ multiplier with respect to the strong topology β(X, X 0 ).

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Proof: A barrelled space always carries the strong topology so the result follows from Theorem 4.5 since the strong topology is a Hellinger-Toeplitz topology. Remark 4.7. Corollary 4.6 applies to any Banach [Frechet ] AK-space. In particular, Corollary 4.6 applies to the spaces λ = c0 , lp (1 ≤ p < ∞), cs or bv0 [Appendix B]. In general, Orlicz-Pettis Theorems do not hold for the strong topology even in the case of subseries convergent series as the following example shows. P Example 4.8. The series j ej is subseries convergent in l ∞ with respect to the weak topology σ(l ∞ , l1 ) = σ(l∞ , (l∞ )β ) but is not subseries convergent with respect to the strong topology β(l ∞ , l1 ) = k·k∞ . We consider Orlicz-Pettis Theorems for the strong topology in Chapter 5. These results require more stringent assumptions on the multiplier space λ. We next establish an AK theorem for a general class of sequence spaces. Recall that a sequence space λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, then there exist a sequence of signs {sj } and a subsequence {nj } P such that the coordinate sum ∞ j=1 sj χInj t ∈ λ ; if the signs can all be chosen to be equal to 1, then λ is said to have the weak gliding hump property (WGHP). For examples, see Appendix B. Theorem 4.9. Assume that λ has signed-WGHP. Then (i) (λ, γ(λ, λβ )) is an AK-space. (ii) (λ, λ(λ, λβ )) is an AK-space. Proof: (i): Since γ(λ, λβ ) is linked to σ(λ, λβ ), it suffices to show that for P∞ every t ∈ λ the series j=1 tj ej is γ(λ, λβ ) Cauchy (Lemma 4.2). Suppose that there exist > 0, K ⊂ λβ which is conditionally σ(λβ , λ) sequentially compact and increasing intervals {Ij } such that X sup u · tj ej > . u∈K j∈Ik

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For each k pick uk ∈ K such that k X j tj e > . (∗) u · j∈Ik

There exists an increasing sequence {nk } such that {unk } is σ(λβ , λ) Cauchy. Define the matrix X tl el ]. M = [mij ] = [uni · l∈Inj

We show that M is a signed K-matrix (Appendix D.3). First the columns of M converge since {uni } is σ(λβ , λ) Cauchy. Next, if {pj } is an increasing sequence, there is a subsequence {qj } of {pj } and a sequence of signs {sj } P∞ P such that v = j=1 sj l∈Inq tl el ∈ λ. Then j

∞ X j=1

P∞

sj miqj = uni · v

so lim j=1 sj miqj exists since {uni } is σ(λβ , λ) Cauchy. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (∗) and establishes (i). (ii): Consider λ with the Mackey topology τ (λ, λβ ) so the dual of (λ, τ (λ, λβ )) is λβ . We claim that (λ, τ (λ, λβ )) is τ (λ, λβ ) separable. This follows since (λ, σ(λ, λβ )) is an AK-space so the σ(λ, λβ ) closure of S = span{ek : k ∈ N} is σ(λ, λβ ) dense in λ. But, S has the same closure in σ(λ, λβ ) and τ (λ, λβ ) so S is τ (λ, λβ ) dense in λ and (λ, τ (λ, λβ )) is τ (λ, λβ ) separable. This implies that σ(λβ , λ) compact sets are sequentially compact ([Wi] 9.5.3). Now, the proof of part (i) may be repeated using a σ(λβ , λ) compact (sequentially compact ) set K ⊂ λβ . From Theorems 4.5 and 4.9, we obtain an Orlicz-Pettis Theorem for λ multiplier convergent series. Corollary 4.10. Assume that λ has signed-WGHP and let X be an P LCTVS. If j xj is λ multiplier convergent with respect to the weak topolP with respect to the ogy σ(X, X 0 ), then j xj is λ multiplier convergent P topologies γ(X, X 0 ) and λ(X, X 0 ). In particular, if j xj is λ multiplier P convergent with respect to σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the Mackey topology τ (X, X 0 ).

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Corollary 4.10 contains an Orlicz-Pettis Theorem with respect to the topology γ(X, X 0 ) for a multiplier space λ which has the signed-WGHP. We observe here that any multiplier space for which the Orlicz-Pettis Theorem holds with respect to the topology γ(X, X 0 ) has the property that the topology σ(λβ , λ) is sequentially complete [thus, Corollary 4.10 implies Stuart’s sequential completeness result in 2.28]. For this observation, let {y k } be σ(λβ , λ) Cauchy. Set yj = limk y k · ej = limk yjk and y = {yj }. P We claim that y ∈ λβ and y k → y in σ(λβ , λ). Then j ej is λ multiplier convergent with respect to σ(λ, λβ ) and is, therefore, λ multiplier convergent with respect to γ(λ, λβ ) by hypothesis. Let t ∈ λ and let > 0. There Pn exists N such that j=m yjk tj < for n > m ≥ N and for all k ∈ N P by the γ(λ, λβ ) convergence. Then nj=m yj tj ≤ for n > m ≥ N so P y t converges and y ∈ λβ . Pick M such that k ≥ M implies that Pj j j N −1 k j=1 (yj tj − yj tj ) < . If k ≥ M , then X X ∞ NX ∞ X ∞ k X ∞ k −1 k + + ≤ < 3 y t y t − y t y t (y t − y t ) j j j j j j j j j j j j j=1 j=1 j=1 j=N j=N so y k → y in σ(λβ , λ). Appendix B gives a list of sequence spaces with signed-WGHP to which Corollary 4.10 applies. In particular, the space m0 has signed-WGHP (being monotone and having WGHP) so Corollary 4.10 applies to subseries convergent series. We give a formal statement of the subseries result. P is subseries convergent Corollary 4.11. Let X be an LCTVS. If j xj P with respect to the weak topology σ(X, X 0 ), then j xj is subseries con0 vergent with respect to the topologies γ(X, X ) and λ(X, X 0 ). In particuP lar, then j xj is subseries convergent with respect to the Mackey topology τ (X, X 0 ).

The usual statement of the Orlicz-Pettis Theorem for subseries convergent series and the Mackey topology was established by McArthur ([Mc]). The statement for the topology λ(X, X 0 ) was established Bennett and Kalton in [BK]. The version for γ(X, X 0 ) is given by Dierolf in [Die]. We give an example covered by Corollary 4.10 but not by Corollary 4.11. P Example 4.12. Consider the series j (1/j)ej in cs, the space of convergent series (Appendix B). This series is obviously not subseries convergent

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in cs with respect to the norm topology. However, if λ = bs, the space P of bounded series (Appendix B), and if t ∈ bs, then the series j (tj /j)ej converges in cs since {tj /j} ∈ cs because {1/j} ∈ bv0 = (bs)β , the space of P null sequences with bounded variation (Appendix B). Thus, j (1/j)ej is bs multiplier convergent in cs but not subseries convergent in cs. Without some assumption on the multiplier space λ, the conclusion of Corollary 4.10 may fail. Example 4.13. Let cc = c0 ⊕span{1, 1, 1, ...}, the space of sequences which are eventually constant (Appendix B). If X is a TVS, then a series P P multiplier convergent in X iff the series j xj converges j xj in X is cc P in X. The series j (ej+1 − ej ) is σ(c0 , l1 ) convergent in c0 (to −e1 ) and, therefore, cc multiplier convergent with respect to σ(c0 , l1 ) but is not cc multiplier convergent with respect to the norm or Mackey topology of c0 . The space l∞ is monotone and, therefore, has WGHP so Corollary 4.10 applies to l∞ or bounded multiplier convergent series. We give a formal statement of this version of the Orlicz-Pettis Theorem. P ∞ multiCorollary 4.14. Let X be an LCTVS. If the series j xj is l 0 plier convergent with respect to the weak topology σ(X, X ), then the series P ∞ multiplier convergent with respect to the topologies γ(X, X 0 ) j xj is l P 0 and λ(X, X ). In particular,if the series j xj is l∞ multiplier convergent with respect to the weak topology, then the series is l ∞ multiplier convergent with respect to the Mackey topology τ (X, X 0 ). We can use Corollary 4.10 to give a generalization of an interesting and useful Orlicz-Pettis Theorem due to Kalton ([Ka3]). Theorem 4.15. Let λ have signed-WGHP. Let X, X 0 be a pair of vector spaces in duality and suppose that τ is a polar topology from this dualP ity which is separable. If j xj is λ multiplier convergent with respect to P σ(X, X 0 ), then j xj is λ multiplier convergent with respect to τ .

Proof: Let D = {dk : k ∈ N} be τ dense in X and let A be a family of σ(X 0 , X) bounded sets which generate the polar topology τ, τ = τA (Appendix A). If A ∈ A, we show that A is conditionally σ(X 0 , X) sequentially compact and the result will follow from Corollary 4.10. Let {yk } ⊂ A. Since {yk } is pointwise bounded on D, a diagonal procedure implies that there exists a subsequence{ynk } of {yk } which converges pointwise on D ([DeS] 26.10). We claim that {ynk } is σ(X 0 , X) Cauchy. Let x ∈ X. There

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exists a net {dα } in D such that {dα } converges to x with respect to τ . If β > 0, there exists β such that y, x − d < /3

for y ∈ A. There exists n such that j, k ≥ n implies that ynj − ynk , dβ < /3. If j, k ≥ n, then

ynj − yn , x ≤ yn , x − dβ + ynj − yn , dβ + ynj , dβ − x < . k k k Hence, lim hynk , xi exists so {ynk } is σ(X 0 , X) Cauchy as claimed. Theorem 4.15 has an application to the strong topology. Corollary 4.16. Let λ have signed-WGHP. If (X, β(X, X 0 )) is separable P P and j xj is λ multiplier convergent with respect to σ(X, X 0 ), then j xj is λ multiplier convergent with respect to β(X, X 0 ). Example 4.8 shows that the separability assumption in Theorem 4.15 and Corollary 4.16 is important. The subseries version of Theorem 4.15 is due to Kalton ([Ka3]). Remark 4.17. If X is a barreled AK-space, Corollary 4.16 applies. In particular, Corollary 4.16 applies to c0 , lp (1 ≤ p < ∞), cs or bv0 with respect to their normed topologies. In Corollary 4.11 we showed that a series which is subseries convergent with respect to the weak topology σ(X, X 0 ) is actually subseries convergent with respect to two stronger polar topologies. Dierolf has shown that there is a strongest polar topology which has the same subseries convergent series as the weak topology ([Die]). He has established a similar result for bounded multiplier convergent series. We now establish both of these results of Dierolf and then give a generalization of his results to λ multiplier convergent series. As established in Theorem 4.5 the conclusion of the Orlicz-Pettis Theorem with respect to a Hellinger-Toeplitz topology is associated with the AK property for the Hellinger-Toeplitz topology on the multiplier space. A series is subseries convergent iff the series is m0 multiplier convergent, but m0 is not an AK-space with respect to its ”natural” topology, the k·k∞ topology. We now define the Dierolf topology and show that mo is an AKspace with respect to this topology. We also show that the Dierolf topology is a Hellinger-Toeplitz topology so Theorem 4.5 is applicable. In treating the Dierolf topology we use some basic properties of l 1 which we now state for convenience. Proposition 4.18. For the space l 1 , we have the following properties:

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(i) The topologies k·k1 , σ(l1 , l∞ ) and σ(l1 , m0 ) have the same convergent sequences, Cauchy sequences, the same bounded sets and the same compact sets. P∞ (ii) A subset K of l 1 is k·k1 relatively compact iff limn j=n |tj | = 0 uniformly for {tj } ∈ K. For Proposition 4.18 see [K1] 22.4. For (ii) see [Sw2] 10.1.15. Part (i) will be established later in Chapter 7. Let X, X 0 be vector spaces in duality. Let M = M(X, X 0 ) be the family of all subsets M ⊂ X 0 such that M is σ(X 0 , X) bounded and for every linear, continuous map T : (X 0 , σ(X 0 , X)) → (l1 , σ(l1 , m0 )),

T M is relatively compact in (l 1 , k·k1 ).

Definition 4.19. The Dierolf topology, δ1 (X, X 0 ), on X is the polar topology, τM , of uniform convergence on the elements of M (Appendix A). From Theorem A.2 of Appendix A, we easily have Theorem 4.20. δ1 (X, X 0 ) is a Hellinger-Toeplitz topology. We now show that m0 is an AK-space under the Dierolf topology δ1 (X, X 0 ). Actually, we have a stronger result. Pn Theorem 4.21. For t ∈ m0 , δ1 (m0 , l1 ) − limn j=1 tj ej = t, uniformly for ktk∞ ≤ 1. In particular, (m0 , δ1 (m0 , l1 )) is an AK-space. Proof: Let M ∈ M (relative to the duality between m0 and l1 ). Then M P∞ is relatively compact in (l 1 , k·k1 ) so by Proposition 4.18, limn j=n |sj | = 0 uniformly for s ∈ M . Thus, for s ∈ M and t ∈ m0 with ktk∞ ≤ 1, we have X X ∞ ∞ ≤ |sj | . s t j j j=n j=n P j Therefore, limn s · ∞ j=n tj e = 0 uniformly for s ∈ M, ktk∞ ≤ 1, so P n 1 j δ1 (m0 , l ) − limn j=1 tj e = t, uniformly for ktk∞ ≤ 1.

From Theorems 4.5 and 4.21, we obtain an Orlicz-Pettis Theorem for the Dierolf topology δ1 (X, X 0 ). P Theorem 4.22. Let X be an LCTVS. If the series j xj is subseries P convergent with respect to σ(X, X 0 ), then j xj is subseries convergent with respect to δ1 (X, X 0 ).

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Actually, from Theorems 4.20, 4.21 and 4.22, we have an improvement of Corollary 4.11. P is subseries Theorem 4.23. Let X be an LCTVS. If the series j xjP n convergent with respect to σ(X, X 0 ), then δ1 (X, X 0 ) − limn j=1 tj xj = P∞ j=1 tj xj uniformly for t ∈ m0 , ktk∞ ≤ 1.

If X is an LCTVS, it is clear that δ1 (X, X 0 ) is stronger than λ(X, X 0 ), the topology of uniform convergence on σ(X 0 , X) compact sets, since any σ(X 0 , X) compact set belongs to the family M [relative to the duality between X and X 0 ] by Proposition 4.18. Thus, Theorem 4.22 gives an improvement to the Orlicz-Pettis Theorem given in Corollary 4.11. Remark 4.24. The topology δ1 (X, X 0 ) is stronger than the topology γ(X, X 0 ), the topology of uniform convergence on the conditionally σ(X 0 , X) sequentially compact sets. For if A ⊂ X 0 is conditionally σ(X 0 , X) sequentially compact and T : (X 0 , σ(X 0 , X))) → (l1 , σ(l1 , m0 ))

is linear and continuous, then T A is conditionally σ(l 1 , m0 ) sequentially compact. By Proposition 4.18, T A is relatively k·k1 compact since k·k1 and σ(l1 , m0 ) have the same Cauchy sequences. Thus, Theorem 4.22 gives an improvement of Corollary 4.11 for the topology γ(X, X 0 ). We now show that δ1 (X, X 0 ) is the strongest polar topology with the same subseries convergent series as σ(X, X 0 ). For this we require a slight refinement of the statement in Proposition 4.18 (i). Proposition 4.25. Let K ⊂ l 1 . The following are equivalent: (i) (ii) (iii) (iv)

K is relatively k·k1 compact, P∞ limn j=n |tj | = 0 uniformly for t ∈ K, P∞ for each s ∈ l ∞ , limn j=n sj tj = 0 uniformly for t ∈ K, P∞ for each s ∈ m0 , limn j=n sj tj = 0 uniformly for t ∈ K.

Proof: (i) and (ii) are equivalent by Proposition 4.18 and clearly (ii) implies (iii) implies (iv). Suppose that (iv) holds but (ii) fails. Then there exists > 0 such that P∞ k for every k there exist mk > k and tk ∈ K such that i=m 1 k ti > 5. In P ∞ 1 particular, there exist m1 > 1, t ∈ K such that i=m1 ti > 5. There Pn1 1 P∞ t exists n1 > m1 such that i=n1 +1 t1i < . Therefore, i=m i < 4. 1 Put I1 = [m1 , n1 ], I1+ = {i ∈ I1 : t1i ≥ 0} and I1− = {i ∈ I1 : t1i < 0}. Either

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P i∈I1− t1i > 2; pick one of these which satisfies this P inequality and label it J1 so i∈J1 t1i > 2. Continuing this construction produces an increasing sequence of finite subsets Jk of N, tk ∈ K, mk < nk < mk+1 < ... with

P i∈I1+ t1i > 2 or

∞ X X k k ti < . ti > 2, Jk ⊂ [mk , nk ], i=nk +1

i∈Jk

Put s =

P∞

i=1

χJi [ coordinate sum ] so s ∈ m0 . ∞ ∞ X X X tki − si tki ≥ i=mk

i=nk +1

i∈Jk

so (iv) fails to hold.

Then

k t i >

Theorem 4.26. δ1 (X, X 0 ) is the strongest polar topology with the same subseries convergent series as σ(X, X 0 ). Proof: From Theorem 4.22, δ1 (X, X 0 ) and σ(X, X 0 ) have the same subseries convergent series. Suppose α is a polar topology with the same subseries convergent series as σ(X, X 0 ) and let α be the topology of uniform convergence on the family A of σ(X 0 , X) bounded subsets of X 0 (Appendix A). Let A ∈ A. We show that A ∈ M [with respect to the duality between X and X 0 ]. Let T : (X 0 , σ(X 0 , X))) → (l1 , σ(l1 , m0 )) be linear and continuous. Then T 0 : (m0 , σ(m0 , l1 )) → (X, σ(X, X 0 )) P is linear and continuous. Now j ej is m0 multiplier convergent with reP spect to σ(m0 , l1 ) so j T 0 ej is m0 multiplier convergent with respect to σ(X, X 0 ) and, therefore, with respect to α. Thus, if s ∈ m0 , lim n

0

∞ X

j=n

0

0 j

sj x , T e

= lim n

∞ X

j=n

s j T x0 , e j = 0

uniformly for x ∈ A. From Proposition 4.25, T A is relatively k·k1 compact so A ∈ M. Thus, A ⊂ M and α is weaker than δ1 (X, X 0 ). As we have seen the Dierolf topology is the strongest polar topology which has the same subseries convergent series as the weak topology. Tweddle has shown that there is a strongest locally convex topology which has

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the same subseries convergent series as the weak topology ([Tw]). We now give a description of the Tweddle topology. Let X be an LCTVS. Let O be P the set of all X valued series j xj which are σ(X, X 0 ) subseries converP∞ P gent. If j xj ∈ O, we write j=1 xj for the σ(X, X 0 ) sum of this series. Let X # be the space of all linear functionals x0 on X such that + * ∞ ∞ X X X 0 0 xj ∈ O. for all xj hx , xj i = x , j=1

j=1

j

The Mackey topology, τ (X, X # ), is the Tweddle topology on X and is denoted by t(X, X 0 ). We have the following important property of the Tweddle topology. Theorem 4.27. The Tweddle topology t(X, X 0 ) is the strongest locally convex topology on X which has the same subseries convergent series as the weak topology σ(X, X 0 ). Proof: Suppose that ν is a locally convex topology on X which has the 0 0 same subseries convergent series as σ(X, X 0 ). Let E . Then for D H P= (X, ν) P P ∞ ∞ 0 # 0 0 0 0 j xj ∈ O and x ∈ H , we have j=1 hx , xj i = x , j=1 xj so x ∈ X

and H 0 ⊂ X # . Therefore, τ (X, H 0 ) is weaker than τ (X, X # ) = t(X, X 0 ). But, ν ⊂ τ (X, H 0 ) so ν is weaker than t(X, X 0 ). We give an example where the Tweddle topology is strictly stronger than the Mackey topology by computing the space X # and comparing it to X 0 . The example uses the Nikodym Boundedness Theorem for countably additive set functions which we prove later in Theorem 4.60 (see also [Sw3] 2.8.8 for a statement).

Example 4.28. Let Σ be a σ-algebra of subsets of a set S. Let B(S, Σ) be the space of all bounded, real valued Σ-measurable functions defined on S. Let ca(Σ) be the space of all real valued, countably additive set functions defined on Σ and let Γ = span{δt : t ∈ S}, where δt is the Dirac measure concentrated at t. The weak topology σ(B(S, Σ), Γ) is just the topology of pointwise convergence, p, on B(S, Σ) so (B(S, Σ), p)0 = Γ. We show that B(S, Σ)# (with respect to p) is ca(Σ) so t(B(S, Σ), Γ) = τ (B(S, Σ), ca(Σ)) is strictly stronger than τ (B(S, Σ), Γ). First, suppose f ∈ B(S, Σ)# . Then f induces a set function on Σ, still denoted by f , defined by f (E) = hf, χE i for E ∈ Σ. We claim that P f ∈ ca(Σ). Let {Ej } ⊂ Σ be pairwise disjoint. Then j χEj is subseries

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P

χEj converges to χ∪∞ j=1 Ej

convergent with respect to p and the series with respect to p. Therefore, D

f, χ∪∞ j=1 Ej

E

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= f (∪∞ j=1 Ej ) =

∞ X

j=1

j

∞ X f (Ej ), f, χEj = j=1

#

and f is countably additive. Thus, B(S, Σ) ⊂ ca(Σ). P Next, let ν ∈ ca(Σ) and let j gj be subseries convergent with respect P to p. We claim that { j∈σ gj : σ finite} is bounded with respect to the sup-norm, k·k∞ , on B(S, Σ). If this is not the case, for every k there exist finite σk and tk ∈ S such that X gj (tk ) > k. (∗) j∈σk P Since j gj is subseries convergent in B(S, Σ) with respect to p, the P ∞ ∞ series with respect to the j {gj (tk )}k=1 is subseries convergent in l ∞ topology of coordinatewise convergence in l . Define µk : 2N → R by P µk (σ) = j∈σ gj (tk ). Note that µk ∈ ca(2N ) and if σ ⊂ N, X sup |µk (σ)| = sup ( gj )(tk ) < ∞. k j∈σ By the Nikodym Boundedness Theorem (Theorem 4.60 and or [Sw3] 2.8.8), X gj )(tk ) < ∞. sup sup |µk (σ)| = sup sup ( σ σ k k j∈σ But, this contradicts (∗). P If {nk } is a subsequence, then { kj=1 gnj } is uniformly bounded on S by the claim established above so by the Bounded Convergence Theorem, Z X k Z ∞ X gnj dν = gnj dν. lim k

j=1

S

S j=1

Therefore, ν ∈ B(S, Σ)# and B(S, Σ)# = ca(Σ). There is an analogous Dierolf topology for bounded multiplier convergent series which we will now describe. The proofs of the various properties of the Dierolf topology for bounded multiplier convergent series are almost identical to the proofs for the Dierolf topology for the subseries case so we will give the appropriate statements but omit the proofs.

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Let X, X 0 be in duality. Let N = N (X, X 0 ) be the family of all σ(X 0 , X) bounded subsets N ⊂ X 0 such that for every continuous linear operator T : (X 0 , σ(X 0 , X)) → (l1 , σ(l1 , l∞ )) T N is relatively compact in (l 1 , k·k1 ) [again note Proposition 4.25]. Definition 4.29. The Dierolf topology for bounded multiplier convergent series, δ2 (X, X 0 ), on X is the polar topology, τN , of uniform convergence on the elements of N = N (X, X 0 ) [Appendix A]. We now state the analogues of Theorems 4.20, 4.21, 4.22 and 4.23 for the Dierolf topology δ2 (X, X 0 ). Theorem 4.30. The topology δ2 (X, X 0 ) is a Hellinger-Toeplitz topology. Theorem 4.31. For t ∈ l ∞ , δ2 (l∞ , l1 ) − lim n

n X

tj e j = t

j=1

uniformly for ktk∞ ≤ 1. In particular, (l ∞ , δ2 (l∞ , l1 )) is an AK-space. P Theorem 4.32. Let X be an LCTVS. If the series j xj is bounded mulP tiplier convergent in the weak topology σ(X, X 0 ), then the series j xj is bounded multiplier convergent in the Dierolf topology δ2 (X, X 0 ). P Theorem 4.33. Let X be an LCTVS. If the series j xj is bounded multiplier convergent in the weak topology σ(X, X 0 ), then δ2 (X, X 0 ) − Pn limn j=1 tj ej = t uniformly for ktk∞ ≤ 1. The analogues of the statements in Remark 4.24 also hold for the Dierolf topology δ2 (X, X 0 ).

Remark 4.34. δ2 (X, X 0 ) is stronger than λ(X, X 0 ), the topology of uniform convergence on the σ(X 0 , X) compact subsets of X 0 . Thus, Theorem 4.32 gives an improvement to the statement in Corollary 4.14 for the topology λ(X, X 0 ). Remark 4.35. δ2 (X, X 0 ) is stronger than γ(X, X 0 ), the topology of uniform convergence on the σ(X 0 , X) bounded sets which are conditionally σ(X 0 , X) sequentially compact. Thus, Theorem 4.32 gives an improvement to the statement in Corollary 4.14 for the topology γ(X, X 0 ).

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Finally, the analogue of Theorem 4.26 holds for the Dierolf topology δ2 (X, X 0 ). Theorem 4.36. The Dierolf topology δ2 (X, X 0 ) is the strongest polar topology on an LCTVS X with the same bounded multiplier convergent series as the weak topology σ(X, X 0 ). The proof of this theorem proceeds as the proof of Theorem 4.26 except that the statement in Proposition 4.25 (iii) is used in place of the statement in Proposition 4.25 (iv). Using the Dierolf topologies for m0 and l∞ multiplier convergent series as models, we show that an analogous topology can be defined for arbitrary sequence spaces of multipliers, and when the multiplier space λ satisfies the signed-WGHP, we compare the general Dierolf topology to the topologies λ(X, X 0 ) and γ(X, X 0 ). Definition 4.37. Let λ be a sequence space containing c00 . A subset P K ⊂ λβ has uniform tails if for every s ∈ λ, limn ∞ j=n sj tj = 0 uniformly for t ∈ K. From Proposition 4.25, if λ = m0 or λ = l∞ , a subset K ⊂ l1 = λβ has uniform tails iff K is relatively k·k1 compact. Let X be an LCTVS. We say that a σ(X 0 , X) bounded subset D ⊂ X 0 belongs to Dλ iff for every continuous linear operator T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ))

the subset T D ⊂ λβ has uniform tails.

Definition 4.38. The general Dierolf topology on X, denoted by Dλ (X, X 0 ), is the polar topology τDλ of uniform convergence on the elements of Dλ . From Proposition 4.25 it follows that δ1 (X, X 0 ) = Dm0 (X, X 0 ) and δ2 (X, X 0 ) = Dl∞ (X, X 0 ) so it is reasonable to refer to Dλ (X, X 0 ) as a Dierolf topology with respect to λ. In order to establish the basic property of the general Dierolf topology we prove the following result. Proposition 4.39. There is a 1-1 correspondence between σ(X, X 0 ) λ mulP tiplier convergent series j xj and continuous linear operators T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ)).

The correspondence is given by xj = T 0 ej .

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P Proof: Suppose that j xj is λ multiplier convergent with respect to P∞ σ(X, X 0 ). Define the summing operator S : λ → X by St = j=1 tj xj [σ(X, X 0 ) sum]. Then S is σ(λ, λβ ) − σ(X, X 0 ) continuous by Theorem 2.2. Therefore, T = S 0 : X 0 → λβ is σ(X 0 , X) − σ(λβ , λ) continuous and T 0 x0 = {hx0 , xj i}. If T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ))

is linear and continuous, then S = T 0 : λ → X is σ(λ, λβ ) − σ(X, X 0 ) P∞ P continuous. Now j ej is σ(λ, λβ ) λ multiplier convergent so j=1 T 0 ej = P∞ P ∞ j 0 j=1 Se = j=1 xj is σ(X, X ) λ multiplier convergent and the correspondence follows. Remark 4.40. Thus, it follows from Proposition 4.39, to check that a subset D ⊂ X 0 belongs to Dλ , it suffices to show that if t ∈ λ, then P∞ P limn j=n tj hx0 , xj i = 0 uniformly for x0 ∈ D whenever j xj is λ multiplier convergent with respect to σ(X, X 0 ). Theorem 4.41. The general Dierolf topology Dλ (X, X 0 ) is the strongest polar topology on X with the same λ multiplier convergent series as σ(X, X 0 ). P Proof: Let j xj be λ multiplier convergent with respect to σ(X, X 0 ). P Let S : λ → X be the summing operator with respect to j xj , St = P∞ 0 β 0 j=1 tj xj [σ(X, X ) sum]. By Theorem 2.2, S is σ(λ, λ ) − σ(X, X ) con0 0 0 β β tinuous and S = T : (X , σ(X , X)) → (λ , σ(λ , λ)) is linear and continuous. Let D ∈ Dλ . Then T D has uniform tails. Therefore, for every s ∈ λ, P∞ limn j=n sj hx0 , xj i = 0 uniformly for x0 ∈ D since T x0 = {hx0 , xj i}. That P is, the series j sj xj converges in Dλ (X, X 0 ). Suppose that α is a polar topology with the same λ multiplier convergent series as σ(X, X 0 ). Let α be the polar topology of uniform convergence on P the family A of σ(X 0 , X) bounded sets (Appendix A). Let A ∈ A. If j xj P is λ multiplier convergent with respect to σ(X, X 0 ), then j xj is λ multiP∞ plier convergent with respect to α so if t ∈ λ, then limn j=n tj hx0 , xj i = 0 uniformly for x0 ∈ A. Then A ∈ Dλ by Remark 4.40 and α is weaker than Dλ (X, X 0 ). As in Theorem 4.20 we can show that Dλ (X, X 0 ) is a Hellinger-Toeplitz topology. Theorem 4.42. Dλ (X, X 0 ) is a Hellinger-Toeplitz topology.

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Proof: From Theorem A.2 of Appendix A, we need to show that if D ∈ Dλ with respect to the dual pair Y, Y 0 and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is linear and continuous, where X, X 0 is another dual pair, then T 0 D ∈ Dλ with respect to the dual pair X, X 0 . Let U : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ)) be linear and continuous. Then U T 0 : (Y 0 , σ(Y 0 , Y )) → (λβ , σ(λβ , λ)) is linear and continuous so U T 0 D has uniform tails in λβ so T 0 D ∈ Dλ with respect to the dual pair X, X 0 . From Theorems 4.5, 4.41 and 4.42, we have Theorem 4.43. (λ, Dλ (λ, λβ )) is an AK-space. We next compare the general Dierolf topology to the topologies λ(X, X 0 ) and γ(X, X 0 ). This will require an additional assumption on the multiplier space λ. We first establish a lemma. P Lemma 4.44. Suppose M ⊂ X 0 is such that there exist a series j xj 0 which is λ multiplier convergent with respect to σ(X, X ) and a t ∈ λ such P∞ that the series j=1 tj hx0 , xj i do not converge uniformly for x0 ∈ M . Then there exist > 0, {x0k } ⊂ M and an increasing sequence of intervals {Ik } with X 0 t hx , x i j j > k j∈Ik for every k.

Proof: If the series do not converge uniformly for x0 ∈ M , then there 0 0 exists > 0 such that for every k there exist mk > k, x = x (k) ∈ M such P∞ 0 that j=mk tj hx , xj i > 2. In particular, there exist m1 , x01 ∈ M with P P ∞ j=m1 tj hx01 , xj i > 2. Since the series j tj xj is σ(X, X 0 ) convergent, P 0 there exists n1 > m1 such that ∞ j=n1 tj hx1 , xj i < . Thus, if I1 = P [m1 , n1 ], then j∈I1 tj hx01 , xj i > . Now just continue the construction.

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We first compare the general Dierolf topology with the topology γ(X, X 0 ) of uniform convergence on σ(X 0 , X) bounded sets which are conditionally σ(X 0 , X) sequentially compact. Theorem 4.45. Let λ have signed-WGHP. Then γ(X, X 0 ) ⊂ Dλ (X, X 0 ). Proof: Suppose that K ⊂ X 0 is σ(X 0 , X) bounded and conditionally σ(X 0 , X) sequentially compact. If K does not belong to Dλ , then by Remark 4.40 and Lemma 4.44 there exist a multiplier convergent series P 0 0 j xj with respect to σ(X, X ), t ∈ λ, > 0, {xk } ⊂ K and an increasing sequence of intervals {Ik } such that X 0 (∗) tj hxk , xj i > j∈Ik

for every k. By the conditional σ(X 0 , X) sequential compactness of K, we may assume that lim hx0k , xi exists for every x ∈ X. Define a matrix X tl hx0i , xl i]. M = [mij ] = [ l∈Ij

We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by the compactness condition. Next, if {pj } is an increasing sequence there is a further subsequence {qj } of {pj } and a sequence of signs P∞ {sj } such that u = {uj } = j=1 sj χIqj t ∈ λ . Then + * ∞ ∞ X X sj miqj = x0i , u j xj j=1

j=1

P∞

so limi j=1 sj miqj exists. Thus, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗). We next consider the topology λ(X, X 0 ) of uniform convergence on σ(X 0 , X) compact subsets of X 0 . Theorem 4.46. Let λ have signed-WGHP. Then λ(X, X 0 ) ⊂ Dλ (X, X 0 ). Proof: Let K ⊂ X 0 be σ(X 0 , X) compact. Assume that K does not belong to Dλ and let the notation be as in the proof of Theorem 4.45. Let X0 = span{xk : k ∈ N}. The set {x0k : k ∈ N} is relatively σ(X00 , X0 ) compact and, therefore, relatively σ(X00 , X0 ) sequentially compact since X0 is separable ([Wi] 9.5.3). Therefore, we may assume that lim hx0k , xi

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exist for every x ∈ X. The proof can now be completed as in the proof of Theorem 4.45. Without the assumption on the multiplier space λ, the inclusions in Theorems 4.45 and 4.46 may fail to hold. Example 4.47. Let λ = cc , the space of sequences which are eventually constant (Appendix B). Then a series in a TVS is λ multiplier convergent iff the series is convergent. If one has a series which is σ(X, X 0 ) convergent but not τ (X, X 0 ) convergent, then the series is λ multiplier convergent with respect to σ(X, X 0 ) but not λ multiplier convergent with respect to τ (X, X 0 ). P [For example, take j (ej+1 − ej ) in c0 .] Thus, τ (X, X 0 ) is not contained in Dλ (X, X 0 ) by Theorem 4.41. Since both topologies λ(X, X 0 ) and γ(X, X 0 ) contain τ (X, X 0 ), this shows that the containments in Theorems 4.45 and 4.46 do not hold. Remark 4.48. Theorems 4.45 and 4.46 contain the results in Corollaries 4.11 and 4.14 as special cases. We now compare the general Dierolf topology with the strong topology when the multiplier space has the ∞-GHP. Recall that λ has ∞-GHP if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate sum of the P∞ series j=1 apj χIpj t ∈ λ (Appendix B; examples are given in Appendix B). Theorem 4.49. Let λ have ∞-GHP. Then β(X, X 0 ) ⊂ Dλ (X, X 0 ).

Proof: Let B ⊂ X 0 be σ(X 0 , X) bounded. Assume that B does not belong to Dλ . Let the notation be as in Theorem 4.45 so X 0 (∗) tj hxk , xj i > j∈Ik

for every k with x0k ∈ B. By the ∞-GHP there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such P that s = {sj } = ∞ k=1 aqk χIqk t ∈ λ. Define a matrix M = [mij ] = [

X

l∈Ij

tl apj hx0i /api , xl i].

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We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since x0i /api → 0 in σ(X 0 , X). Next, given a subsequence of {pj } let {qj } be a subsequence as above. Then + * ∞ ∞ ∞ X X X X 0 0 sl xl → 0, sl hxi /api , xl i = xi /api , miqj = j=1

j=1 l∈Iqj

P∞

l=1

0

where l=1 sl xl is the σ(X, X ) sum of the series. Thus, M is a K-matrix so the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). There is also an analogue of the Tweddle topology for λ multiplier convergent series when the multiplier space has signed-WGHP. If X is an LCTVS, let X # (= Xλ# ) be the space of all linear functionals x0 on X satisfying * + ∞ ∞ X X 0 0 tj hx , xj i = x , t j xj j=1

j=1

0

P for every t ∈ λ and every σ(X, X ) λ multiplier convergent series j xj , P∞ where j=1 tj xj is the σ(X, X 0 ) sum of the series. We define the Tweddle topology on X to be Dλ (X, X # ) and denote the topology by tλ (X, X 0 ). Thus, tm0 (X, X 0 ) = t(X, X 0 ). We show that tλ (X, X 0 ) is the strongest locally convex topology on X with the same λ multiplier convergent series as σ(X, X 0 ). Theorem 4.50. Let λ have signed-WGHP. Then tλ (X, X 0 ) is the strongest locally convex topology on X with the same λ multiplier convergent series as σ(X, X 0 ). P Proof: If j xj is λ multiplier convergent with respect to σ(X, X 0 ), then P with respect to σ(X, X # ) by the definition j xj is λ multiplier convergent P # of X . By Theorem 4.41, j xj is λ multiplier convergent with respect to tλ (X, X 0 ) = Dλ (X, X #). Suppose α is a locally convex topology with the same λ multiplier convergent series as σ(X, X 0 ). Put H 0 = (X, α)0 . Then H 0 ⊂ X # . By Theorem 4.41, α ⊂ τ (X, H 0 ) ⊂ Dλ (X, H 0 ) ⊂ Dλ (X, X # ) = tλ (X, X 0 ). We now present several applications of the Orlicz-Pettis theorems to various topics in functional analysis and measure theory. As a first application we present the original application by Pettis to vector valued measures.

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Theorem 4.51. Let Σ be a σ-algebra of subsets of a set S and let X be an LCTVS. If µ : Σ → X is such that x0 ◦ µ = x0 µ : Σ → R is countably additive for every x0 ∈ X 0 , then µ is countably additive with respect to the original topology of X. That is, if µ is countably additive with respect to the weak topology, then µ is countably additive with respect to the original topology. Proof: Let {Aj } ⊂ Σ be pairwise disjoint. If {Anj } isa subsequence, P∞ then A = ∪∞ Σ and hx0 , µ(A)i = j=1 x0 , µ(Anj ) for every x0 ∈ j=1 Anj ∈ P X 0 . Thus, the series j µ(Aj ) is subseries convergent with respect to the weak topology σ(X, X 0 ). By the Orlicz-Pettis Theorem in Corollary 4.11, the series is subseries convergent with respect to the original topology of X. That is, µ is countably additive with respect to the original topology of X. For vector valued set functions defined on algebras, we have Theorem 4.52. Let A be an algebra of subsets of a set S and let X be a weakly sequentially complete LCTVS. If µ : A → X is such that x0 µ is countably additive for every x0 ∈ X 0 , then µ is countably additive with respect to the original topology of X. ∈ A. Then Proof: Let {Aj } ⊂ A be pairwise disjoint with A = ∪∞ j=1 AjP P∞ 0 0 0 0 hx , µ(A)i = j=1 hx , µ(Aj )i for every x ∈ X and the series j µ(Aj ) is unconditionally convergent with respect to σ(X, X 0 ) since the union ∪∞ j=1 Aj is independent of the ordering of the {Aj } [note that we cannot assert that not belong to A for the series is subseries convergent since ∪∞ j=1 Anj mayP arbitrary subsequences]. By Corollary 2.60 the series j µ(Aj ) is subseries convergent with respect to σ(X, X 0 ). By the Orlicz-Pettis Theorem in P Corollary 4.11, the series j µ(Aj ) is subseries convergent with respect to the original topology of X. Thus, µ is countably additive with respect to the original topology of X. As an application of Theorem 4.52 and the Orlicz-Pettis Theorem, we have Theorem 4.53. Let X be a weakly sequentially complete LCTVS. Then X contains no subspace isomorphic to c0 . P Proof: Let j xj be c0 multiplier convergent in X. By Proposition 3.8, P∞ 0 ∞ 0 0 j=1 tj hx , xj i converges for every t ∈ l , x ∈ X . Hence, the partial

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P sums of the series j tj xj are σ(X, X 0 ) Cauchy and, therefore, σ(X, X 0 ) P ∞ convergent for every t ∈ l ∞ . That is, the series multiplier j xj is l convergent in the weak topology. By the Orlicz-Pettis Theorem in Corollary P 4.14 the series j xj is l∞ multiplier convergent in the original topology of X. Thus, condition (iv) of Theorem 3.15 holds. The proofs that (iv) implies (viii) implies (i) in Theorem 3.15 do not employ the sequential completeness of X so the result follows from these proofs. Corollary 4.54. If X is a semi-reflexive space, then X contains no subspace isomorphic to c0 . Proof: (X, σ(X, X 0 )) is boundedly complete and, therefore, sequentially complete ([Wi] 10.2.4) so the result follows from Theorem 4.53. Corollary 4.55. If X is a barrelled LCTVS, then (X 0 , σ(X 0 , X)) contains no subspace isomorphic to c0 . Proof: (X 0 , σ(X 0 , X)) is sequentially complete ([Wi] 9.3.8). But, (X 0 , σ(X 0 , X))0 = X so (X 0 , σ(X 0 , X)) is weakly sequentially complete. As another application of the Orlicz-Pettis Theorem, we derive a result similar in spirit to Theorem 3.2 but with different hypothesis ([KG] 3.10.5). Theorem 4.56. Let X be weakly sequentially complete and let λ be monoP tone. Then a series j xj in X is λ multiplier convergent in X iff 0 β {hx , xj i} ∈ λ for every x0 ∈ X 0 . P Proof: If j xj is λ multiplier convergent, then {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 by Proposition 3.1. Conversely, suppose that {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Let t = {tj } ∈ λ. Since λ is monotone, the partial sums of any subseries P of j tj xj is σ(X, X 0 ) Cauchy and, therefore, σ(X, X 0 ) convergent by hyP pothesis. That is, the series j tj xj is σ(X, X 0 ) subseries convergent and, therefore, convergent in X by the Orlicz-Pettis Theorem in Corollary 4.11. Note that the assumption that λ is monotone in Theorem 4.56 is an algebraic condition whereas the assumptions in Theorem 3.2 are topological. Theorem 4.56, however, has the weak sequential completeness assumption. We can also use the Orlicz-Pettis Theorem to derive a result of Pelczynski on weakly compact and unconditionally converging operators (recall

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Theorem 3.18). A continuous linear operator T between normed spaces X and Y is weakly compact if T carries bounded sets into relatively weakly compact sets and T is unconditionally converging if T carries wuc series into subseries convergent series. Theorem 4.57. ([Pl]) If T : X → Y is a weakly compact operator from the normed space X into the normed space Y , then T is unconditionally converging. P P Proof: Let j xj be wuc. Then { j∈σ xj : σ f inite} is bounded by P Proposition 3.8. Thus, { j∈σ T xj : σ f inite} is relatively σ(Y, Y 0 ) comP pact. By Theorem 2.48, the series j T xj is σ(Y, Y 0 ) subseries convergent. P By the Orlicz-Pettis Theorem in Corollary 4.11, the series j T xj is subseries convergent in Y . The identity operator on l 1 shows that the converse of Theorem 4.57 is false, in general; however, there are spaces for which the converse does hold (see [Pl]). As another application of the Orlicz-Pettis Theorem, we derive a version of the Nikodym Boundedness Theorem for countably additive set functions. Dunford and Schwartz refer to the Nikodym Boundedness Theorem as a ”striking improvement of the principle of uniform boundedness” ([DS] p. 309). The theorem states that a family of countably additive signed measures defined on a σ-algebra which is pointwise bounded on the σ-algebra is uniformly bounded on the entire σ-algebra. For the proof of the theorem, we first establish a result which is central to most of the proofs of the theorem (see, however, Dunford and Schwartz where there is a proof based on the Baire Category Theorem ([DS] IV.9.8)). Let A be an algebra of subsets of a set S, and let ba(A) be the space of all real valued, bounded, finitely additive set functions defined on A. Lemma 4.58. Let M ⊂ ba(A) be pointwise bounded on A. Then sup{|µ(A)| : µ ∈ M, A ∈ A} < ∞ iff sup{|µi (Ai )| : i ∈ N} < ∞ for every pairwise disjoint sequence {Ai } from A and every sequence {µi } ⊂ M. Proof: Suppose sup{|µ(A)| : µ ∈ M, A ∈ A} = ∞ . Note that for each r > 0 there exist a partition (E, F ) of S with E, F ∈ A and µ ∈ M such that min{|µ(E)| , |µ(F )|} > r. [ This follows since |µ(E)| > r + sup{|ν(S)| : ν ∈ M } implies |µ(S \ E)| ≥ |µ(E)| − |µ(S)| > r. ] Hence, there exist µ1 ∈ M and a partition (E1 , F1 ) of S such that min{{|µ1 (E1 )| , |µ1 (F1 )|} > 1.

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Now either sup{|µ(E1 ∩ A)| : µ ∈ M, A ∈ A} = ∞ or sup{|µ(F1 ∩ A)| : µ ∈ M, A ∈ A} = ∞. Pick whichever of E1 or F1 satisfies this condition and label it B1 and set A1 = S \ B1 . Now treat B1 as S above to obtain a partition (A2 , B2 ) of B1 and µ2 ∈ M such that |µ2 (A2 )| > 2 and sup{|µ(B2 ∩ A)| : µ ∈ M, A ∈ A} = ∞. Continuing this construction produces a sequence {µi } ⊂ M and a pairwise disjoint sequence {Ai } from A such that |µi (Ai )| > i. This establishes the sufficiency; the necessity is clear. Let Σ be a σ-algebra of subsets of a set S. Let S(Σ) be the vector space of all real valued Σ-simple functions and let ca(Σ) be the space of all countably additive signed measures µ : Σ → R.R Then S(Σ), ca(Σ) form a dual pair via the integration pairing hµ, f i = f dµ, f ∈ S(Σ), µ ∈ca(Σ). We now give our proof of the Nikodym Boundedness Theorem. Theorem 4.59. Let M ⊂ ca(Σ) be such that sup{|µ(E)| : µ ∈ M } < ∞ for every E ∈ Σ. Then sup{|µ(E)| : µ ∈ M, E ∈ Σ} < ∞. Proof: By Lemma 4.58 it suffices to show that sup{|µi (Ai )| : i ∈ N} < ∞ for every {µi } ⊂ M and pairwise disjoint sequence {Ai } from Σ or that P (1/i)µi (Ai ) → 0. The series i χAi is σ(S(Σ),ca(Σ)) subseries convergent by the countable additivity of the members of ca(Σ). By the version of the P Orlicz-Pettis Theorem for the topology λ(S(Σ),ca(Σ)) the series i χ Ai P∞ converges in λ(S(Σ), ca(Σ)). In particular, limn j=n (1/k)µk (Aj ) = 0 uniformly for k ∈ N since {(1/k)µk } is σ(ca(Σ), S(Σ)) convergent to 0 by the pointwise boundedness assumption. In particular, (1/i)µi (Ai ) → 0 as desired. It should be noted that a version of the theorem for countably additive set functions defined on σ-algebras with values in an LCTVS follows immediately from Theorem 4.59. Corollary 4.60. Let X be an LCTVS and let M be an family of countably additive set functions defined on Σ with values in X. If {µ(E) : µ ∈ M } is bounded in X for every E ∈ Σ, then {µ(E) : µ ∈ M, E ∈ Σ} is bounded.

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Proof: Let x0 ∈ X 0 . Then the subset {x0 µ : µ ∈ M } ⊂ ca(Σ) is pointwise bounded on Σ so the set {x0 µ : µ ∈ M } is uniformly bounded on Σ by Theorem 4.59. That is, the set {x0 µ : µ ∈ M, E ∈ Σ} is σ(X, X 0 ) bounded and, therefore, bounded in X. A few remarks pertaining to the Nikodym Boundedness Theorem are in order. First, the local convex assumption in Corollary 4.60 is important. Turpin has given an example of a countably additive set function defined on a σ-algebra with values in a (non-locally convex) TVS which is unbounded ([Rol]). Theorem 4.59 actually holds for bounded, finitely additive set functions defined on σ-algebras; we will give a proof of this version of the Nikodym Boundedness Theorem in Chapter 7 based on the Hahn-Schur Theorem. As the following example shows, the conclusion of Theorem 4.59 is false for set functions defined on algebras. Example 4.61. Let A be the algebra of finite/co-finite subsets of N. Let δn be the Dirac measure concentrated at n; δn (E) = 1 if n ∈ E and δn (E) = 0 if n ∈ / E. Define µn (E) = n(δn+1 (E) − δn (E)) if E is finite and µn (E) = −n(δn+1 (E) − δn (E)) if E c = N \ E is finite. Then {µn } is pointwise bounded on A but not uniformly bounded on A [µn ({n}) = n]. Despite Example 4.61, there are algebras for which the conclusion of Theorem 4.59 holds. The treatise by Schachermeyer contains examples, references and other discussions concerning the Nikodym Boundedness Theorem ([Sm]); see also Diestel and Uhl ([DU]). Finally, we present several versions and applications of the Orlicz-Pettis Theorem in an abstract setting. Let E, F be vector spaces such that there is a bilinear mapping from · : E × F → X, (x, y) → x · y, x ∈ E, y ∈ F , where X is an LCTVS. Of course, an example of this situation is when E, F are two vector spaces in duality ; we give other examples in the applications which follow. Let w(E, F ) [w(F, E)] be the weakest topology on E [F ] such that the linear maps x → x · y [y → x · y] from E into X [F into X] are continuous for all y ∈ F [x ∈ E]. If E, F are 2 vector spaces in duality, then w(E, F ) [w(F, E)] is just the weak topology σ(E, F ) [σ(F, E)]. A subset K ⊂ F is said to be conditionally w(F, E) sequentially compact if for every sequence {yj } ⊂ K, there is a subsequence {ynj } such that limj x·ynj exists for every x ∈ E. In this setting we have the analogue of Corollary 4.10 for the topology γ(X, X 0 ). Again, if E, F are 2 vector spaces in duality, this agrees with previous terminology.

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We establish a version of Corollary 4.10 for the topology γ(E, F ) in this setting. P Theorem 4.62. Let λ have signed-WGHP. If the series j xj is λ multiplier convergent in E with respect to w(E, F ), then for each t ∈ λ and each conditionally w(F, E) sequentially compact subset K ⊂ F , the series P∞ j=1 tj xj · y converge uniformly for y ∈ K.

Proof: If the conclusion fails to hold, there exists a neighborhood of 0, W , in X, yk ∈ K and an increasing X sequence of intervals {Ik } such that (#) t l xl · y k ∈ /W l∈Ik

for every k. We may assume, by passing to a subsequence if necessary, that limk x · yk exists for every x ∈ E. Consider X the matrix tl xl · yi ]. M = [mij ] = [ l∈Ij

We claim that M is a signed K-matrix (Appendix D). First, the columns of M converge. Next, given an increasing sequence of positive integers there is a subsequence {nj } and a sequence of signs {sj } such that u = P∞ j χInj t ∈ λ. Then j=1 s (∞ ) ∞ ∞ X X X X = t l xl · y i u l xl · y i = sj sj minj j=1

i

j=1

l∈Inj

i

l=1

i

converges. Hence, M is a signed K-matrix so the diagonal of M converges to 0 by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3). But, this contradicts (#). We now derive an analogue of Corollary 4.10 for λ multiplier convergent series and the topology λ(X, X 0 ). P Theorem 4.63. Let λ have signed-WGHP. If the series j xj is λ multiplier convergent in E with respect to w(E, F ), then for each w(F, E) compact (countably compact) subset K ⊂ F and each t ∈ λ, the series P∞ j=1 tj xj · y are convergent uniformly for y ∈ K.

Proof: Let p be a continuous semi-norm on X. We need to show that P the series j tj xj · y converge uniformly for y ∈ K with respect to p. This will follow if we can show that this property holds in the quotient space X/p. Hence, we may assume that p is actually a norm. Define an equivalence relation ∼ on F by y ∼ z iff xj · y = xj · z for all j. If

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P P∞ E0 = { ∞ j=1 sj xj : s ∈ λ, where j=1 sj xj is the w(E, F ) sum of the series}, then x · y = x · z for every x ∈ E0 when y ∼ z. Let y − be the equivalence class of y ∈ F and set F − = {f − : f ∈ F }. Define a metric d on F − by d(y − , z − ) =

∞ X j=1

p(xj · (y − z))/2j (1 + p(xj · (y − z)));

note that d is a metric since p is a norm. Define a bilinear mapping · : E0 × F − → (X, p) by x · y − = x · y so we may consider the triple E0 , F − , (X, p) as above. The quotient map F → F − is w(F, E)−w(F − , E0 ) continuous and the inclusion (F − , w(F − , E0 )) ⊂ (F − , d) is continuous so K − is compact (countably compact) with respect to w(F − , E0 ) and d and, therefore, w(F − , E0 ) = d on K − and K − is w(F − , E0 ) sequentially compact. Since the series P P j xj j xj is λ multiplier convergent with respect to w(E, F ), the series is λ multiplier convergent with respect to w(E0 , F − ) in the abstract triple E0, F − , (X, p). Since K − is sequentially compact in w(F − , E0 ), by Theorem P∞ P∞ 4.62 the series j=1 tj xj · y − = j=1 tj xj · y converge uniformly for y − ∈ K − with respect to p. Theorems 4.62 and 4.63 have as immediate corollaries the results in Corollary 4.10. We now present another corollary related to Theorem 4.9. Corollary 4.64. Let λ have signed-WGHP. Then (λ, γ(λ, λβ )) and (λ, λ(λ, λβ )) are AK-spaces. P j Proof: The series j e is λ multiplier convergent with respect to β σ(λ, λ ) and, therefore, is λ multiplier convergent with respect to γ(λ, λβ ) and λ(λ, λβ ) by Theorems 4.62 and 4.63. The result is now immediate. Note that the results above were derived in the other order previously. We now give several applications of Theorems 4.62 and 4.63. Example 4.65. Let Σ be a σ-algebra of subsets of a set S and let ca(Σ, X) be the space of all X valued countably additive R set functions from Σ into X. If E = S(Σ) and F = ca(Σ, X), then f · µ = S f dµ, f ∈ E, µ ∈ F , defines a bilinear map from E × F into X (note that we are only integrating simple functions so no elaborate integration theory is involved). If {Ej } ⊂ Σ P is pairwise disjoint, then the series j χEj is w(S(Σ), ca(Σ, X)) subseries

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P convergent. By Theorem 4.62 above, the series ∞ j=1 µ(Ej ) converge uniformly for µ belonging to any conditionally w(ca(Σ, X), S(Σ)) sequentially compact subset of ca(Σ, X). In particular, we have as a special case the Nikodym Convergence Theorem. Theorem 4.66. Let {µj } ⊂ ca(Σ, X) be such that limj µj (E) = µ(E) exists for every E ∈ Σ. Then {µj } is uniformly countably additive and µ ∈ ca(Σ, X). Proof: By the observation above, since {µj } is conditionally w(ca(Σ, X), S(Σ)) sequentially compact, {µj } is uniformly countably additive. That µ ∈ ca(Σ, X) then follows. From Theorem 4.63, we can also derive a result of Graves and Ruess ([GR]) Lemma 6). Theorem 4.67. If K ⊂ ca(Σ, X) is w(ca(Σ, X), S(Σ)) compact, then K is uniformly countably additive. Next, we derive a version of a theorem of Thomas ([Th]). Example 4.68. Let S be a sequentially compact Hausdorff space. Let E = SC(S, X) be the space of sequentially continuous functions from S into X and let F = span{δt : t ∈ S}, where δt is the Dirac measure concentrated at t. Then f · t = f (t) defines a bilinear mapping from E × F into X. Note that S is conditionally w(span{δt : t ∈ S}, SC(S, X)) sequentially compact since S is sequentially compact [here we are identifying t with δt ]. Thus, P from Theorem 4.62 above if λ has signed-WGHP and j fj is λ multiplier convergent in SC(S, X)with respect the topology of pointwise convergence P on S, then for each t ∈ λ the series j tj fj converges uniformly on S. Similarly, if S is compact, then S is w(span{δt : t ∈ S}, C(S, X)) compact P so from Theorem 4.63 if λ has signed-WGHP and the series j fj is λ multiplier convergent in C(S, X) with respect to the topology of pointwise P convergence on S, then for each t ∈ λ the series j tj fj converges uniformly on S. The subseries version of this result is due to Thomas ([Th]). We can also use Theorems 4.62 and 4.63 above to derive a version of the Orlicz-Pettis Theorem for continuous linear operators. Example 4.69. Let Z be an LCTVS. Set E = L(Z, X) and F = Z and define a bilinear mapping from E × F into X by T · x = T x. Then w(E, F ) is just the topology of pointwise convergence on Z or Ls (Z, X). If K ⊂ Z is sequentially compact (compact), then K is conditionally w(Z, L(Z, X))

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sequentially compact (w(Z, L(Z, X)) compact) so if K (C) denotes the set of all sequentially compact (compact) subsets of Z, from Theorem 4.62 (Theorem 4.63) above, we have P Theorem 4.70. Let λ have signed-WGHP. If j Tj is λ multiplier conP vergent in Ls (Z, X), then j Tj is λ multiplier convergent in LK (Z, X) (LC (Z, X)). We will obtain some similar results for operator valued series later in Chapter 6. An operator T ∈ L(Z, X) is completely continuous if T carries weakly convergent sequences into convergent sequences; denote all such operators by CC(Z, X). Note that if T is completely continuous, then T carries weak Cauchy sequences into Cauchy sequences. Now consider the abstract triple E = CC(Z, X), F = Z and the bilinear map · : E × F → X defined by · : (T, z) → T · z = T z. If a subset K ⊂ Z is conditionally weakly sequentially compact, then K is conditionally w(CC(Z, X), Z) sequentially compact. If CW denotes the set of all conditionally weakly sequentially compact subsets of Z, then from Theorem 4.62 we have P Theorem 4.71. Let λ have signed-WGHP. If the series j Tj is λ mulP tiplier convergent in CCs (Z, X), then j Tj is λ multiplier convergent in CCCW (Z, X). An operator T ∈ L(Z, X) is weakly compact if T carries bounded sets to relatively weakly compact sets; denote all such operators by W (Z, X). The space Z has the Dunford-Pettis property if every weakly compact operator from Z into any locally convex space X carries weak Cauchy sequences into convergent sequences. Consider the abstract triple E = W (Z, X), F = Z and the bilinear map · : E × F → X defined by · : (T, z) → T · z = T z. If K ⊂ Z is conditionally weakly sequentially compact and Z has the Dunford-Pettis property, then K is conditionally w(W (Z, X), Z) sequentially compact. If CW denotes the set of all conditionally weakly compact subsets of Z, then from Theorem 4.62 we have Theorem 4.72. Let λ have signed-WGHP and assume that Z has the P Dunford-Pettis property. If the series j Tj is λ multiplier convergent in P Ws (Z, X), then j Tj is λ multiplier convergent in WCW (Z, X).

A space Z is almost reflexive if every bounded sequence contains a weak Cauchy subsequence ([LW]). For example, Banach spaces with separable

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duals, quasi-reflexive Banach spaces and c0 (S) are almost reflexive ([LW]). If Z is almost reflexive and has the Dunford-Pettis property, then every bounded set is conditionally w(W (Z, X), Z) sequentially compact so from Theorem 4.62, we have Theorem 4.73. Let λ have signed-WGHP and assume that Z is almost P reflexive with the Dunford-Pettis property. If the series j Tj is λ mulP tiplier convergent in Ws (Z, X), then j Tj is λ multiplier convergent in Wb (Z, X). As another application of Theorem 4.63, we derive an Orlicz-Pettis result of Stiles for a locally convex TVS with a Schauder basis ([Sti]). Stiles’ version of the Orlicz-Pettis Theorem is for subseries convergent series with values in an F-space with a Schauder basis and his proof uses the metric properties of the space. Other proofs of Stiles’ result have been given in [Bs] and [Sw5]. We will establish a version of Stiles’ result for multiplier convergent series which requires no metrizability assumptions. Later in Chapter 9 we will establish a version of the result for non-locally convex spaces using the Antosik Interchange Theorem. Let X be an LCTVS with a Schauder basis {bj } and associated coordinate functionals {fj }. That is, every x ∈ X has a unique series repreP sentation x = ∞ j=1 tj bj and fj : X → R is defined by hfj , xi = tj . We do not assume that the coordinate functionals are continuous although this is the case when X is an F-space ([Sw2] 10.1.13). Define Pi : X → X by Pi Pi x = j=1 hfj , xi bj . Let E = X, F = span{Pi : i ∈ N} and let the bilinear mapping from E × F into X be the extension to F of the mapping x · Pi = Pi x. Let G = span{fi : i ∈ N}. P Theorem 4.74. Let λ have signed-WGHP. If j xj is λ multiplier conP vergent with respect to σ(X, G), then j xj is λ multiplier convergent with respect to the original topology of X. Proof: Since a sequence in X is σ(X, G) convergent iff the sequence is P w(E, F ) convergent, the series j xj is λ multiplier convergent with respect to w(E, F ). Now {Pi : i ∈ N} is conditionally w(F, E) sequentially compact since P x → x for every x ∈ X. By Theorem 4.62, for every t ∈ λ the series P∞ i N. Let U be a closed neighborhood j=1 tj Pi xj converge uniformly for i ∈ P∞ P∞ of 0 in X. There exists N such that j=m tj Pi xj = Pi ( j=m tj xj ) ∈ U P∞ for m ≥ N, i ∈ N. Let i → ∞ gives j=m tj xj ∈ U for m ≥ N . Note that we did not use the continuity of the coordinate functionals in the proof so the topology of X may not even be comparable to σ(X, G).

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We next consider a more general situation than that encountered in Stiles’ result. Assume that there exists a sequence of linear operators Pi : P∞ X → X such that for each x ∈ X, x = i=1 Pi x [convergence in X]. When each Pi is continuous, {Pi } is called a Schauder decomposition ([LT]). If X has a Schauder basis {bi } with coordinate functionals {fi }, then Pi x = hfi , xi bi is an example of this situation. Let E = X, F = span{Pi : i ∈ N} and let the bilinear mapping from E × F into X be the extension of the map x · Pi = Pi x. Theorem 4.75. Let λ have signed-WGHP and assume that each Pi is P w(E, F ) − X continuous. If the series x is λ multiplier convergent P j j with respect to w(E, F ), then the series j xj is λ multiplier convergent in X with respect to the original topology. Pn Proof: Define Sn : X → X by Sn = i=1 Pi . Then {Sn : n ∈ N} is conditionally w(F, E) sequentially compact so by Theorem 4.62 for each P∞ t ∈ λ the series j=1 tj Sn xj converge uniformly for n ∈ N. Let U be a P∞ closed neighborhood of 0 in X. There exists N such that j=m tj Sn xj = P∞ P∞ Sn ( j=m tj xj ) ∈ U for m ≥ N, n ∈ N. Letting n → ∞ gives j=m tj xj ∈ U for m ≥ N . We give an example where the theorem above is applicable. Example 4.76. Let Y be an LCTVS and let X be a vector space of Y valued sequences containing the space of sequences which are eventually 0. Then X is an AK-space if the coordinate functionals fj : X → Y , fj ({xj }) = xj are continuous for every j and each x = {xj } has a repreP j j sentation x = ∞ j=1 e ⊗ xj [Appendix C; here e ⊗ x denotes the sequence with x in the j th coordinate and 0 in the other coordinates]. The space X has the property (I) if the injections x → ej ⊗ x are continuous from Y into X. If Pj : X → X is defined by Pj ({xj }) = ej ⊗ xj , then {Pj } is a Schauder decomposition for X. If X has property (I), then the topology of coordinatewise convergence is equal to w(E, F ) so the result above applies and if λ has signed-WGHP, then any series which is λ multiplier convergent in the topology of coordinatewise convergence converges in the topology of X. For examples where the result above applies let Y be a normed space. If 1≤ p < ∞, then l p (Y ) and c0 (Y ) are AK-spaces satisfying the conditions in the example above. We will consider non-locally convex versions of these results later in Chapter 9.

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Finally, we show that the result in Theorem 2.26 can be obtained from Theorem 4.62. P Theorem 4.77. Let λ have signed-WGHP. Assume that j xij is λ mulP∞ tiplier convergent for every i ∈ N and that lim i j=1 tj xij exists for every t ∈ λ with xj = limi xij for every j. Then for every t ∈ λ the series P∞ j=1 tj xij converge uniformly for i ∈ N.

Proof: For every i ∈ N define a linear map fi : λ → X by fi (t) = P∞ j=1 tj xij and set F = span{fi : i ∈ N}. Consider the abstract triple E = λ, F and X and let the bilinear mapping from E × F into X be the extension of the map (t, fi ) → t · fi = fi (t). We first claim that the P j convergent with respect to w(E, F ). For if series j e is λ multiplier P P∞ P∞ ∞ j j t ∈ λ, t e · f = j i j=1 j=1 tj fi (e ) = j=1 tj xij converges for every i. Now {fi } is conditionally w(F, E) sequentially compact since {t · fi } = P∞ { j=1 tj xij } converges for every t ∈ λ. Theorem 4.62 implies that the P∞ P∞ series j=1 tj fi (ej ) = j=1 tj xij converge uniformly for i ∈ N. Recall that Theorem 4.77 (Theorem 2.26) and the convergence result in Lemma 2.27 were used to derive Stuart’s completeness result in Theorem 2.28. We can also derive a version of Kalton’s Theorem on subseries convergence in the space of compact operators. Let X and Y be normed spaces and let K(X, Y ) be the space of all compact operators from X into Y (an operator T ∈ L(X, Y ) is compact if T carries bounded sets into relatively compact sets). The space X has the DF property if every weak* subseries convergent series in X 0 is k·k subseries convergent ([DF]; Diestel and Faires have shown that for B-spaces this is equivalent to X 0 containing no subspace isomorphic to l∞ ). Theorem 4.78. Let X and Y be normed spaces and let X have the DF P property. If the series j Tj is subseries convergent in the weak operator topology of K(X, Y ), then the series is subseries convergent in the norm topology of K(X, Y ).

Proof: Each Tj has separable range so we may assume that Y is separable by replacing Y with span ∪∞ j=1 Tj X. By Lemma A.6 of Appendix

0 A or

T → 0. Lemma 4.4, it suffices to show that kT k → 0 or, equivalently, j

j

Pick yj0 ∈ Y 0 , yj0 = 1, such that Tj0 ≤ Tj0 yj0 + 1/j. By the separability of Y there exists a subsequence {yn0 j } which is weak* convergent to some y 0 ∈ Y 0 ; for convenience assume that the sequence {yj0 } is weak* convergent to y 0 . Consider the abstract triple E = {T 0 : T ∈

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K(X, Y )}, F = Y 0 and (X 0 , k·k) with the bilinear map E × F → (X 0 , k·k) P defined by (T 0 , y 0 ) → T 0 · y 0 = T 0 y 0 . For each z 0 ∈ Y 0 , the series j Tj0 z 0 is weak* subseries convergent in X 0 and is, therefore, subseries convergent in P (X 0 , k·k) by the DF property. Hence, the series j Tj0 is w(E, F ) subseries convergent. The sequence {yj0 } is w(F, E) relatively sequentially compact since k·k − lim T 0 yj0 = T 0 y 0 for every T ∈ K(X, Y ) ([DS] VI.5.6). By P∞ Theorem 4.62 the series j=1 Tj0 yi0 converge uniformly for i ∈ N. In particular, Tj0 yj0 → 0 so Tj0 = kTj k → 0 as desired. Kalton’s Theorem will also be considered in Chapter 6. We next show that the conclusions in Theorems 4.62 and 4.63 can be strengthened if the multiplier space has signed-SGHP instead of signedWGHP. P Theorem 4.79. Let Λ ⊂ λ have signed-SGHP. If j xj is λ multiplier convergent with respect to w(E, F ), then for each conditionally w(F, E) sequentially compact (w(F, E) compact, w(F, E) countably compact) subset P∞ K ⊂ F and each bounded subset B ⊂ Λ, the series j=1 tj xj · y converge uniformly for y ∈ K, t ∈ B. Proof: If the conclusion fails to hold, there exist a neighborhood, W , in X ,yk ∈ K, tk ∈ B and an increasing sequence of intervals {Ik } such that X (#) tkl xl · yk ∈ /W l∈Ik

for every k. We may assume, by passing to a subsequence if necessary, that limk x · yk exists for every x ∈ E. Consider the matrix X j M = [mij ] = [ tl xl · yi ]. l∈Ij

We claim that M is a signed K matrix as in Theorem 4.62 (Appendix D.3). First, the columns of M converge. Next given an increasing sequence of positive integers, there exist a sequence of signs {sj } and a subsequence P nk {nj } such that u = ∞ ∈ Λ. Then k=1 sk χInk t {

∞ X j=1

sj minj }i = {

∞ X j=1

sj

X

l∈Inj

n

t l j xl · y i } i = {

∞ X l=1

u l xl · y i } i

converges. Hence, M is a signed K matrix so the diagonal of M converges to 0 by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3). But, this contradicts (#).

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The proof of the statements in parentheses follow as in the proof of Theorem 4.63. If the multiplier space Λ ⊂ λ has signed-SGHP, then the conclusion of Corollary 4.10 can be improved (see Theorem 2.16). Corollary 4.80. Let Λ have signed-SGHP and let E, F be in duality. If P the series j xj is λ multiplier convergent with respect to σ(E, F ), then P∞ the series j=1 tj xj converge uniformly for t belonging to bounded subsets of Λ with respect to both λ(E, F ) and γ(E, F ). Corollary 4.80 covers the case of subseries convergent series (Λ = {χ σ : σ ⊂ N} ⊂ m0 = λ) and bounded multiplier convergent series (Λ the unit ball of l∞ ). Using Theorem 4.79 we can also obtain an improved conclusion in Theorem 4.66. In particular, if {Ej } is a pairwise disjoint sequence from Σ, then P the series ∞ χ (j)µi (Ej ) converge uniformly for i ∈ N, E ∈ Σ. That is, Pj=1 E the series ∞ j=1 µi (Ej ) are uniformly unordered convergent for i ∈ N. Similarly, we can obtain an improvement to the statements in Example 4.68 if the multiplier space λ has signed-SGHP. If λ has signed-SGHP and P the series j fj is λ multiplier convergent in SC(S, X) (C(S, X)) with respect to the topology of pointwise convergence on S, then the series P∞ j=1 tj fj (s) converge uniformly for s ∈ S and t belonging to bounded subsets of λ (Theorem 4.79). We can also obtain a strengthened version of the result given in Theorem 4.77. P Theorem 4.81. Let λ have signed-SGHP. Assume that j xij is λ mulP∞ tiplier convergent for each i ∈ N and that limi j=1 tj xij exists for each P∞ t ∈ λ with xj = limi xij for every j. Then the series j=1 tj xij converge uniformly for t belonging to bounded subsets of λ and i ∈ N, and the series P j xj is λ multiplier convergent. The proof of Theorem 4.77 carries forward using Theorem 4.79 in place of Theorem 4.62.

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Chapter 5

Orlicz-Pettis Theorems for the Strong Topology

In this chapter we consider Orlicz-Pettis Theorems for the strong topology. As the following example shows, in general, an Orlicz-Pettis Theorem does not hold for the strong topology. Recall that if X, X 0 is a pair of vector spaces in duality, the strong topology β(X, X 0 ) is the polar topology of uniform convergence on the family of σ(X 0 , X) bounded subsets of X 0 (Appendix A, Example A.2). As before, throughout this chapter λ will denote a scalar sequence space which contains c00 , the space of sequences which are eventually 0. P Example 5.1. The series j ej is subseries convergent in l ∞ with respect to the weak topology σ(l ∞ , l1 ) but is not subseries convergent in the strong topology β(l∞ , l1 ) = k·k∞ . In order to obtain an Orlicz-Pettis Theorem for the strong topology, we will impose stronger conditions on the multiplier space λ. Before proceeding in this direction, we use the Nikodym Boundedness Theorem to show that although a weak subseries convergent series may fail to be subseries convergent in the strong topology, the partial sums of the series are strongly bounded. P Theorem 5.2. Let X be an LCTVS. If j xj is σ(X, X 0 ) subseries conP P vergent, then P = { j∈σ xj : σ ⊂ N} is β(X, X 0 ) bounded, where j∈σ xj is the σ(X, X 0 ) sum of the series. Proof: Let P be the power set of N and define µ : P → X by µ(σ) = 0 0 0 j∈σ xj [ σ(X, X ) sum of the series ]. Let B ⊂ X be σ(X , X) bounded. 0 0 The family M = {x µ : x ∈ B} is a family of scalar valued, signed measures which is pointwise bounded on P since B is σ(X 0 , X) bounded. By the

P

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Nikodym Boundedness Theorem [Theorem 4.59], M is uniformly bounded on P. Since the range of µ is P , P is β(X, X 0 ) bounded. A similar result for barrelled AB-spaces λ is given in Corollary 2.6. First, we recall an Orlicz-Pettis result relative to the strong topology which was established in Corollary 4.6. Corollary 5.3. Let λ have signed-WGHP. Let X, X 0 be in duality. If P (X, β(X, X 0 )) is separable, then any series j xj which is λ multiplier convergent with respect to the weak topology σ(X, X 0 ) is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Example 5.1, where X = l ∞ and X 0 = l1 , shows that the separability condition in Corollary 5.3 is important. We next establish an Orlicz-Pettis Theorem for the strong topology which requires strong topological assumptions on the multiplier space. Theorem 5.4. Assume that λ is a barrelled AK-space and X is an LCTVS. P multiplier convergent with respect to the weak topology If j xj is λ P σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Proof: By Proposition 2.5, λ0 = λβ so the original topology of λ is β(λ, λβ ) and λ is an AK-space with respect to β(λ, λβ ). Since the strong topology is a Hellinger-Toeplitz topology, the result follows from Theorem 4.5. Example 5.1 where λ = m0 shows that the AK assumption in Theorem 5.4 is important. Although the assumptions on the multiplier space in Theorem 5.4 are quite restrictive, the result covers a large number of multiplier spaces. Example 5.5. If λ is a Banach [Frechet] AK-space, Theorem 5.4 applies. For example, λ = c0 , lp (1 ≤ p < ∞), cs or bv0 are Banach AK-spaces. Likewise, if λ = s or if λ is a K¨ othe echelon space ([K1] 30.8), then λ is a Frechet AK-space. The spaces (l p , k·k1 ), 0 < p < 1, are barrelled AKspaces ([Be]) so Theorem 5.4 applies. Examples of barrelled subspaces of (l1 , k·k1 ) are given in [RS] and more examples of barrelled AK-spaces are given in [BK].

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Whereas the assumptions on the multiplier space in Theorem 5.4 are topological in nature, we now consider a gliding hump assumption which is purely algebraic. We recall a gliding hump property which will be employed (Appendix B.36) Definition 5.6. The space λ has the infinite gliding hump property (∞-GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate P sum ∞ j=1 apj χIpj t ∈ λ. The term ”infinite gliding hump” is used to suggest that the ”humps”, χIpj t, are multiplied by a sequence of scalars which converges to ∞; there are other gliding hump properties where the humps are multiplied by elements of classical sequence spaces [see Appendix B for the µ-gliding hump property]. Examples of multiplier spaces with ∞-GHP are given in Appendix B. For example, λ = l p , 0 < p < ∞, or λ = cs have ∞-GHP. We now establish an Orlicz-Pettis Theorem for spaces with ∞-GHP.

P Theorem 5.7. Let λ have ∞-GHP and let X be an LCTVS. If j xj is λ P multiplier convergent with respect to the weak topology σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Proof: If the conclusion fails to hold, there exist t ∈ λ, a σ(X 0 , X) bounded subset B ⊂ X 0 and > 0 such that for every k there exist P ∞ x0k ∈ B, mk > k such that j=mk hx0k , tj xj i > 2. For k = 1, let 0 x condition. There exists n1 > m1 such that 1 , m1 satisfy the previous P ∞ 0 j=n1 +1 hx1 , tj xj i < . Then

X X ∞ ∞ X n1 0 0 0 hx1 , tj xj i − hx1 , tj xj i > . hx1 , tj xj i = j=m1 j=m1 j=n1 +1

Continuing this construction produces {x0k } ⊂ B, increasing sequences {mk }, {nk } with mk < nk < mk+1 satisfying nk X (∗) tj hx0k , xj i > . j=mk

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Set Ik = [mk , nk ]. Since λ has ∞-GHP, there exists {pj }, apj > 0, apj → ∞ such that every subsequence of {pj } has a further subsequence {qj } such P∞ that the coordinate sum j=1 aqj χIqj t ∈ λ. Define matrix M = [mij ] = [

X

l∈Ipj

apj tl hx0i /api , xl i].

We claim that M is a K-matrix [Appendix D.2]. First, the columns of M converge to 0 since {x0i } is σ(X 0 , X) bounded and api → ∞. Next, given a subsequence there exists a further subsequence {qj } such that s = P∞ j=1 aqj χIqj t ∈ λ. Then + * ∞ ∞ ∞ X X X X 0 0 sl hxi /api , xl i = xi /aqi , sl xl → 0, miqj = j=1

j=1 l∈Iqj

l=1

P∞ where l=1 sl xl is the σ(X, X 0 ) sum of the series. Thus, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem [Appendix D.2], the diagonal of M converges to 0. But, this contradicts (∗) and establishes the result. 1/k

1/k

The spaces d = {t : supk |tk | < ∞} and δ = {t : lim |tk | = 0} furnish examples of spaces to which Theorem 5.7 applies but Theorem 5.4 does not [the natural metric on d does not give a vector topology ([KG] p.68)]. Diestel and Faires have established an interesting Orlicz-Pettis Theorem for the weak* topology on the dual of a Banach space. They have shown that if X is a Banach space such that X 0 contains no subspace isomorphic to P P l∞ and if j x0j is weak* subseries convergent, then j x0j is norm subseries convergent ([DU] I.4.7). We give statements of Theorems 5.4 and 5.7 for the weak* topology where the emphasis is on the multiplier space λ instead of topological assumptions on the dual space as in the Diestel-Faires Theorem. Corollary 5.8. Assume that λ either has ∞-GHP or is a barrelled AKP 0 xj is λ multiplier convergent in the space. Let X be an LCTVS. If jP weak* topology σ(X 0 , X) of X 0 , then j x0j is λ multiplier convergent with respect to the strong topology β(X 0 , X). Recall that if X is a Banach space, then the strong topology β(X 0 , X) is just the dual norm topology so Corollary 5.8 can be compared to the Diestel-Faires result in this case.

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Finally, we observe that the proof of Theorem 5.7 shows directly that the strong topology β(X, X 0 ) is weaker than the Dierolf topology Dλ (X, X 0 ) [Definition 4.38]. Theorem 5.7 will then follow from Theorem 4.41. Theorem 5.9. Let λ have ∞-GHP and let X be an LCTVS. Then the strong topology β(X, X 0 ) is weaker than the Dierolf topology Dλ (X, X 0 ). Proof: Let B be σ(X 0 , X) bounded. If B does not belong to Dλ , there P exists t ∈ λ and a λ multiplier convergent series j xj such that the series P∞ 0 0 j=1 tj hx , xj i do not converge uniformly for x ∈ B [Remark 4.40]. The proof of Theorem 5.7 then yields the result. Theorem 5.9 and Theorem 4.41 then yield Theorem 5.7 as a corollary and furnishes an alternate proof of Theorem 5.7.

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Orlicz-Pettis Theorems for Linear Operators

In this chapter we consider multiplier convergent series of continuous linear operators and establish Orlicz-Pettis Theorems for such series. Throughout this chapter let X and Y 6= {0} be LCTVS and L(X, Y ) the space of continuous linear operators from X into Y . We first describe the topologies on L(X, Y ) which will be considered. Let A be a family of bounded subsets of X whose union is all of X and let Y be the family of all continuous semi-norms on Y . The pair (A, Y) generate a locally convex topology on L(X, Y ) defined by the family of semi-norms (1) pA,q (T ) = sup{q(T x) : x ∈ A}, q ∈ Y, A ∈ A. We denote by LA (X, Y ) the locally convex topology on L(X, Y ) generated by the semi-norms in (1); this notation suppresses the dependence of the topology on the semi-norms in Y . A net {Tδ } in L(X, Y ) converges to 0 in LA (X, Y ) iff for every A ∈ A lim Tδ x = 0 uniformly for x ∈ A; for this reason the topology LA (X, Y ) is called the topology of uniform convergence on A (Appendix A). If A is the family of all bounded subsets of X, the topology LA (X, Y ) is denoted by Lb (X, Y ). In the case when X and Y are normed spaces, the topology Lb (X, Y ) is called the uniform operator topology and is generated by the semi-norm kT k = sup{kT xk : kxk ≤ 1}. If A is the family of all finite subsets of X, the topology LA (X, Y ) is denoted by Ls (X, Y ) and is just the topology of pointwise convergence on X. When Y has its original topology, the topology Ls (X, Y ) is called the strong operator topology of L(X, Y ). When Y has the weak topology 89

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σ(Y, Y 0 ), the topology Ls (X, Y ) is called the weak operator topology of L(X, Y ). Thus, a net {Tδ } converges to 0 in the strong operator topology (weak operator topology) iff Tδ x → 0 in Y for every x ∈ X ( hy 0 , Tδ xi → 0 for every x ∈ X, y 0 ∈ Y 0 ). See Appendix A for more details. Throughout this chapter let λ be a sequence space containing c00 , the space of all sequences which are eventually 0. We begin by establishing an Orlicz-Pettis Theorem for the weak and strong operator topologies. Recall that λ has the signed weak gliding hump property (signed-WGHP) if for every t ∈ λ and every increasing sequence of intervals {Ij } there exist a sequence of signs {sj } and a subsequence {nj } such that the coordinate P sum ∞ j=1 sj χInj t ∈ λ; if the signs can all be chosen equal to 1, then λ has the weak gliding hump property (WGHP). Examples of spaces with these properties are given in Appendix B. For example, any monotone space has WGHP while bs has signed-WGHP but not WGHP. P Theorem 6.1. Let λ have signed-WGHP. If the series j Tj is λ multiplier convergent in L(X, Y ) with respect to the weak operator topology, then P the series j Tj is λ multiplier convergent in L(X, Y ) with respect to the strong operator topology. P Proof: For every x ∈ X the series j Tj x is λ multiplier convergent in Y with respect to σ(Y, Y 0 ). By the Orlicz-Pettis Theorem in Corollary P 4.10, the series j Tj x is λ multiplier convergent in Y with respect to the P original topology of Y . That is, the series j Tj is λ multiplier convergent in the strong operator topology. Since subseries convergence is just m0 multiplier convergence as a special case of Theorem 6.1, we have P Corollary 6.2. If the series j Tj is subseries convergent with respect to the weak operator topology of L(X, Y ), then the series is subseries convergent with respect to the strong operator topology. If the multiplier space λ does not satisfy the signed-WGHP, then the conclusion of Theorem 6.1 may fail. Example 6.3. Let λ = cc , the space of all sequences which are eventually P constant (Appendix B). Then a series j zj is λ multiplier convergent in P a TVS Z iff the series j zj converges in Z. Define continuous linear operators Tj : R → c0 by Tj s = s(ej+1 − ej ) = −s ej − ej+1 . Then P∞ for every s ∈ R, the series j=1 Tj s converges in σ(c0 , l1 ) but does not

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P converge in (c0 , k·k∞ ). That is, the series j Tj is λ multiplier convergent in the weak operator topology of L(R, c0 ) but is not λ multiplier convergent in the strong operator topology of L(R, c0 ). First, we have the analogue of Lemma A.6 of Appendix A for λ multiplier convergent series. Lemma 6.4. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. Let λ have signed-WGHP. Suppose that P every series j xj which is λ multiplier convergent with respect to σ is such that for every t ∈ λ and increasing sequence of intervals {Ik } there is P a subsequence {nk } such that τ − lim j∈In tj xj = 0. Then every series k P j xj which is λ multiplier convergent with respect to σ is also λ multiplier convergent with respect to τ . Proof: By Lemma A.4, it suffices to show that if t ∈ λ, then the partial P sums of the series j tj xj are τ Cauchy. Suppose the partial sums of P the series j tj xj are not τ Cauchy. Then there exist a symmetric τ neighborhood of 0,U, and an increasing sequence of intervals {Ik } such that X (∗) t j xj ∈ / U. j∈Ik

Since λ has signed-WGHP, there exist a sequence of signs {sk } and a P∞ subsequence {nk } such that u = k=1 sk χInk t [coordinate sum] belongs P∞ P P∞ to λ. The series j=1 uj xj = k=1 sk j∈In tj xj is σ convergent so by k P hypothesis, {nk } has a subsequence {pk } such that τ −lim j∈Ip tj xj = 0. k But, this contradicts (∗). We now use Lemma 6.4 to establish a general Orlicz-Pettis Theorem for linear operators. Let C = {C ⊂ X : if {xj } ⊂ C, then lim T xj exists for every T ∈ L(X, Y )}. P Theorem 6.5. Let λ have signed-WGHP. If j Tj is λ multiplier converP gent in L(X, Y ) with respect to the weak operator topology, then j Tj is λ multiplier convergent in LC (X, Y ). Proof: By Lemma 6.4 it suffices to show that if t ∈ λ and {Ik } is an increasing sequence of intervals and {xi } ∈ C, then (#) lim( k

X

l∈Ik

tl Tl )(xi ) = 0 uniformly for i ∈ N.

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Define the matrix M = [mij ] = [

X

(tl Tl )(xi )].

l∈Ij

We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge since {xi } ∈ C. Next, if {pj } is an increasing sequence of integers, then there exist a subsequence {qk } of {pk } and a sequence of signs {sk } such that the coordinate sum of the series u = P∞ k=1 sk χIqk t belongs to λ. Then ∞ X j=1

sj miqj =

∞ X j=1

sj

X

t l T l xi = (

l∈Iqj

∞ X

ul Tl )xi

l=1

P∞ converges as i → ∞, where l=1 ul Tl ∈ L(X, Y ) is the weak operator sum of the series. Hence, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem the condition (#) is satisfied [Appendix D.3]. We use Theorem 6.5 to establish Orlicz-Pettis Theorems for two of the common topologies employed on L(X, Y ). If A is the family of all sequences in X which converge to 0, the topology LA (X, Y ) is denoted by L→0 (X, Y )[Appendix A]; this topology is studied in [GDS]. From Theorem 6.5, we have P Corollary 6.6. Let λ have signed-WGHP. If conj Tj is λ multiplier P vergent in L(X, Y ) with respect to the weak operator topology, then j Tj is λ multiplier convergent in L→0 (X, Y ). Proof: If xj → 0 in X, then {xj } ∈ C. We next consider the topology of uniform convergence on precompact subsets. If A is the family of all precompact subsets of X, the topology LA (X, Y ) is denoted by Lpc (X, Y ). In order to establish an Orlicz-Pettis Theorem for Lpc (X, Y ), we require the following representation theorem for precompact sets. Proposition 6.7. Let Z be a dense subspace of X and assume that X is metrizable. If K is a precompact subset of X, then there exists a null seP∞ quence {xk } ⊂ Z such that every x ∈ K has a representation x = j=1 tj xj P∞ with j=1 |tj | ≤ 1.

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Proof: Let k·k1 ≤ k·k2 ≤ ... be a sequence of semi-norms which generate the topology of X. Since Z is dense and K is precompact, there is a finite set F1 ⊂ Z such that for every x ∈ K there exists z 1 (x) ∈ F1 such that

x − z 1 (x) ≤ (1/2)(1/23). 1

Now K − F1 is precompact so there is a finite set F2 ⊂ Z such that for every x ∈ K there exists z 2 (x) ∈ F2 such that

x − z 1 (x) − z 2 (x) ≤ (1/3)(1/24). 2

Continuing this construction produces a sequence of finite subsets F1 , F2 , ... of Z such that every x ∈ K there exist z i (x) ∈ Fi satisfying

(1) x − z 1 (x) − ... − z i (x) i ≤ (1/(i + 1))(1/2i+2 ).

Therefore,

(2) z i (x) i−1

≤ x − z 1 (x) − ... − z i (x) i−1 + x − z 1 (x) − ... − z i−1 (x) i−1 ≤ (1/i)(1/2i )

Set y i (x) = 2i z i (x) for x ∈ K, i ∈ N. Arrange the elements of 2F1 , 22 F2 , ... in a sequence with the elements of F1 first, those of F2 second and so on. By (2) this sequence converges to 0 and by (1) x = z 1 (x) + z 2 (x) + ... = (1/2)y 1 (x) + (1/22 )y 2 (x) + ... This gives the desired representation. Theorem 6.8. Let λ have signed-WGHP and let X be metrizable or the P regular strict inductive limit of a sequence of metrizable LCTVS. If j Tj is λ multiplier convergent in L(X, Y ) with respect to the weak operator P topology, then j Tj is λ multiplier convergent in Lpc (X, Y ).

Proof: First suppose that X is metrizable and let K ⊂ X be precompact. By Proposition 6.7 there exists {xj } ⊂ K, xj → 0 such that every P∞ P∞ x ∈ K has a representation x = j=1 tj xj with j=1 |tj | ≤ 1. Let U be a closed, absolutely convex neighborhood of 0 in Y and let s ∈ λ. By P Corollary 6.6 there exists N such that nj=m sj Tj xk ∈ U for n > m ≥ N and k ∈ N. If x ∈ K has the representation above, then for n > m ≥ N we have n ∞ ∞ n n X X X X X s j T j xk ∈ U tk t k xk = s j Tj s j Tj x = j=m

j=m

k=1

k=1

j=m

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P since U is closed and absolutely convex. By Theorem 6.1, ∞ j=m sj Tj x ∈ U for m ≥ N and x ∈ K. This establishes the result in the metrizable case. If X is a regular strict inductive limit of metrizable spaces {Xk } and K ⊂ X is precompact, then K ⊂ Xk for some k and K is precompact in Xk . Thus, the first part gives the result in this case. A similar result was derived in Theorem 4.70. We next consider Orlicz-Pettis Theorems for the topology Lb (X, Y ). As was the case for the strong topology β(X, X 0 ) considered in Chapter 5 such results require strong hypotheses on the multiplier space λ. We present an example which shows that, in general, a series which is subseries convergent in the strong operator topology may not be subseries convergent in Lb (X, Y ). The example also suggests that to obtain Orlicz-Pettis theorems connecting the weak (strong) operator topology to the topology of Lb (X, Y ), one should consider the space K(X, Y ) of compact operators from X to Y . Example 6.9. Let X be a Banach space with an unconditional Schauder basis {bj }, i.e., every x ∈ X has a unique expansion ∞ X tj b j , x= j=1

where the series is unconditionally (subseries) convergent. Let fj be the j th coordinate functional associated with the basis {bj } defined by hfj , xi = tj in the expansion above. Each fj is linear and continuous ([Sw2] 10.1.13). Let Pj x = hfj , xi bj . If T ∈ L(X, Y ), where Y is a Banach space, then P∞ P∞ T x = j=1 hfj , xi T bj = j=1 T Pj x, where the series is norm convergent P∞ in Y . That is, the series j=1 T Pj is subseries convergent in the strong operator topology of L(X, Y ) to T . If L(X, Y ) has the property that any series which is subseries convergent in the strong operator topology (or weak operator topology) is subseries convergent in the norm topology of L(X, Y ), it follows that every T ∈ L(X, Y ) is compact, being the norm limit Pn of a sequence of compact operators, { j=1 T Pj }, with finite dimensional range. That is, if L(X, Y ) has this property, then L(X, Y ) = K(X, Y ), where K(X, Y ) is the space of all compact operators from X into Y . In particular, if X = Y , then the identity operator is compact and X must be finite dimensional ([Sw2] 7.8).

Before considering Orlicz-Pettis theorems for Lb (X, Y ), we observe a necessary condition for such a result to hold for subseries convergent series of operators.

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P Definition 6.10. The space X has the DF property iff every series j x0j in X 0 which is subseries convergent in σ(X 0 , X) is also subseries convergent in β(X 0 , X). For Banach spaces, Diestel and Faires have shown that X 0 has the DF property iff X 0 contains no subspace isomorphic to l ∞ ([DF]). P Theorem 6.11. If every series j Tj in L(X, Y ) which is subseries convergent in the weak operator topology is also subseries convergent in Lb (X, Y ), then X has the DF property. P 0 0 Proof: Let j xj be subseries convergent in X with respect

0 to 0 σ(X , X). Pick y ∈ Y, y 6= 0. Define Tj ∈ L(X, Y ) by Tj x = xj , x y. P Then so by j Tj is subseries convergent in the strong operator topology P∞

hypothesis the series converges in Lb (X, Y ). Thus, the series j=1 x0j , x converge uniformly for x belonging to bounded subsets of X. That is, the P series j x0j converges in (X 0 , β(X 0 , X)).

For our first Orlicz-Pettis Theorem for Lb (X, Y ), we establish the analogue of Theorem 5.4 for operators. P Theorem 6.12. Let λ be a barrelled AK space. If the series j Tj is λ P multiplier convergent in the weak operator topology of L(X, Y ), then j Tj is λ multiplier convergent in Lb (X, Y ). Proof: If x ∈ X and y 0 ∈ Y 0 , define a continuous linear functional x ⊗ y 0 on L(X, Y ) by hx ⊗ y 0 , T i = hy 0 , T xi. Let X ⊗ Y 0 = span{x ⊗ y 0 : x ∈ X, y 0 ∈ Y 0 }. Note that the weak operator topology on L(X, Y ) is just P σ(L(X, Y ), X ⊗ Y 0 ). From Theorem 5.4 it follows that the series j Tj is λ multiplier convergent in the strong topology β(L(X, Y ), X ⊗ Y 0 ). Thus, it suffices to show that the strong topology β(L(X, Y ), X ⊗ Y 0 ) is stronger than Lb (X, Y ). Let {Sδ } be a net in L(X, Y ) which converges to 0 in β(L(X, Y ), X ⊗ Y 0 ). Let A ⊂ X be bounded, B ⊂ Y 0 be equicontinuous and set C = {x ⊗ y 0 : x ∈ A, y 0 ∈ B}. Since A is β(X, X 0 ) bounded from the barrelledness assumption, sup{|hx ⊗ y 0 , T i| : x ∈ A, y 0 ∈ B} < ∞

for every T ∈ L(X, Y ); that is, C is σ(X ⊗ Y 0 , L(X, Y )) bounded. Thus, sup{|hx ⊗ y 0 , Sδ i| : x ∈ A, y 0 ∈ B} → 0

so Sδ → 0 in Y uniformly for x ∈ A or Sδ → 0 in Lb (X, Y ). Thus, β(L(X, Y ), X ⊗ Y 0 ) is stronger than Lb (X, Y ) as desired.

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An extensive list of multiplier spaces satisfying the assumptions of Theorem 6.12 is given in Remark 4.7. We next establish the analogue of Theorem 5.7 for linear operators. P Theorem 6.13. Let λ have ∞-GHP. If the series j Tj is λ multiplier P convergent in the strong operator topology, then j Tj is λ multiplier convergent in Lb (X, Y ). Proof: If the conclusion fails to hold, there exist > 0, A ⊂ X bounded,t ∈ λ, a continuous semi-norm p on Y and sequences {mk }, {nk } with m1 < n1 < m2 < ... and nk nk X X sup p( tl Tl x) = pA ( tl Tl ) > . x∈A

l=mk

l=mk

For every k there exists xk ∈ A such that (∗) p(

nk X

tl Tl xk ) > .

l=mk

Set Ik = [mk , nk ]. Since λ has ∞-GHP, there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such P∞ that s = k=1 aqk χIqk t ∈ λ. Define a matrix X (tl apj )Tl (xi /api )]. M = [mij ] = [ l∈Ij

We claim that M is a K-matrix [Appendix D.2]. First, the columns of M converge to 0 since xi /api → 0 and each Tl is continuous. Next, given a subsequence there is a further subsequence {qk } such that s = P∞ P∞ l=1 sl Tl converges in the strong operator k=1 aqk χIqk t ∈ λ. The series topology to an operator T ∈ L(X, Y ). Hence, ∞ X j=1

miqj =

∞ X X

j=1 l∈Iqj

sl Tl (xi /api ) = T (xi /api ) → 0.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem [Appendix D.2], the diagonal of M converges to 0. But, this contradicts (∗). Remark 6.14. If λ also has signed-WGHP in Theorem 6.13, we may reP place the assumption that the series j Tj is λ multiplier convergent in the strong operator topology with the assumption that the series is λ multiplier convergent in the weak operator topology (Theorem 6.1).

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Example 6.9 suggests that if one wishes to establish Orlicz-Pettis theorems for λ multiplier convergent series with respect to Lb (X, Y ) without imposing strong conditions on the multiplier space λ, one should consider the space K(X, Y ) of compact operators. The major result in this area is a result of Kalton. Kalton has shown that if X has the DF property (or, P if X 0 contains no subspace isomorphic to l ∞ ) and if j Tj is a series of compact operators from a Banach space X into a Banach space Y which is subseries convergent in the weak operator topology of K(X, Y ), then the P series j Tj is subseries convergent in the uniform operator topology of K(X, Y ) ([Ka]). We first establish a result of Wu and Lu which characterizes the OrliczPettis property for the space of compact operators ([WL]). Their result contains Kalton’s result as a special case. Let Kb (X, Y ) be the topology on K(X, Y ) induced by Lb (X, Y ). Theorem 6.15. Let λ have signed-WGHP. The following are equivalent: P (i) Every series j Tj which is λ multiplier convergent in the weak operator topology of K(X, Y ) is λ multiplier convergent in Kb (X, Y ). (ii) Every continuous linear operator S : X → (λβ , σ(λβ , λ)) is sequentially compact (an operator is sequentially compact if it carries bounded sets into relatively sequentially compact sets). P Proof: Suppose (ii) holds. Let j Tj be λ multiplier convergent in the weak operator topology of K(X, Y ). By Theorem 6.1 the series is λ multiplier convergent in the strong operator topology of K(X, Y ). SupP pose there exists t ∈ λ such that the series j tj Tj is not convergent in Kb (X, Y ). Then there exist T ∈ K(X, Y ) and a bounded set A ⊂ X P∞ such that j=1 tj Tj x = T x for every x ∈ X but the series do not converge uniformly for x ∈ A. Thus, there exist a continuous semi-norm p on Y , increasing sequences {mk } and {nk } with mk < nk < mk+1 ,xk ∈ A and > 0 such that ! nk X p t l T l xk > l=mk

for all k. By the Hahn-Banach Theorem there is a sequence {yk0 } ⊂ Y 0 such that + * nk X t l T l xk > (∗) yk0 , l=mk

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and sup{|hyk0 , yi| : p(y) ≤ 1} ≤ 1. Let Y0 be the closure in Y of span{Ti xj : i, j ∈ N}. Then (Y0 , p) is a separable semi-norm space. By the Banach-Alaoglu Theorem for separable semi-norm spaces, {yk0 } has a subsequence {yn0 k } and y 0 ∈ Y 0 such that

lim yn0 k , y = hy 0 , yi for every y ∈ Y0 and

sup{|hy 0 , yi| : p(y) ≤ 1} ≤ 1.

For notational convenience, assume that nk = k. Define a semi-norm q on X 0 by q(x0 ) = sup{|hx0 , xk i| : k ∈ N}. We claim that if U ∈ K(X, Y ) satisfies U xk ∈ Y0 , then (∗∗)

lim q(U 0 yk0 − U 0 y 0 ) = 0.

If (∗∗) fails to hold, there exist δ > 0, a subsequence {yn0 k } and a subsequence {xnk } such that

(∗ ∗ ∗) U 0 yn0 k − U 0 y 0 , xnk > δ.

Since U is compact, {U xnk } is a relatively compact subset of Y and, therefore, a relatively compact subset of (Y0 , p). Without loss of generality, we may assume that there exists y ∈ Y0 such that p(U xnk − y) → 0. Then 0

yn − y 0 , U xn ≤ yn0 − y 0 , U xn − y + yn0 − y 0 , y k k k k k

≤ sup{ yn0 k − y 0 , z : p(z) ≤ 1}p(U xnk − y)

+ yn0 k − y 0 , y

≤ 2p(U xn − y) + y 0 − y 0 , y → 0. nk

k

This contradicts (∗ ∗ ∗) and establishes the claim. P∞ If s ∈ λ, x ∈ X and z 0 ∈ Y 0 , the series j=1 sj hz 0 , Tj xi converges so we define a linear operator S(= Sz0 ) : X → (λβ , σ(λβ , λ)) by Sx = {hz 0 , Tj xi}. Since S is obviously continuous, S is sequentially compact by condition (ii). Thus, SA is sequentially compact with respect to σ(λβ , λ). By Corollary P 2.29, if s ∈ λ, then the series ∞ s hz 0 , T xi converge uniformly for x ∈ A P∞j=1 j 0 0 j or, equivalently, the series j=1 sj Tj z , x converge uniformly for x ∈ A. Now consider the matrix nj X M = [mij ] = tl Tl0 yi0 . l=mj

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We show that M is a signed K-matrix with values in the semi-norm space (X 0 , q) [Appendix D.3]. First, the columns of M converge by condition (∗∗). Next, given any subsequence {pj } there is a further subsequence P∞ Pnql {qj } and a sequence of signs {j } such that s = l=1 l j=m tj ∈ λ. P∞ P∞ Pnqqll There exist U ∈ K(X, Y ) such that s T = j=mql tj Tj j=1 j j l=1 l converges to U in the strong operator topology. By the paragraph above P∞ P n ql 0 0 0 0 j=mql tj Tj yk converges to U yk uniformly for x ∈ A. In particul=1 l lar, n ql ∞ X X q l tj Tj0 yk0 − U 0 yk0 → 0. l=1

j=mql

Thus, M is a signed K-matrix (with respect to (X 0 , q). By the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0 in (X 0 , q). This contradicts (∗) and establishes that (ii) implies (i). Suppose that (i) holds. Let S : X → (λβ , σ(λβ , λ)) be linear and continuous. So Sx = {Sx · ej }. Let y ∈ Y, y 6= 0. Define Tj ∈ K(X, Y ) by Tj x = (Sx · ej )y. Let t ∈ λ. Define T (= Tt ) ∈ K(X, Y ) by T x = (Sx · t)y. P∞ P Then j=1 tj Tj x = T x for every x ∈ X, i.e., the series j tj Tj converges to T in the strong operator topology of K(X, Y ). By (i) the series ∞ X j=1

t j Tj x =

∞ X j=1

tj (Sx · ej )y = T x = (Sx · t)y

converge uniformly for x belonging to bounded subsets of X or the series P∞ j j=1 tj (Sx · e ) = Sx · t converge uniformly for x belonging to bounded subsets of X. Now to show S is sequentially compact, let {xk } be a bounded sequence in X. Then {Sxk } is coordinatewise bounded in λβ since S is bounded. By the diagonal method ([DeS] 26.10), there is a subsequence {nk } such that limk Sxnk · ej exists for every j and since the series P∞ j j=1 tj (Sxnk · e ) converge uniformly for k ∈ N, limk t · Sxnk exists. Thus, β {Sxnk } is σ(λ , λ) Cauchy. By Corollary 2.28 there exists u ∈ λβ such that Sxnk → u in σ(λβ , λ). Therefore, {Sxnk } is relatively sequentially compact in (λβ , σ(λβ , λ)). Remark 6.16. Wu Junde has shown that subsets of λβ are σ(λβ , λ) sequentially compact iff they are σ(λβ , λ) compact so condition (ii) of Theorem 6.15 can be replaced with the hypothesis that the operator S is compact ([Wu]).

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For the case of subseries convergent series, that is, when λ = m0 , we have Theorem 6.17. Let X be a barrelled LCTVS. The following are equivalent: P (i) Every series j Tj which is subseries convergent in the weak operator topology of K(X, Y ) is subseries convergent in Kb (X, Y ). (ii) Every continuous linear operator S : X → (l 1 , σ(l1 , m0 )) is compact. (iii) Every continuous linear operator S : X → (l 1 , k·k1 ) is compact. (iv) (X 0 , β(X 0 , X)) contains no subspace isomorphic to c0 . (v) X has the DF property. Proof: Since subsets of l 1 are σ(l1 , m0 ) [k· k1 ] sequentially compact iff they are compact (Proposition 4.18), (i), (ii) and (iii) are equivalent by Theorem 6.15. Since X is barrelled (iii), (iv) and (v) are equivalent by Theorem 3.20. Remark 6.18. If X and Y are Banach spaces, the equivalence of (i) and (v) is Kalton’s result except that Kalton uses the hypothesis that X 0 contains no subspace isomorphic to l ∞ which is equivalent to the DF property by the Diestel/Faires result ([DF]). For Banach spaces the equivalence of (i) and (iv) was established by Bu and Wu ([BW]). We used the abstract set-up preceding Theorem 4.62 to establish a version of Kalton’s theorem for normed spaces (Theorem 4.78) which we restate. Theorem 6.19. Let X and Y be normed spaces and let X have the DF P property. If the series j Tj is subseries convergent in the weak operator topology of K(X, Y ), then the series is subseries convergent in the norm topology of K(X, Y ).

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Chapter 7

The Hahn-Schur Theorem

In this chapter we establish vector versions of the classical Hahn-Schur Theorem for multiplier convergent series. For later reference, we first give a statement of one version of the Schur Theorem for absolutely convergent scalar valued series. Theorem 7.1. (Schur) For each i ∈ N, let vergent series of scalars. Assume

P

j tij

be an absolutely con-

P ∞ (h) limi ∞ and let limi tij = tj for every j=1 sj tij exists for every {sj } ∈ l j ∈ N. Then P∞ P∞ is absolutely convergent and limi j=1 sj tij = j=1 sj tj for every {sj } ∈ l∞ , P∞ (ii) limi j=1 |tij − tj | = 0, P∞ (iii) the series j=1 |tij | converge uniformly for i ∈ N. (i)

P

j tj

The statement in Theorem 7.1 is often referred to as the Schur Lemma ([Sr]). In particular, Theorem 7.1 implies that a sequence in l 1 which is weak (σ(l1 , l∞ )) Cauchy is norm (k·k1 ) convergent. The Hahn version of Theorem 7.1 relaxes the hypothesis (h) and retains conclusions (ii) and (iii) with a slight restatement of condition (i). Again for later reference, we give a statement of Hahn’s theorem ([Ha]). Theorem 7.2. (Hahn) For each i ∈ N, vergent series of scalars. Assume 101

let

P

j tij

be an absolutely con-

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P (h’) limi j∈σ tij exists for every σ ⊂ N and let limi tij = tj for every j ∈ N. Then P P P (i) j tj is absolutely convergent and limi j∈σ tij = j∈σ tj for every σ ⊂ N, (ii) and (iii). In particular, the Hahn version in Theorem 7.2 implies that a sequence in l1 which is σ(l1 , m0 ) Cauchy is k·k1 convergent ([Ha]). Hahn’s theorem can also be used to establish an important result from summability. Namely, (S) A matrix T = [tij ] maps the sequence space m0 into the space c of convergent sequences iff the matrix T maps l ∞ into c. ([Sw2] 9.5.3; see Theorem 7.29 for a vector version of statement (S)). Both the Schur and Hahn theorems have numerous applications to various topics in analysis; in particular, Schur’s theorem was used in the proofs of the versions of the Orlicz-Pettis Theorem given by both Orlicz and Pettis ([Or], [Pe]). The conclusions in (ii) and (iii) (as well as those in the hypothesis and condition (i)) involve absolutely convergent series and, therefore, do not represent interesting suggestions for vector valued generalizations of either Theorem 7.1 or Theorem 7.2. However, conditions (ii) and (iii) can be restated in forms which do not involve absolute convergence and which do suggest possible vector valued generalizations. We first give a restatement of conditions (ii) and (iii) for hypothesis (h). P Proposition 7.3. Let j tij be absolutely convergent for every i ∈ N. Condition (ii) is equivalent to: P P∞ (ii)’ limi ∞ j=1 sj tij = j=1 sj tj uniformly for k{sj }k∞ ≤ 1.

Condition (iii) is equivalent to: P∞ (iii)’ the series j=1 sj tij converge uniformly for i ∈ N and k{sj }k∞ ≤ 1. Proof: Assume (ii). If {sj } ∈ l∞ , then ∞ ∞ X X s (t − t ) ≤ k{s }k |tij − tj | j ij j j ∞ j=1 j=1

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so (ii)’ follows immediately. P∞ Assume (ii)’. Fix i ∈ N. Define sj = sign(tij − tj ). Then j=1 sj (tij − P∞ tj ) = j=1 |tij − tj |. Since k{sj }k∞ ≤ 1, (ii) follows. The equivalence of (iii) and (iii)’ are established similarly. We next give a restatement of conditions (ii) and (iii) for hypothesis (h)’. For this we use Lemma 3.37. Proposition 7.4. Let

P

j tij

be absolutely convergent for every i ∈ N.

Condition (ii) is equivalent to: Proposition 7.5. P P (ii)” limi j∈σ tij = j∈σ tj uniformly for σ ⊂ N. Condition (iii) is equivalent to:

(iii)” for P every > 0 there exist N such that σ ⊂ N, min σ ≥ N implies j∈σ tij < for all i ∈ N.

Proof: Clearly (ii) implies (ii)” and (iii) implies (iii)”. that i ≥ N implies P Assume (ii)” holds. Let > 0. There exists N such P∞ (t − t ) < /2 for σ ⊂ N. By Lemma 3.37, j∈σ ij j j=1 |tij − tj | ≤ for i ≥ N . Thus, (ii) holds. Similarly, (iii)” implies (iii). The hypothesis (h) [(h)’] and conclusions (ii)’ and (iii)’ [(ii)” and (iii)”] suggest generalizations of the Schur and Hahn theorems for multiplier convergent series with values in a TVS. Hypothesis (h) and conditions (ii)’ and (iii)’ [(h)’ and conditions (ii)” and (iii)”] use bounded multiplier convergent series [subseries convergent series] or Λ multiplier convergent series where Λ = l∞ [where Λ = {χσ : σ ⊂ N}]. Thus, it would be natural to seek versions of Theorems 7.1 and 7.2 for Λ multiplier convergent series. We pursue these versions in this chapter. Theorems 7.1 and 7.2 will follow directly from our general results for multiplier convergent series. Let λ be a scalar sequence space which contains c00 , the space of sequences which are eventually 0 and let Λ ⊂ λ. Let X be a TVS. The analogue of hypotheses (h) and (h)’ for Λ multiplier convergent series would then be:

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P • (H) Let Λ multiplier convergent in X for every i ∈ N. j xij be P∞ Assume that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N. The analogues of conditions (ii)’ [(ii)”], and (iii)’ [(iii)”] for multiplier convergent series require topological assumptions on the multiplier space Λ. Assume that λ is a K-space. The analogues of conditions (i), (ii)’ [(ii)”], and (iii)’ [(iii)”] for multiplier convergent series would then be: P∞ P • (C1) the series j xj is Λ multiplier convergent and limi j=1 tj xij = P∞ every t ∈ Λ. j=1 tj xj for P∞ P∞ • (C2) limi j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ. P∞ • (C3) the series j=1 tj xij converge uniformly for t belonging to bounded subsets of Λ. We first consider conclusion (C1). Under the hypothesis (H) conclusion (C1) follows from Lemma 2.27 and Theorem 2.26 preceding Stuart’s weak completeness result. Recall that Λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ Λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence P∞ {nj } such that the coordinate sum of the series j=1 sj χInj t belongs to Λ [Appendix B.6]. Theorem 7.6. Assume that Λ has signed-WGHP. Then condition P (H) Let ij be Λ multiplier convergent in X for every i ∈ N. Assume jx P ∞ that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N implies the conclusion P P (C1) the series xj is Λ multiplier convergent and limi ∞ j j=1 tj xij = P∞ t x for every t ∈ Λ. j=1 j j The result in Theorem 7.5 may fail if λ does not have signed-WGHP.

Example 7.7. Let λ = c. Define xij by xij = 1 if i = j and xij = 0 P∞ otherwise. If t ∈ c, then limi j=1 tj xij = limi ti and xj = limi xij = 0. P∞ P∞ But, limi j=1 tj xij = limi ti 6= j=1 tj xj = 0 if limi ti 6= 0.

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This example shows that if {ej } ⊂ l1 , then {ej } is σ(l1 , c) Cauchy but does not have a σ(l 1 , c) limit. That is, σ(l1 , c) is not sequentially complete. We next consider conclusions (C2) and (C3). This will require stronger assumptions on the multiplier space λ. Let λ be a K-space and let Λ ⊂ λ. Recall that Λ has the signed strong gliding hump property (signed-SGHP) if whenever {tk } ⊂ Λ is bounded and {Ik } is an increasing sequence of intervals, there exist a sequence of signs {sk } and a subsequence {nk } such P∞ that the coordinate sum of the series k=1 sk χInk tnk belongs to Λ. If all of the signs can be chosen equal to 1, then Λ is said to have the strong gliding hump property (SGHP)(Appendix B.17). For example, the space l∞ has SGHP while the subset Λ = M0 = {χσ : σ ⊂ N} ⊂ m0 has SGHP but the space m0 does not have SGHP; the space bs of bounded series has signed-SGHP but not SGHP [see Appendix B for these and additional examples]. We first establish a lemma which is a special case of condition (C2). The proof of the lemma uses a property of TVS which we now establish. Lemma 7.8. Let X be a TVS. If lim xj = 0 in X, then lim txj = 0 uniformly for |t| ≤ 1. Proof: Let U be a balanced neighborhood of 0 in X. There exists N such that k ≥ N implies that xk ∈ U . Therefore, if k ≥ N and |t| ≤ 1, txk ∈ tU = U . Remark 7.9. There is another proof of Lemma 7 in Yosida ([Y] I.2.2) which uses Egoroff’s Theorem and properties of Lebesgue measure and one in [Sw1] 8.2.4 which uses the Banach-Steinhaus Theorem. Lemma 7.10. Assume that Λ ⊂ λ and Λ has signed-SGHP and B ⊂ Λ P is bounded. If j xij is Λ multiplier convergent for every i ∈ N, P∞ limi j=1 tj xij = 0 for every {tj } ∈ Λ and limi xij = 0 for every j, then P∞ limi j=1 tj xij = 0 uniformly for t ∈ B.

P∞ Proof: It suffices to show that limi j=1 tij xij = 0 for any sequence {ti } ⊂ B. Let U be a neighborhood of 0 in X and pick a symmetric neighborhood of 0,V , in X such that V +V +V ⊂ U . Set n1 = 1 and pick N1 P n1 i such that ∞ j=N1 tj xn1 j ∈ V . Since limi xij = 0 for every j and {tj : i ∈ N} i is bounded from the K-space assumption, limi tj xij = 0 for every j by PN1 −1 i Lemma 7.7. Therefore, there exists n2 > n1 such that j=1 tj xij ∈ V for P∞ every i ≥ n2 . Pick N2 > N1 such that j=N2 tnj 2 xn2 j ∈ V . Continuing

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this construction produces increasing sequences {nk }, {Nk } such that ∞ X

l=Nj

Nj −1 n

X

tl j xnj l ∈ V and

l=1

ti xil ∈ V f or i ≥ nj .

Set Ij = {l : Nj−1 ≤ l < Nj }. Define a matrix X n M = [mij ] = [ tl j xni l ]. l∈Ij

We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge to 0 since limi xil = 0 for every l. Given an increasing sequence {pj }, there is a sequence of signs {sj } and a subsequence P∞ q {qj } of {pj } such that t = j=1 sj χIqj tj j ∈ Λ (coordinate sum). Then ∞ X

sj miqj =

j=1

∞ X

sj

j=1

X

q

t l j xn i l =

∞ X

t j xn i j

j=1

l∈Iqj

P∞ and j=1 tj xni j → 0 by hypothesis. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). Thus, there exists N such that mii ∈ V for i ≥ N . If i ≥ N, then ∞ X

Ni−1 −1

tnl i xni l =

l=1

X l=1

P∞

tnl i xni l +

X

l∈Ii

tnl i xni l +

∞ X

l=Ni

tnl i xni l ∈ V + V + V ⊂ U

so limi l=1 tnl i xni l = 0. Since the same argument can be applied to any P∞ subsequence, it follows that limi j=1 tij xij = 0.

We can now establish the result with conclusions (C2) and (C3) under hypothesis (H).

Theorem 7.11. Assume that Λ ⊂ λ and Λ has signed-SGHP. If P (H) Let ij be Λ multiplier convergent in X for every i ∈ N. Assume jx P ∞ that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N, then the following conclusions hold: P P∞ (C1) the series j xj is Λ multiplier convergent and limi j=1 tj xij = P∞ t x for every t ∈ Λ, j j j=1

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P P∞ (C2) limi ∞ j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ and P∞ (C3) the series j=1 tj xij converge uniformly for t belonging to bounded subsets of Λ Proof: Since Λ has signed-WGHP, conclusion (C1) holds by Theorem 7.5. P∞ Since limi j=1 tj (xij − xj ) = 0 by conclusion (C1), Lemma 7.9 now applies and gives conclusion (C2) immediately. Suppose that (C3) fails to hold. Then there exist a closed, symmetric neighborhood of 0,U , in X and a bounded set B ⊂ Λ such that for every P i i there exist ki > i, ni > i, ti ∈ B with ∞ / U . There exists j=ni tj xki j ∈ Pmi i P mi > ni such that j=ni tj xki j ∈ / U . Set I = [ni , mi ] so j∈Ii tij xki j ∈ / U. By the condition above for i1 = 1, there exist k1 , a finite interval I1 P with min I1 > i1 , t1 ∈ B with j∈I1 t1j xk1 j ∈ / U . By Theorem 2.35, there P∞ exists j1 such that k=j tk xik ∈ U for every t ∈ B, 1 ≤ i ≤ k1 , j ≥ j1 . Set i2 = max{I1 + 1, j1 }. Again by the condition above there exist k2 > i2 , P / U. a finite interval I2 with min I2 > i2 , t2 ∈ B such that k∈I2 t2k xk2 k ∈ Note that k2 > k1 by the definition of i2 . Continuing this construction produces an increasing sequence {ki }, an increasing sequence of intervals {Ii } and {ti } ⊂ B such that X tik xki k ∈ / U. (∗) k∈Ii

Define a matrix M = [mij ] = [

X

tjk xki k ].

k∈Ij

We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Next, given any increasing sequence {pj }, there exist a sequence of signs {sj } and a subsequence {qj } of {pj } P∞ q such that the coordinate sum t = j=1 sj χIqj tj j ∈ Λ. Then the sequence ∞ X j=1

sj miqj =

∞ X j=1

sj

X

l∈Iqj

q

t l j xk i l =

∞ X

t j xk i j

j=1

converges by hypothesis. Hence, M is a signed K-matrix so by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (∗).

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If the multiplier space Λ does not have signed-SGHP, then the hypothesis (H) may not imply (C2) and (C3) even if Λ has WGHP or 0-GHP. Example 7.12. Let 1 ≤ p < ∞ and λ = Λ = l p . Define xij = ej if P 1 ≤ j ≤ i and xij = 0 if j > i. Then j xij is λ multiplier convergent for every i, limi xij = ej = xj for every j and lim i

∞ X

tj xij = lim i

j=1

i X

tj e j =

j=1

∞ X

tj e j =

j=1

∞ X

t j xj

j=1

in lp for every t ∈ l p so (H) holds and (C1) holds. However, both (C2) and P∞ (C3) fail to hold. [Take tk = ek so {tk } is bounded in lp but j=1 tkj xij = Pi k j k j=1 tj e = e if i ≥ k so (C2) and (C3) fail.]

We next consider the hypothesis (H) as a conclusion. P Proposition 7.13. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. P (1) If for every t ∈ Λ the series ∞ j xij converge uniformly for i ∈ N, j=1 tP then for every t ∈ Λ the sequence { ∞ j=1 tj xij }i is Cauchy. P∞ (2) If the series j=1 tj xij converge uniformly for i ∈ N and t belonging P∞ to bounded subsets of Λ, then the sequences { j=1 tj xij }i satisfy a Cauchy condition uniformly for t belonging to bounded subsets of Λ. Proof: (1): Let t ∈ Λ. Let U be a neighborhood of 0 in X. Pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists P∞ N such that n ≥ N implies j=n tj xij ∈ V for every i ∈ N. There exists PN n > N such that i, k ≥ n implies j=1 tj (xij − xkj ) ∈ V . If i, k ≥ n, then (∗)

∞ X j=1

N X j=1

tj (xkj − xij ) +

∞ X

j=N +1

tj xkj −

tj xkj −

∞ X

tj xij =

j=1

∞ X

j=N +1

tj xij ∈ V + V + V ⊂ U.

(2): Let B ⊂ Λ be bounded. Let U be a neighborhood of 0 in X. Pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists P∞ N such that n ≥ N implies j=n tj xij ∈ V for every i ∈ N, t ∈ B. By the K-space assumption {tj : t ∈ B} is bounded for every j. By Lemma 7.7 PN there exists n > N such that j=1 tj (xij − xkj ) ∈ V for every t ∈ B. If i, k ≥ n, then (∗) holds.

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Proposition 7.14. Assume that λ is an AB-space (Appendix B.3). Let P be Λ multiplier convergent for every i ∈ N and assume that j xij P∞ limi xij = xj exists for every j ∈ N. If limi j=1 tj xij exists for every P∞ t ∈ Λ and the series j=1 tj xij converge uniformly for i ∈ N and t beP longing to bounded subsets of Λ, then j xj is Λ multiplier convergent and P∞ P∞ limi j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ. P Proof: Let t ∈ Λ. Put z = limi ∞ j=1 tj xij . Let U be a neighborhood of 0 in X. Pick a closed, symmetric neighborhood of 0, V , such that V + V + P V ⊂ U . There exists k such that ∞ j=1 tj xkj − z ∈ V and by Proposition 7.12(2) since V is closed and {Pm t : m ∈ N} is bounded by the ABPm Pm assumption (Pm is the section map Pm t = j=1 tj ej ), j=1 tj (xj − xkj ) ∈ P∞ V for every m. There exists M such that m ≥ M implies j=m+1 tj xkj ∈ V . Then if m ≥ M , we have m X j=1

tj xj −z =

∞ X j=1

tj xkj −z +

m X j=1

tj (xj −xkj )−

∞ X

j=m+1

tj xkj ∈ V +V +V ⊂ U.

P∞ Pm Hence, z = limm j=1 tj xj = j=1 tj xj . P∞ Let B ⊂ Λ be bounded. Since j=1 tj xij converges uniformly for i ∈ P∞ N, t ∈ B, the series j=1 tj xij satisfy a Cauchy condition uniformly for i ∈ N, t ∈ B. Therefore, there exists N such that n > m ≥ N implies that Pn P∞ j=m tj xij ∈ V for i ∈ N, t ∈ B. Hence, j=m tj xj ∈ V for m ≥ N, t ∈ B. Since {tj : t ∈ B} is bounded for every j, by Lemma 7.7 there exists M P such that N j=1 tj (xij − xj ) ∈ V for i ≥ M, t ∈ B. If i ≥ M , then ∞ X j=1

tj (xij −xj ) =

N X j=1

tj (xij −xj )+

∞ X

j=N +1

tj xij −

∞ X

j=N +1

tj xj ∈ V +V +V ⊂ U

for every t ∈ B. Corollary 7.15. Let Λ have signed-WGHP and let X be sequentially comP plete. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. The following are equivalent: P∞ (i) limi j=1 tj xij exists for every t ∈ Λ, P∞ P∞ P and limi j=1 tj xij = j=1 tj xj , (ii) j xj is Λ multiplier convergent P∞ (iii) for every t ∈ Λ the series j=1 tj xij converge uniformly for i ∈ N.

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Proof: That (ii) implies (i) is clear; (iii) implies (i) by Proposition 7.12(1); (i) implies (ii) by Lemma 2.27; (ii) implies (iii) by Theorem 2.26. Corollary 7.16. Let Λ have signed-SGHP and let X be sequentially comP plete. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. The following are equivalent: P (I) limi ∞ j=1 tj xij exists for every t ∈ Λ, P P P∞ x is Λ multiplier convergent and limi ∞ (II) j j j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ, P∞ (III) the series converge uniformly for i ∈ N, t belonging to j=1 tj xij bounded subsets of Λ. P∞ (IV) for every t ∈ Λ the series j=1 tj xij converge uniformly for i ∈ N. Proof: Clearly (II) implies (I); (I) implies (II) and (III) by the HahnSchur Theorem 7.10; that (III) implies (IV) is clear; (IV) implies (I) by Proposition 7.12(2).

We also obtain a boundedness result. P Proposition 7.17. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. If B ⊂ Λ is bounded P∞ and the series j=1 tj xij converge uniformly for i ∈ N, t ∈ B, then ∞ X S= tj xij : i ∈ N, t ∈ B j=1

is bounded.

Proof: Let U be a balanced neighborhood of 0 in X and pick a balanced neighborhood of 0, V , such that V + V ⊂ U . There exists N such that P∞ for i ∈ N, t ∈ B. Since {xij : i ∈ N} and {tj : t ∈ B} j=N +1 tj xij ∈ V P are bounded for every j, there exists t > 1 such that { N j=1 tj xij : i ∈ N, t ∈ B} ⊂ tV . Hence, S ⊂ V + tV ⊂ tU and S is bounded. Since both M0 = {χσ : σ ⊂ N} ⊂ m0 = spanM0 and l∞ have SGHP, the previous results hold for both M0 multiplier (=subseries) convergent series and l∞ multiplier (=bounded multiplier) convergent series. We record these special cases for the previous results. We begin with the subseries case. From Theorem 7.10, we have

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P Theorem 7.18. Let j xij be subseries convergent for every i ∈ N. AsP sume that limi j∈σ xij exists for every σ ⊂ N and limi xij = xj for every j. Then P (1) j is subseries convergent, j xP P (2) limi j∈σ xij = j∈σ xj uniformly for σ ⊂ N and P (3) the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N. Note that the scalar case of Theorem 7.17 gives the scalar version of the Hahn Theorem stated in Theorem 7.2. From Proposition 7.12, we have P Proposition 7.19. Let j xij be subseries convergent for every i ∈ N and limi xij = xj for every j. P (1) If for every σ ⊂ N the series j∈σ xij converge uniformly for i ∈ N, P then for every σ ⊂ N the sequence { j∈σ xij }i is Cauchy. P (2) If the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, then the P sequences { j∈σ xij }i satisfy a Cauchy condition uniformly for σ ⊂ N. From Corollary 7.15, we have P Corollary 7.20. Let X be sequentially complete. Let j xij be subseries convergent for every i ∈ N and limi xij = xj for every j. The following are equivalent: P (1) limi j∈σ xij exists for every σ ⊂ N, P P P (2) j xj is subseries convergent and lim i j∈σ xij = j∈σ xj uniformly for σ ⊂ N, P (3) the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, P (4) for every σ ⊂ N the series j∈σ xij converge uniformly for i ∈ N. From Proposition 7.16, we have P Proposition 7.21. Let j xij be subseries convergent for every i ∈ N and P limi xij = xj for every j. If the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, then X S= xij : i ∈ N, σ ⊂ N j∈σ

is bounded.

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We now show that Theorem 7.17 can be improved for sequentially complete spaces by replacing the family of all subsets of N by a smaller family of subsets. Recall that a family of subsets F of N is an FQσ family if F contains the finite subsets and whenever {Ik } is a pairwise disjoint sequence of finite subsets there is a subsequence {Ink } such that ∪∞ k=1 Ink ∈ F (see Appendix B for examples). If Λ = {χσ : σ ∈ F} ⊂ m0 where F is an FQσ family, then Λ has SGHP (Appendix B) so Theorem 7.10 applies. We first establish a lemma. Lemma 7.22. Let X be sequentially complete and F be an FQσ family P with Λ = {χσ : σ ∈ F} ⊂ m0 . If j xj is Λ multiplier convergent in X, P then j xj is subseries convergent. P∞ Proof: Let {nj } be a subsequence and σ = {nj : j ∈ N}. If j=1 xnj does not converge, there exist a neighborhood of 0, U , in X and an increasP P ing sequence of intervals {Ik } such that j∈Ik xnj = j∈Ik ∩σ xj ∈ / U . By the FQσ property there is a subsequence {mk } such that I = ∪∞ I k=1 mk ∩σ ∈ P F. But, then j∈I xj does not converge since the series fails the Cauchy condition. We now give the improvement of Theorem 7.17 for sequentially complete spaces. Theorem 7.23. Let X be sequentially complete and F be an FQσ famP ily with Λ = {χσ : σ ∈ F} ⊂ m0 . Assume that j xij is Λ multiplier P convergent for every i and that limi j∈σ xij exists for every σ ∈ F with P xj = limi xij for every j. Then limi j∈σ xij exists for every σ ⊂ N so conclusions (1),(2) and (3) of Theorem 7.17 hold. P Proof: We first claim that for each σ ⊂ N, the series j∈σ xj converges. P∞ We show that j=1 xj converges; the same argument can be applied to any P subseries j∈σ xj . Let U be a neighborhood of 0 in X and pick a closed, symmetric neighborhood of 0, V , such that V +V ⊂ U . By (C2) of Theorem P 7.10 there exists n such that j∈σ (xij − xj ) ∈ V for i ≥ n, σ ∈ F. By Pl Lemma 7.21 there exists m such that j=k xnj ∈ V for l ≥ k ≥ m. If l ≥ k ≥ m, then l X j=1

so

P∞

j=1

xj −

k X j=1

xj =

l X

j=k+1

(xj − xnj ) +

l X

j=k+1

xnj ∈ V + V ⊂ U

xj converges by the sequential completeness of X.

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P Since j∈σ (xij − xj ) ∈ V for i ≥ n and finite σ, Lemma 7.21 gives that j∈σ (xij − xj ) ∈ V for i ≥ n and any σ ⊂ N. Therefore, the hypothesis of Theorem 7.17 holds and (1), (2) and (3) of Theorem 7.17 follow.

P

We next give a scalar corollary of Theorem 7.22 due to Samaratanga and Sember ([SaSe]) which will be used later. Corollary 7.24. Let F be an FQσ family with Λ = {χσ : σ ∈ F} ⊂ m0 and set λ = spanΛ. Assume that ti ∈ l1 and limi t i · s exists for each s ∈ λ with tj = limi tij . Then t = {tj } ∈ l1 and ti − t 1 → 0. In particular, if ti → 0 in σ(l1 , λ), then ti → 0. 1

We next show that we can relax the hypothesis in Theorem 7.22 and retain part of the conclusion of the theorem. Recall that a family F of subsets of N is an IQσ family if F contains the finite subsets of N and whenever {Ik } is an increasing sequence of intervals there is a subsequence {Ink } such that ∪∞ k=1 Ink ∈ F (see Appendix B for examples). If F is an IQσ family and Λ = {χσ : σ ∈ F} ⊂ m0 , then Λ has SGHP so Theorem 7.10 applies and gives Theorem 7.25. Let F be an IQσ family which contains N, let Λ = {χσ : P σ ∈ F} ⊂ m0 and assume that j xij is Λ multiplier convergent for every P P i. If limi j∈σ xij exists for every σ ∈ F and xj = limi xij , then j xj is P∞ P∞ P∞ Λ multiplier convergent and limi j=1 xij = j=1 xj . [Note that j=1 xij converges since N ∈ F.] We use this result later in Chapter 9 when we consider iterated series. For the bounded multiplier case we have the following result as a special case of Theorem 7.10. P Theorem 7.26. Let be bounded multiplier convergent for every j xij P ∞ i ∈ N. Assume that limi ∞ t and limi xij = xj j=1 j xij exists for every t ∈ l for every j. Then P (1) j is bounded multiplier convergent, j xP P∞ ∞ (2) limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k∞ ≤ 1 and P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1.

Note that the scalar version of Theorem 7.25 gives the scalar version of the Schur Theorem stated in Theorem 7.1. From Proposition 7.12, we have

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P Proposition 7.27. Let j xij be bounded multiplier convergent for every i ∈ N and limi xij = xj for every j. P (1) If for every t ∈ l ∞ , the series ∞ j xij converge uniformly for i ∈ N, j=1 tP ∞ then for every t ∈ l the sequence { ∞ j=1 tj xij }i is Cauchy. P∞ t x converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, (2) If the series j ij j=1 P ∞ then the sequences { j=1 tj xij }i satisfy a Cauchy condition uniformly for k{tj }k∞ ≤ 1. From Corollary 7.15, we have P Corollary 7.28. Let X be sequentially complete. Let j xij be bounded multiplier convergent for every i ∈ N and limi xij = xj for every j. The following are equivalent: P ∞ (1) limi ∞ j=1 tj xij exists for every t ∈ l , P∞ P (2) = j xj is bounded multiplier convergent and limi j=1 tj xij P∞ t x uniformly for k{t }k ≤ 1, j j j j=1 ∞ P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, P ∞ (4) for every t ∈ l ∞ the series j=1 tj xij converge uniformly for i ∈ N. From Proposition 7.16, we have P Proposition 7.29. Let j xij be bounded multiplier convergent for every P∞ i ∈ N and limi xij = xj for every j. If the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, then ∞ X S= tj xij : i ∈ N, k{tj }k∞ ≤ 1 j=1

is bounded.

In a sequentially complete LCTVS we can obtain a stronger conclusion in Theorem 7.17 for the subseries convergent version of the Hahn-Schur Theorem. Recall that the inequality of McArthur/Rutherford (Lemma 2.53) implies that a series in a sequentially complete LCTVS is subseries convergent iff the series is bounded multiplier convergent (Theorem 2.54). Also, from the inequality, we obtain Theorem 7.30. Let X be a sequentially complete LCTVS. Assume that P subseries convergent for every j and limi xij = xj exists for every j xij is P j. If limi j∈σ xij exists for every σ ⊂ N, then

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P (1) j is bounded multiplier convergent, j xP P∞ ∞ (2) limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k∞ ≤ 1 and P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1.

Proof: The first statement in (1) follows from Theorems 7.17(1) and 2.54. For (2) let p be a continuous semi-norm on X. By Theorem 7.17(2), for P > 0 there exists n such that p( j∈σ (xij − xj )) < for i ≥ N, σ ⊂ N. If {tj } ∈ l∞ and i ≥ n, then from Lemma 2.53 for σ ⊂ N finite X X p( tj (xij − xj )) ≤ 2 k{tj }k∞ sup p( (xij − xj )) ≤ k{tj }k∞ . σ 0 ⊂σ

j∈σ

j∈σ 0

Therefore, if i ≥ n, then p(

∞ X j=1

tj (xij − xj )) ≤ k{tj }k∞

and (2) follows. (3) follows from the McArthur/Rutherford inequality in a similar fashion. Theorem 7.29 gives a generalization of the summability result stated in (S) following Theorem 7.2. Namely, we have: (S)’ If X is sequentially complete, the vector valued matrix [xij ] maps m0 into c(X), the space of X valued convergent sequences, iff [xij ] maps l∞ into c(X). Corollaries 7.19 and 7.27 give necessary and sufficient conditions for a vector valued matrix [xij ] to map m0 or l∞ into c(X) analogous to the scalar case (see [Sw2] 9.5.3). We next consider a generalization of the compactness result in Theorem 2.45. For this we require a preliminary lemma. Lemma 7.31. Let S be a compact Hausdorff space and gi : S → X continuous functions for i = 0, 1, 2, .... Suppose that lim gi (t) = g0 (t) uniformly for t ∈ S. Then R = ∪∞ i=0 Rgi is compact, where Rgi is the range of gi . Proof: Let G be an open cover of R. For each x ∈ R there exists Ux ∈ G such that x ∈ Ux . Then −x + Ux is an open neighborhood of 0 so there is an open neighborhood of 0, Vx , such that Vx + Vx ⊂ −x + Ux . Then G 0 = {x + Vx : x ∈ R} is an open cover of R.

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Since Rg0 is compact, there exist finite x1 + Vx1 , ..., xk + Vxk covering Rg0 . Put V = ∩kj=1 Vxj so V is an open neighborhood of 0. There exists n such that gi (t) − g0 (t) ∈ V for i ≥ n, t ∈ S. For t ∈ S there exists j such that g0 (t) ∈ xj + Vxj so gi (t) ∈ g0 (t) + V ⊂ xj + Vxj + Vxj ⊂ Uxj for i ≥ n. Hence, Ux1 , ..., Uxk covers ∪∞ i=n Rgi . Since Rgi , i = 0, ...n − 1, are compact, a finite subcover of G covers the union of these sets, and, hence, G has a finite subcover covering R. From Theorem 7.10, Corollary 2.43 and Lemma 7.30, we have Theorem 7.32. Assume that Λ ⊂ λ is bounded and has signed-SGHP and is compact with respect to p, the topology of pointwise convergence on Λ. If P x be Λ multiplier convergent in X for every i ∈ N. Assume (H) Let P∞ j ij that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N, P∞ P∞ then B = { j=1 tj xij : i ∈ N, t ∈ Λ} ∪ { j=1 tj xj : t ∈ Λ} is compact.

P Proof: Let Si (S0 ) be the summing operator with respect to j xij ( j xj ). If (C2) holds, then Si and S0 are continuous with respect to p and the topology of X (Corollary 2.43). If (C3) holds, then Si → S0 uniformly on Λ so it follows from Lemma 7.30 that B is compact. P

In particular, if Λ = {t ∈ l ∞ : k{tj }k∞ ≤ 1} ⊂ λ = l∞ or if Λ = {χσ : σ ⊂ N} ⊂ λ = m0 , Theorem 7.31 applies. We next consider another property of the multiplier space which implies vector versions of the Hahn-Schur Theorem. Definition 7.33. Let λ be a K-space. The multiplier space λ has the Hahn-Schur property if si ∈ λβ and si · t → 0 for every t ∈ λ implies that lim si · t = 0 uniformly for t belonging to bounded subsets of λ. From Corollary 7.23 we have the following example of a multiplier space with the Hahn-Schur property. Example 7.34. Let F be an FQσ family with Λ = {χσ : σ ∈ F} ⊂ m0 and set λ = spanΛ. Then λ with the sup-norm has the Hahn-Schur property. For multiplier spaces with the Hahn-Schur property we have a vector Hahn-Schur Theorem.

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Theorem 7.35. Assume that λ has the Hahn-Schur property. If {xk } ⊂ λβX is such that lim xk · t = 0 for every t ∈ λ, then lim xk · t = 0 uniformly for t belonging to bounded subsets of λ. Proof: Let B ⊂ λ be bounded. Suppose that the conclusion of the theorem fails for B. Then there exist δ > 0, a continuous semi-norm p on X, tk ∈ B and an increasing sequence {nk } such that p(xnk · tk ) > δ.

By the Hahn-Banach Theorem, for every k there exist x0k ∈ X 0 such that sup{|hx0k , xi| : p(x) ≤ 1} ≤ 1 and * + ∞ X 0 k nk (∗) xk , t j xj > δ. j=1

Since p(

1,

P∞

nk j=1 tj xj )

→ 0 for every t ∈ λ and sup{|hx0k , xi| : p(x) ≤ 1} ≤ lim k

∞ X j=1

tj x0k , xnj k = 0

uniformly for t belonging to bounded subsets of λ by the Hahn-Schur property. This contradicts (∗). Corollary 7.36. Assume that λ has the Hahn-Schur property, (λβX , ω(λβX , λ)) is sequentially complete and {xk } ⊂ λβX . If lim xk · t exists for every t ∈ λ and xj = limk xkj , then x ∈ λβX and lim xk · t = x · t uniformly for t belonging to bounded subsets of λ. Proof: By the sequential completeness assumption, there exists y ∈ λβX such that xk → y in ω(λβX , λ). Since xkj → yj for every j, y = {yj } = {xj } = x. If t ∈ λ, then xk · t → x · t so the result follows from Theorem 7.34. We now consider some Hahn-Schur type results for normed spaces due to the Spanish school in Cadiz ([AP1])). Let X be a Banach space. The space X is a Grothendieck space if every weak* convergent sequence in X 0 is weakly convergent. For example, the space l∞ is a Grothendieck space by Phillips’ Lemma (Lemma 7.52 or [Sw2] 15.16). Definition 7.37. Let M be a subspace of X 0 such that X ⊂ M ⊂ X 00 . Then X is an M Grothendieck space if every σ(X 0 , X) convergent sequence in X 0 is σ(X 0 , M ) convergent.

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Thus, X is a Grothendieck space iff X is an X 00 Grothendieck space. There are closed subspaces λ of l ∞ with λ 6= l∞ such that λ is a Grothendieck space not containing a subspace isomorphic to l ∞ . For example, if H is the algebra of Hayden (Appendix B.21), and if H is the Stone space of H , then C(H) can be isometrically identified with a closed subspace of l∞ such that c0 ⊂ C(H) and which is a Grothendieck space not containing a copy of l ∞ (see [AP1]). When X is a Banach space and c0 ⊂ λ ⊂ l∞ , we define a norm on λβX by 0

kxk = sup{kt · xk : t ∈ λ, ktk∞ ≤ 1}. 0

Note that k·k is finite by Proposition 3.8 since for any x = {xi } ∈ λβX , P the series j xj is c0 multiplier convergent. Note that we also have 0

kxk = sup{kt · xk : t ∈ c0 , ktk∞ ≤ 1}

= sup{kt · xk : t ∈ c00 , ktk∞ ≤ 1} = sup{

∞ X j=1

|hx0 , xj i| : kx0 k ≤ 1}.

Theorem 7.38. Let λ be a subspace of l ∞ containing c0 which is an l∞ P Grothendieck space. Let j xij be λ multiplier convergent for every i. If P∞ lim t x exists for every t ∈ λ and xj = limi xij for every j, then P i j=1 j ij P∞ P∞ x is λ multiplier convergent and limi j=1 tj xij = j=1 tj xj for every j j t ∈ λ. 0

Proof: Put xi = {xij }j ∈ λβX and x = {xj }. We claim that {xi } is k·k Cauchy in λβX . If not, there exist a subsequence

{n

0k } and δ > 0 such that 0 kxnk+1 − xnk k > δ . Put z k = xnk+1 − xnk so z k > δ and z k · t → 0 for every t ∈ λ. For each k pick x0k ∈ X 0 such that kx0k k ≤ 1 and (∗)

∞ X 0 k xk , zj > δ. j=1

The series j zjk is λ multiplier convergent so let Sk be the summing opP∞ P erator with respect to the series j zjk , Sk t = j=1 tj zjk for t ∈ λ (Theorem E D P∞ 2.2). Then x0k Sk ∈ λ0 and x0k Sk (t) = x0k , j=1 tj zjk → 0 for t ∈ λ. That P

space, is, {x0k Sk } is σ(λ0 , λ) convergent to 0. Since λ is an l ∞DGrothendieck E P∞ 0 0 ∞ 0 0 k {xk Sk } is σ(λ , l ) convergent to 0. Thus, xk Sk (t) = xk , j=1 tj zj → 0

for t ∈ l∞ . By the classical Hahn-Schur Theorem 7.1, { x0k , zjk }j → 0 in P∞ 0 k k·k1 or j=1 xk , zj → 0. This contradicts (∗).

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xj is λ multiplier convergent and xi · t → x · t for

0 every t ∈ λ. It will then follow that xi − x → 0 and the result follows. Let > 0 and

t ∈ λ. There exists N such that k, l ≥ N implies

P n k l

j=m tj (xj − xj ) < for all n > m by the part above. Let k → ∞ to obtain We claim that

P

j

X

n l t (x − x ) (∗∗) j j j ≤ for all n > m, l ≥ N.

j=m P P This shows that the series j tj (xj −xN j ) is Cauchy so j tj xj is Cauchy and, therefore, convergent since X is complete. Condition (∗∗) also shows that xl · t → x · t as desired. Example 7.11 shows that even when the multiplier space λ has the 0GHP and the signed-WGHP, the summing operators in Theorem 7.37 may not converge uniformly on bounded subsets of λ. However, we show that in this case we do have uniform convergence on null sequences. P Theorem 7.39. Assume that λ has 0-GHP and signed-WGHP. Let j xij P∞ be λ multiplier convergent for every i. If limi j=1 tj xij exists for every t ∈ λ and limi xij = xj for every j, then P∞ P (C1) the series j xj is λ multiplier convergent and limi j=1 tj xij = P∞ t x and j j j=1 P∞ P∞ (C4) if tk → 0 in λ, then limi j=1 tkj xij = j=1 tkj xj uniformly for k ∈ N. Proof: (C1) follows from Theorem 7.5. For (C4) let U be a neighborhood of 0 in X and pick a neighborhood of 0, V , such that V + V ⊂ U . By Theorem 2.39 there exists n such that ∞ X

j=n+1

tkj (xij − xj ) ∈ V

for i, k ∈ N. Since limi (xij − xj ) = 0 for every j and {tkj : k ∈ N} is bounded for every j, then for every j limi tkj (xij − xj ) = 0 uniformly for k ∈ N by Lemma 7.7. Therefore, there exists m such that i ≥ m implies Pn k j=1 tj (xij − xj ) ∈ V for every k ∈ N. If i ≥ m, then ∞ X j=1

tkj (xij

− xj ) =

n X j=1

tkj (xij

− xj ) +

∞ X

j=n+1

tkj (xij − xj ) ∈ V + V ⊂ U

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and (C4) follows. As noted earlier Example 7.11 shows that conclusion (C4) cannot be improved to uniform convergence on bounded subsets of λ (conclusion (C2)). The following example shows that the signed-WGHP in Theorem 7.38 cannot be dropped even in the presence of 0-GHP. P Example 7.40. Let λ = c and X = R. Then is λ multij δij plier convergent for every i and limi δij = 0 for every j. If t ∈ c, then P∞ P∞ limi j=1 tj δij = limi ti exists. However, if tk = ek , then j=1 tkj δij = 1 so (C4) fails. Note that λ = c has 0-GHP but not signed-WGHP. We now establish a Hahn-Schur Theorem in the spirit of Li’s Lemma 3.29. These theorems are useful in treating operator valued series with vector valued multipliers. Let Ω be a non-empty set and G be an Abelian topological group. Let fij : Ω → G for i, j ∈ N and assume that Ω has a distinguished element w0 such that fij (w0 ) = 0 for every i, j. P∞ Theorem 7.41. Assume that the series j=1 fij (wj ) converges for every P∞ i and every sequence {wj } ⊂ Ω and that limi j=1 fij (wj ) exists for every sequence {wj } ⊂ Ω. Then (1) limi fij (w) = fj (w) exists for every w ∈ Ω, j ∈ N, P∞ (2) the series j=1 fij (wj ) converge uniformly for i ∈ N and all sequences {wj } ⊂ Ω, P∞ P∞ (3) limi j=1 fij (wj ) = j=1 fj (wj ) for every sequence {wj } ⊂ Ω.

Proof: Let w ∈ Ω and j ∈ N. Define a sequence in Ω by wj = w and P∞ wi = w0 for i 6= j. Then fij (wj ) = 0 if i 6= j so limi j=1 fij (wj ) = limi fij (w) exists by hypothesis and (1) holds. P We first show that for each {wj } the series ∞ j=1 fij (wj ) converge uniformly for i ∈ N. If this fails to hold, there exists a neighborhood of 0, U , in G such that

(∗) for every k there exist p > k and q such that

∞ X j=p

Hence, there exist n1 > 1, i1 such that ∞ X

j=n1

fi1 j (wj ) ∈ / U.

fqj (wj ) ∈ / U.

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Pick a neighborhood of 0, V , such that V + V ⊂ U . There exists m1 > n1 such that ∞ X

j=m1 +1

Hence,

m1 X

fi1 j (wj ) ∈ V.

fi1 j (wj ) ∈ / V.

j=n1

P∞ Since j=1 fij (wj ) converge for i = 1, ..., i1 by (∗) there exist n2 > m1 , i2 > i1 such that ∞ X fi2 j (wj ) ∈ /U j=n2

and as above there exists m2 > n2 such that m2 X

j=n2

fi2 j (wj ) ∈ / V.

Continuing this construction produces increasing sequences {ip }, {mp } and {np } with np+1 > mp > np such that (∗∗)

mp X

j=np

fip j (wj ) ∈ / V.

Now consider the matrix M = [mpq ] = [

mq X

fip j (wj )].

j=nq

We claim that M is a K-matrix (Appendix D.2). The columns of M converge by (1). If {kq } is an increasing sequence, set vj = wj if nkq ≤ j ≤ mkq and vj = w0 otherwise. Then lim p

∞ X q=1

mpkq = lim p

∞ X

fip j (vj )

j=1

exists by hypothesis. Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗∗).

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If (2) fails to hold, then as above there exist increasing sequences {ik }, {mk } and {nk } with nk < mk < nk+1 , a matrix {wij } ⊂ Ω and a neighborhood, V , with mk X (∗ ∗ ∗) fik j (wkj ) ∈ / V. j=nk

Now define a sequence {wj } ⊂ Ω by wj = wkj if nk ≤ j ≤ mk and P∞ wj = w0 otherwise. But, then the series j=1 fij (wj ) do not satisfy the Cauchy condition uniformly for i ∈ N by (∗ ∗ ∗) and, therefore, violates the condition established above. For (3), let U be a neighborhood of 0 and {wj } ⊂ Ω. Pick a neighP borhood of 0,V , such that V + V + V ⊂ U . Put g = limi ∞ j=1 fij (wj ). P∞ We show that the series j=1 fj (wj ) converges to g. By (2) there exists P n such that ∞ ≥ n and i ∈ N. Suppose m > n. j=m fij (wj ) ∈ V for m Pn Then by (1) there exists i such that j=1 (fij (wj ) − fj (wj )) ∈ V and g− P∞ f (w ) ∈ V . So ij j j=1 g−

n X j=1

fj (wj ) = g −

∞ X

fij (wj ) +

∞ X

fij (wj ) +

j=1

j=n+1

j=1

∈ V + V + V ⊂ U.

n X

(fij (wj ) − fj (wj ))

Concerning the converse of Theorem 7.40, we have P∞ Theorem 7.42. Assume that the series j=1 fij (wj ) converges for every i and every sequence {wj } ⊂ Ω and that limi fij (w) = fj (w) exists for every P∞ j and w ∈ Ω. If for every {wj } ⊂ Ω the series j=1 fij (wj ) converge P∞ uniformly for i ∈ N, then { j=1 fij (wj )}i is Cauchy. If G is sequentially complete, then the stronger conclusion (2) of Theorem 7.40 holds. Proof: Let {wj } ⊂ Ω and let U be a neighborhood of 0. Pick a symmetric neighborhood of 0, V , such that V +V +V ⊂ U . By hypothesis there exP∞ ists n such that j=n fij (wj ) ∈ V for all i. Since limi fij (w) = fj (w) exists Pn−1 for every j and w ∈ Ω there exists m such that j=1 (fij (wj )−fkj (wj )) ∈ V for all i, k ≥ m. Then for all i, k ≥ m, ∞ X j=1

fij (wj ) − =

n−1 X j=1

∞ X

fkj (wj )

j=1

(fij (wj ) − fkj (wj )) +

∈ V + V + V ⊂ U.

∞ X

j=n

fij (wj ) −

∞ X

j=n

fkj (wj )

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The last statement follows from Theorem 7.40. Under stronger assumptions we establish a stronger convergence conclusion than condition (3) in Theorem 7.40. P∞ Theorem 7.43. Assume that the series j=1 fij (wj ) converges for every i and every sequence {wj } ⊂ Ω. If for each j ∈ N limi fij (w) = fj (w) P∞ converges uniformly for w ∈ Ω and if the series j=1 fij (wj ) converge uniformly for all sequences {wj } ⊂ Ω and i ∈ N, then the sequences P∞ { j=1 fij (wj )}i satisfy a Cauchy condition uniformly for all sequences P∞ {wj } ⊂ Ω. If G is sequentially complete, then limi j=1 fij (wj ) = P∞ j=1 fj (wj ) uniformly for all sequences {wj } ⊂ Ω.

Proof: Let U be a closed neighborhood of 0 in G and pick a symmetric neighborhood of 0,V , such that V + V + V ⊂ U . P∞ There exists n such that fij (wj ) ∈ V for all {wj } ⊂ Ω and i ∈ N. Pn−1j=n There exists m such that j=1 (fij (w) − fkj (w)) ∈ V for all i, k ≥ m and w ∈ Ω by the uniform convergence assumption. Hence, if i, k ≥ m and {wj } ⊂ Ω, we have (∗)

∞ X j=1

n−1 X j=1

(fij (wj ) − fkj (wj )) +

∞ X

j=n

fij (wj ) − fij (wj ) −

∞ X

fkj (wj ) =

j=1

∞ X

j=n

fkj (wj ) ∈ V + V + V ⊂ U

so the first part of the statement is established. P∞ If G is sequentially complete, then limi j=1 fij (wj ) exists by (∗). The last statement then follows from (3) of Theorem 7.40 and (∗) above. We can also obtain a version of Lemma 2.42. Proposition 7.44. Let Ω be a topological space with gj : Ω → G continuous P and assume that the series ∞ ) converges for every {wj } ⊂ Ω. If j=1 gj (wjP F : ΩN → G is defined by F ({wj }) = ∞ j=1 gj (wj ), then F is continuous with respect to the product topology. Proof: Let wk = {wjk } be a net in ΩN which converges to w = {wj } in the product topology. Let U be a neighborhood of 0 in G and pick a symmetric neighborhood,V , such that V + V + V ⊂ U . By Lemma 3.29

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P there exists n such that ∞ gj (vj ) ∈ V for all {vj } ⊂ Ω. There exists k0 Pj=n n−1 such that k ≥ k0 implies j=1 (gj (wjk ) − gj (wj )) ∈ V . If k ≥ k0 , then F (wk ) − F (w) =

n−1 X j=1

(gj (wjk ) − gj (wj )) +

Thus, F is continuous.

∞ X

j=n

gj (wjk ) −

∞ X

j=n

gj (wj ) ∈ V + V + V ⊂ U.

From Proposition 7.43, we have Corollary 7.45. Let Ω be a compact topological space with gj : Ω → G P∞ continuous and assume that the series j=1 gj (wj ) converges for every {wj } ⊂ Ω.Then ∞ X gj (wj ) : {wj } ⊂ Ω S= j=1

is compact.

From Theorem 7.42, Lemma 7.30 and Proposition 7.43, we also obtain

Corollary 7.46. Let Ω be a compact topological space. Assume that each P∞ fij is continuous, the series j=1 fij (wj ) converge uniformly for {wj } ⊂ Ω and i ∈ N and for each j ∈ N limi fij (w) = fj (w) converges uniformly for w ∈ Ω, then ∞ X S= fij (wj ) : {wj } ⊂ Ω, i ∈ N j=1

is compact.

Proof: As in Proposition 7.43 define Fi : ΩN → G (F0 : ΩN → G) by P∞ P∞ Fi ({wj }) = j=1 fij (wj ) (F0 ({wj }) = j=1 fj (wj )). By Proposition 7.43 each Fi is continuous and by Theorem 7.42, Fi → F0 uniformly on ΩN . The result follows from Lemma 7.30. The results above cover the cases of subseries convergent series and bounded multiplier convergent series given in Theorems 7.17 and 7.25. In the subseries case, we take Ω = {0, 1} and in the bounded multiplier conP vergent case, we take Ω = [0, 1]. If j xij are the series in these statements, we define fij (t) = txij and take for the distinguished element w0 = 0. That limi fij (w) = fj (w) converges uniformly for w ∈ Ω follows from Lemma 7.7.

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Throughout the remainder of this chapter we give applications of the Hahn-Schur results to various topics in functional analysis and measure theory. The original proofs of the Orlicz-Pettis Theorem for normed spaces given by both Orlicz and Pettis used the version of the Schur Theorem stated in Theorem 7.1. We indicate how this version of the Orlicz-Pettis Theorem can easily be obtained from Hahn’s Theorem 7.2. P Theorem 7.47. Let X be a normed space. If the series j xj is subseries convergent in the weak topology of X, then the series is subseries convergent in the norm topology. Proof: By replacing X by the span of {xj : j ∈ N}, we

may

assume 0 0

x0 = 1 and that X is separable. For every j pick x ∈ X such that j j

0 xj , xj = kxj k. Since X is separable, {x0j } has a subsequence {x0nj } which P is weak* convergent to some x0 ∈ X 0 . We have limi j∈σ x0ni , xnj = E D P P x0 , j∈σ xnj for every σ ⊂ N, where j∈σ xnj is the weak sum of the P∞

series. By Theorem 7.2, the series j=1 x0ni , xnj converge uniformly for D E i ∈ N. In particular, x0nj , xnj = xnj → 0. Since the same argument

can be applied to any subsequence of {xj }, it follows that kxj k → 0 so P j xj is norm subseries convergent by Lemma 4.4. We next indicate several applications of the Hahn-Schur results to topics in vector valued measure theory. Let Σ be a σ-algebra of subsets of a set S. P ∞ A set function µ : Σ → X is countably additive if ∞ j=1 µ(Aj ) = µ(∪j=1 Aj ) for every pairwise disjoint sequence {Aj } ⊂ Σ. A family of countably additive set functions {µa : a ∈ I} is uniformly countably additive if for P∞ every pairwise disjoint sequence {Aj } ⊂ Σ, the series j=1 µa (Aj ) converge uniformly for a ∈ I. We have the following result due to Nikodym. Theorem 7.48. (Nikodym Convergence Theorem) Let µj : Σ → X be countably additive for every j ∈ N. If lim µj (A) = µ(A) exists for every A ∈ Σ , then (1) µ is countably additive and (2) {µj } is uniformly countably additive. Proof: Let {Aj } ⊂ Σ be pairwise disjoint. For any σ ⊂ N, we have X µi (Aj ) = µi (∪j∈σ Aj ) → µ(∪j∈σ Aj ). j∈σ

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P By Theorem 7.17 it follows that the series ∞ j=1 µi (Aj ) converge uniformly for i ∈ N so {µj } is uniformly countably additive. Also, from Theorem 7.17, ∞ X

µ(Aj ) = lim i

j=1

∞ X j=1

∞ µi (Aj ) = lim µi (∪∞ j=1 Aj ) = µ(∪j=1 Aj ) i

so µ is countably additive. A theorem closely related to the Nikodym Convergence Theorem is the Vitali-Hahn-Saks Theorem which we now derive. Let ν : Σ → [0, ∞] be a measure. If µ : Σ → X is countably additive, then µ is ν continuous if limν(A)→0 µ(A) = 0. If {µj } is a sequence of countably additive set functions, then {µj } is uniformly ν continuous if limν(A)→0 µj (A) = 0 uniformly for j ∈ N. We have the following result which connects uniform countable additivity and uniform ν continuity. Theorem 7.49. Let {µj } be countably additive, µj : Σ → X, such that each µj is ν continuous. If {µj } is uniformly countably additive, then {µj } is uniformly ν continuous. Proof: If the conclusion fails to hold, there exists a neighborhood, U , of 0 such that for every δ > 0 there exist k ∈ N, E ∈ Σ such that µk (E) ∈ / U and ν(E) < δ. In particular, there exists E1 ∈ Σ, n1 such that µn1 (E1 ) ∈ / U and ν(E1 ) < 1. Pick a neighborhood of 0, V , such that V + V ⊂ U . There exists δ1 > 0 such that µn1 (E) ∈ V when ν(E) < δ1 . There exist / U and ν(E2 ) < δ1 /2. Continuing this E2 ∈ Σ, n2 > n1 such that µn2 (E2 ) ∈ construction produces sequences {Ek } ⊂ Σ, δk+1 < δk /2 , {nk } such that / U, ν(Ek+1 ) < δk /2 and µnk (E) ∈ V when ν(E) < δk . Note that µnk (Ek ) ∈ ν(∪∞ j=k+1 Ej ) ≤

∞ X

ν(Ej ) < δk /2 + δk+1 /2 + ... < δk /2 + δk /22 + ... = δk

j=k+1

so that µnk (Ek ∩ ∪∞ j=k+1 Ej ) ∈ V.

Now set Ak = Ek \ ∪∞ j=k+1 Ej . The {Ak } are pairwise disjoint and / V. µnk (Ak ) = µnk (Ek ) − µnk (Ek ∩∞ j=k+1 Ej ) ∈

However, by the uniform countable additivity of {µj } we have limk µj (Ak ) = 0 uniformly for j ∈ N. This gives the desired contradiction.

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From the Nikodym Convergence Theorem, we can now obtain the VitaliHahn-Saks Theorem. Theorem 7.50. (Vitali-Hahn-Saks) Let {µj } be countably additive, µj : Σ → X, such that each µj is ν continuous. If lim µj (A) = µ(A) exists for every A ∈ Σ, then (1) {µj } is uniformly ν continuous and (2) µ is countably additive and ν continuous. Proof: The result is an immediate consequence of Theorems 7.47 and 7.48. As noted earlier there is a notion between the concepts of finite additivity and countable additivity called strong boundedness. If µ : Σ → X is finitely additive, then µ is strongly bounded (strongly additive, exhaustive) if µ(Aj ) → 0 whenever {Aj } is a pairwise disjoint sequence from Σ (3.36). A family {µa : a ∈ A} of finitely additive set functions is uniformly strongly bounded (strongly additive, exhaustive) if whenever {Aj } is a pairwise disjoint sequence from Σ, limj µa (Aj ) = 0 uniformly for a ∈ A. We have the analogue of Theorem 3.42 for uniformly strong bounded set functions. Theorem 7.51. For a ∈ A let µa : Σ → X be finitely additive. The following are equivalent: (i) {µa : a ∈ A} is uniformly strongly bounded, P∞ (ii) for any pairwise disjoint sequence {Aj } ⊂ Σ the series j=1 µa (Aj ) satisfy a Cauchy condition uniformly for a ∈ A. Proof: Clearly (ii) implies (i). If (ii) fails to hold, there exist a neighborhood of 0,U , an increasing P / sequence of intervals {Ik } and a sequence ak ∈ A such that j∈Ik µak (Aj ) ∈ / U so U . Set Bk = ∪j∈Ik Aj . Then {Bk } is pairwise disjoint and µak (Bj ) ∈ (i) fails. We now establish a version of the Nikodym Convergence Theorem for strongly bounded set functions. Theorem 7.52. Let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (A) = µ(A) exists for every A ∈ Σ, then (1) µ is strongly bounded and (2) {µi } is uniformly strongly bounded.

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Proof: Suppose that (2) fails. Then there exist a pairwise disjoint sequence {Aj } and a neighborhood of 0, U , such that for every k there exist jk > k, ik such that µik (Ajk ) ∈ / U . For k = 1 there exist j1 > 1, i1 such that / U . There exist J1 > j1 such that µi (Aj ) ∈ U for 1≤ i ≤ i1 and µi1 (Aj1 ) ∈ j ≥ J1 . For k = J1 there exist j2 > J1 and i2 such that µi2 (Aj2 ) ∈ / U. Note that i2 > i1 . Continuing this construction produces increasing sequences {jk }, {ik } such that / U. (∗) µik (Ajk ) ∈

By Drewnowski’s Lemma (Appendix E.2) there is a subsequence {nk } such that each µik is countably additive on the σ-algebra Σ0 generated by the {Ajnk }. By the Nikodym Convergence Theorem 7.47, {µink } is uniformly countably additive on Σ0 . In particular, lim µink (Ajnk ) = 0. This contradicts (∗). (1) follows from (2) since if {Aj } ⊂ Σ is pairwise disjoint, lim µ(Aj ) = lim lim µi (Aj ) = lim lim µi (Aj ) = 0 j

j

i

i

j

by the uniform convergence of limj µi (Aj ) = 0. We next establish a vector version of a lemma due to Phillips which he used to show that there is no continuous projection from l ∞ onto c0 . We first state the scalar version of Phillips’ Lemma. Let ba be the space of all finitely additive, bounded real valued set functions defined on 2N equipped with the variation norm, kνk = var(ν)(N). ba equipped with this norm is the dual space of (l ∞ , k·k∞ ) ([DS] IV.5.1, [SW3] 6.3). If j ∈ N, we write ν({j}) = ν(j) for ν ∈ ba. Lemma 7.53. (Phillips) Let νk ∈ ba for every k ∈ N and suppose that P∞ lim νk (E) = 0 for every E ⊂ N. Then limk j=1 |νk (j)| = 0. Phillips’ Lemma has the following duality interpretation. Let J be the canonical imbedding of c0 into its bidual l∞ . Then the transpose operator J 0 : (l∞ )0 = ba → (c0 )0 = l1 is given by J 0 ν = {ν(j)}. Phillips’ Lemma asserts that if {νi } converges to 0 in the weak topology σ(ba, m0 ), then {J 0 νi } converges to 0 in k·k1 . In particular, if {νi } converges to 0 in the weak* topology σ(ba, l ∞ ), then kJ 0 νi k1 → 0. This also shows that l ∞ is a Grothendieck space. We next show how Phillips’ Lemma can be used to show that there is no continuous projection of l ∞ onto c0 . Theorem 7.54. There is no continuous projection of l ∞ onto c0 .

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Proof: If P were such a projection, then for y ∈ l ∞ , P y ∈ c0 so ek ·P y = P e · y → 0. Hence, P 0 ek → 0 weak* in ba. By the observation above,

0 0 k

J P e = sup{ J 0 P 0 ek · x : x ∈ c0 , kxk ≤ 1} ∞ 1 = sup{ ek · P Jx : x ∈ c0 , kxk∞ ≤ 1}

= sup{ ek · x : x ∈ c0 , kxk ≤ 1} = ek = 1 → 0 0 k

∞

1

an obvious contradiction.

We now establish a vector version of Phillips’ Lemma which yields Lemma 7.52 as a special case. Theorem 7.55. Let X be sequentially complete and let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (E) = 0 for every E ∈ Σ, then P for every pairwise disjoint sequence {Ej } from Σ, limi j∈σ µi (Ej ) = 0 uniformly for σ ⊂ N. P Proof: By Theorem 7.17 it suffices to show that lim i j∈σ µi (Ej ) = 0 for every σ ⊂ N. If this fails to hold, we may assume, by passing to a subsequence if necessary, that there exists a closed neighborhood of 0, U , P∞ such that j=1 µi (Ej ) ∈ / U for every i. Pick a symmetric neighborhood of P n1 0, V , such that V + V ⊂ U . There exists n1 such that j=1 µ1 (Ej ) ∈ / U. Pn1 µ (E ) ∈ V for i ≥ m . There exists There exists m1 such that i j 1 j=1 Pn2 n2 > n1 such that j=1 µm1 (Ej ) ∈ / U . Hence, n2 X

µm1 (Ej ) =

j=n1 +1

n2 X j=1

µm1 (Ej ) −

n1 X j=1

µm1 (Ej ) ∈ / V.

Continuing this construction produces increasing sequences {mi }, {ni } Pni+1 ni+1 such that j=n µmi (Ej ) ∈ / V . Set Fi = ∪j=n Ej so i +1 i +1 and {Fi } is pairwise disjoint. Consider the matrix

/V (∗) µmi (Fi ) ∈

M = [mij ] = [µmi (Fj )]. We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 by hypothesis. If {rj } is an increasing sequence, by Drewnowski’s Lemma (Appendix E.2), there is a subsequence {sj } of {rj } such that each µi is countably additive on the σ-algebra generated by {Fsj }. Thus, ∞ X µmi (Fsj ) = µmi (∪∞ j=1 Fsj ) → 0. j=1

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Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). We indicate how Theorem 7.54 yields Phillips’ Lemma 7.52 and, therefore, can be viewed as a vector version of Phillips’ Lemma. Let > 0 and let the notation be as in Lemma 7.52. By Theorem 7.54 there exists P N such that j∈σ νi (j) < for every σ ⊂ N, i ≥ N . By Lemma 3.37, P∞ j=1 |νi (j)| ≤ 2 for i ≥ N. We next give a generalization of Theorem 7.54 which in turn yields a generalization of Phillips’ Lemma 7.52. Theorem 7.56. Let X be sequentially complete and let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (E) = µ(E) exists for every E ∈ P Σ, then for every pairwise disjoint sequence {Ej } ⊂ Σ, lim j∈σ µi (Ej ) = P j∈σ µ(Ej ) uniformly for σ ⊂ N. [In particular, µ is strongly bounded.] P Proof: By Theorem 7.17 it suffices to show that limi j∈σ µi (Ej ) exists for every σ ⊂ N. Since this is trivial for finite σ, assume that σ is infinite P with σ = {m1 < m2 < ...}. We claim that { ∞ j=1 µi (Emj )} is a Cauchy sequence in X. For this, assume that {pi } and {qi } are increasing sequences with pi < qi < pi+1 . Then lim(µpi (E) − µqi (E)) = 0 for every E ∈ Σ so by P∞ P∞ Theorem 7.54, limi j=1 (µpi (Emj ) − µqi (Emj )) = 0 so { j=1 µi (Emj )} is a Cauchy sequence and the result follows. The scalar case of Theorem 7.55 gives an improvement to Phillips’ Lemma 7.52. In particular, as in the proof of Phillips’ Lemma from Theorem 7.54 indicated above, we have Corollary 7.57. Let νk ∈ ba for every k ∈ N and suppose that lim νk (E) = P∞ ν(E) exists for every E ⊂ N. Then ν ∈ ba and j=1 |νk (j) − ν(j)| → 0. We show that Theorem 7.55 can be used to derive a version of the Nikodym Boundedness Theorem for strongly bounded set functions.

Theorem 7.58. (Nikodym Boundedness Theorem) Let νi : Σ → R be bounded and finitely additive. If {νi (E) : i ∈ N} is bounded for every E ∈ Σ, then {νi (E) : i ∈ N, E ∈ Σ} is bounded. Proof: Let {Ej } ⊂ Σ be pairwise disjoint. By Lemma 4.58 it suffices to show that {νi (Ei ) : i ∈ N} is bounded.

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Let ti → 0 in R. Then limi ti νi (E) = 0 for every E ∈ Σ. Since each ν ∈ ba is strongly bounded (Proposition 3.38) Theorem 7.55 implies that {ti νi } is uniformly strongly bounded so limi ti νi (Ei ) = 0. Therefore, {νi (Ei ) : i ∈ N} is bounded. From Theorem 7.57 and the Uniform Boundedness Principle, we can immediately obtain a version of the Nikodym Boundedness Theorem for LCTVS (see the proof of Corollary 4.60). Theorem 7.59. (Nikodym Boundedness Theorem) Let X be an LCTVS. Let νi : Σ → X be strongly bounded. If {νi (E) : i ∈ N} is bounded for every E ∈ Σ, then {νi (E) : i ∈ N, E ∈ Σ} is bounded. Recall the local convex assumption in Theorem 7.58 cannot be dropped; see Remark 4.61. We can use the version of the Nikodym Boundedness Theorem in Theorem 7.57 to show that (m0 , k·k∞ ) is barrelled. More generally, let S(Σ) be the space of all real valued Σ simple functions equipped with the sup-norm. Then the dual of S(Σ) is the space ba(Σ) of all bounded, finitely additive, real valued set functions ν defined on Σ with the variation 0 norm, kνk = var(ν)(S); the pairing R between f ∈ S(Σ) and ν ∈ ba(Σ) is given by integration, hf, gi = S gdν, g ∈ S(Σ) [no elaborate integration theory is used since we are only integrating simple functions] (see [DS], [Sw3] 6.3). A norm equivalent to the variation norm is given by 0 kνk = sup{|ν(E)| : E ∈ Σ} ([DS], [Sw3] 2.2.1.7). From the Nikodym Boundedness Theorem, we have Theorem 7.60. (S(Σ), k·k∞ ) is barrelled. That is, if {νj } ⊂ ba(Σ) is σ(ba(Σ), S(Σ)) bounded, then {νj } is norm bounded. In particular, (m0 , k·k∞ ) is barrelled. Proof: Since {νj } is pointwise bounded on S(Σ), {νj (E)} is bounded for each E ∈ Σ. Theorem 7.57 then implies that {kνj k0 : j ∈ N} is bounded so {kνj k : j ∈ N} is bounded.

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Chapter 8

Spaces of Multiplier Convergent Series and Multipliers

In this chapter we consider topological properties of the space of λ multiplier convergent series. Throughout this chapter let λ be a sequence space containing c00 and let X be an LCTVS. Recall that λβX is the space of all X valued λ convergent series. If x = {xj } ∈ λβX and t = {tj } ∈ λ, P∞ we write x · t = j=1 tj xj . We define a locally convex topology on λβX induced by X and λ when λ is a K-space. Assume that λ is a K-space and let B = {B ⊂ λ : B is bounded and {x · t : t ∈ B} is bounded in X ∀ t ∈ λ}. Let X be the family of all continuous semi-norms on X. For B ∈ B and p ∈ X , define a semi-norm on λβX by pB (x) = sup{p(x · t) : t ∈ B}.

Let τB be the locally convex topology on λβX generated by the semi-norms pB for B ∈ B and p ∈ X . Remark 8.1. The family B is equal to the family of all bounded subsets of λ if either λ has 0-GHP (Corollary 2.12) or if λβ ⊂ λ0 (Corollary 2.4). If the maps t → x · t from λ into X are continuous for all x ∈ λβX , then λβX ⊂ L(λ, X) and the topology τB is just the relative topology from Lb (λ, X). We consider the sequential completeness of τB . For this let Pi be the Pi sectional operator on λ defined by Pi t = j=1 tj ej . Recall the following property from Appendix B.4. Definition 8.2. The space λ has the sections uniformly bounded property (SUB) if {Pi t : i ∈ N, t ∈ B} is bounded for every bounded subset B of λ. 133

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Theorem 8.3. Assume that X is sequentially complete, λ has SUB and for each y ∈ λβX the operator t → y · t from λ into X is bounded. Then τB is sequentially complete. Proof: Let {xk } be τB Cauchy, B ∈ B , p ∈ X and > 0. First, put t = ej . Then p{t} (xk − xl ) = p(xkj − xlj ) so {xkj }k is Cauchy in X for each j. Let xj = limk xkj . Set x = {xj }. Set B 0 = B ∪ {Pi B : i ∈ N} so B 0 is bounded by the SUB assumption. Also, B 0 ∈ B by the boundedness assumption of the maps t → y · t for y ∈ λβX . There exists N such that k, l ≥ N implies pB 0 (xk − xl ) < . Pn Thus, if k, l ≥ N and n > m, then p( i=m ti (xki − xli )) < for t ∈ B. Pn Hence, p( i=m ti (xki − xi )) ≤ for t ∈ B, k ≥ N, m > n. This implies that x ∈ λβX and pB (xk − x) ≤ for k ≥ N . Remark 8.4. The boundedness assumption on the maps t → y · t for y ∈ λβX means that B is equal to the family of all bounded subsets of λ. This condition is satisfied, for example, if λ has 0-GHP (Corollary 2.12) or if λβ ⊂ λ0 (Corollary 2.4). The completeness assumption in Theorem 8.3 is necessary. Proposition 8.5. If λβX is τB sequentially complete, then X is sequentially complete. Proof: Let {xk } be Cauchy in X. Put xkj = xk if j = 1 and xkj = 0 otherwise and set xk = {xkj }j . Then {xk } is τB Cauchy. Therefore, there exists y = {yj } ∈ λβX such that xk → y in τB . In particular, xk1 = xk → y1 in X. From the uniform convergence result in Theorem 2.16, we have Theorem 8.6. Let λ have signed-SGHP. Then for every x ∈ λβX , n X τB − lim xj ej = x, j=1

i.e., λ

βX

is a vector valued AK-space (Appendix C).

Recall that ω(λβX , λ) is the weakest topology on λβX such that the maps x → x · t from λβX into X are continuous for all t ∈ λ. From Corollary 2.28, we have Theorem 8.7. If λ has signed-WGHP and X is sequentially complete, then ω(λβX , λ) is sequentially complete.

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We now give conditions which guarantee sequential convergence in ω(λβX , λ). Proposition 8.8. Let {xk } ⊂ λβX . (1) If xk → 0 in ω(λβX , λ), then limk xkj = 0 for every j. P k (2) If limk xkj = 0 for every j and if for every t ∈ λ the series j t j xj k βX converge uniformly for k ∈ N, then x → 0 in ω(λ , λ). (3) If λ has signed-WGHP, the converse of (2) holds. Proof: (1) follows since limk xk · ej = limk xkj = 0. For (2), let t ∈ λ and consider (∗) xk · t =

n X j=1

tj xkj +

∞ X

tj xkj .

j=n+1

Let U be a neighborhood of 0 in X and pick a neighborhood of 0,V , such that V + V ⊂ U . By hypothesis, there exists n such that the last term in (∗) belongs to V for every k. Since limk xkj = 0 for every j, for large k the first term on the right hand side of (∗) belongs to V . Therefore, for large k, xk · t ∈ U . The statement in (3) follows from (1) and Theorem 2.26. One of the scalar versions of the Hahn-Schur theorem asserts that if the sequence {ti } in l1 is σ(l1 , l∞ ) convergent, then the sequence {ti } is k·k1 convergent. The vector version of the Hahn-Schur Theorem given in Theorem 7.10 can be given a similar interpretation. Theorem 8.9. Let λ have signed-SGHP and let X be sequentially complete. If {xk } is ω(λβX , λ) Cauchy, then there exists x ∈ λβX such that lim xk ·t = x · t uniformly for t belonging to bounded subsets of λ. The scalar version of Theorem 8.9 gives a generalization of the classical scalar version of the Hahn-Schur Theorem for l 1 described above. Corollary 8.10. Let λ have signed-SGHP. If {tk } is σ(λβ , λ) Cauchy, then there exists t ∈ λβ such that β(λβ , λ) − lim tk = t. For λ = l∞ , this is the Hahn-Schur result for l 1 described above. We next establish a Banach-Steinhaus equicontinuity type result for λβX . This will lead to another sequential completeness result for ω(λβX , λ).

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Theorem 8.11. Let λ have 0-GHP and {xk } ⊂ λβX . If lim xi · t = Lt exists for every t ∈ λ, then {xi } is sequentially equicontinuous, i.e., if tj → 0 in λ, then limj xi · tj = 0 uniformly for i ∈ N. Proof: If the conclusion fails to hold, we may assume that there exist > 0, p ∈ X and tj → 0 in λ such that p(xj · tj ) > for all j. Set m1 = 1 and pick n1 such that n1 X 1 m1 p( tm k xk ) > . k=1

limi xik

i

Note that = limi x · ek exists for every k so {xik }i is bounded for each k. Thus, since limi tik = 0 for every k, limi tik xik = 0 (Lemma 7.7) so there exists m2 > m1 such that n1 X 2 m2 p( tm k xk ) < /2. k=1

There exists n2 > n1 such that n2 X 2 m2 p( tm k xk ) > . k=1

Hence,

p(

n2 X

2 m2 tm k xk ) > /2.

k=n1 +1

Continuing this construction produces increasing sequences {mk }, {nk } such that X m m (∗) p( tk j xk j ) > /2 k∈Ij

for all j, where Ij = [nj−1 + 1, nj ]. Define a matrix M = [mij ] = [

X

m

i t k j xm k ].

k∈Ij

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge by the observation above. Next, if {pj } is an increasing sequence, by 0-GHP, there is a further subsequence, still denoted by {pj }, such that P∞ s = j=1 χIpj tmpj ∈ λ. Then ∞ X j=1

mipj =

∞ X X

j=1 k∈Ipj

mp j

tk

mi i xm ·s k =x

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so limi xmi · s = Ls exists. Thus, M is a K-matrix and by the AntosikMikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). We can now use Theorem 8.11 to obtain another sequential completeness result for ω(λβX , λ). Corollary 8.12. Let λ be an AK-space with 0-GHP and X be sequentially complete. If {xi } is ω(λβX , λ) Cauchy, then there exists x ∈ λβX such that lim xi · t = x · t for every t ∈ λ, i.e., ω(λβX , λ) is sequentially complete. Proof: If Lt = lim xi · t for t ∈ λ, then by Theorem 8.11, L : λ → λ is linear and sequentially continuous. Set xk = limi xi ·ek and x = {xk }. If t ∈ P P∞ P∞ j j j λ, by the AK-property, t = ∞ j=1 tj e so Lt = j=1 tj Le = j=1 tj x = x · t. We next establish a uniform boundedness result for λβX . Theorem 8.13. Let λ have 0-GHP. If Γ ⊂ λβX is pointwise bounded on λ, then Γ is uniformly bounded on bounded subsets of λ. Proof: Suppose the conclusion fails to hold. Then there exist p ∈ X , > 0, {xk } ⊂ Γ, a bounded sequence {tk } ⊂ λ and sk → 0, sk > 0, such that p(sk xk · tk ) >

for all k. Put k1 = 1 and pick m1 such that m1 X p(sk1 tkj 1 xkj 1 ) > . j=1

{tkj

Since λ is a K-space, : k ∈ N} is bounded for each j and {xkj : k ∈ N} is bounded for each j by hypothesis. Therefore, limk sk tkj xkj = 0 for every j (Lemma 7.7) so there exists k2 > k1 such that m1 X tkj 2 xkj 2 ) < /2. p(sk2 j=1

Pick m2 > m1 such that

p(sk2

m2 X

tkj 2 xkj 2 ) > .

j=1

Set I2 = [m1 + 1, m2 ] and note p(sk2

X

j∈I2

tkj 2 xkj 2 ) > /2.

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Continuing this construction produces an increasing sequence {kp } and an increasing sequence of intervals {Ip } such that X k k (∗) p(skp tj p xj p ) > /2 j∈Ip

for all p. Define a matrix

√ √ M = [mpq ] = [ skp xkp · skq χIq tkq ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M √ converge to 0 since {xk } is pointwise bounded on λ. Next, since sk tk → 0, by 0-GHP if {rq } is any subsequence, there is a further subsequence, still P∞ denoted by {rq }, such that t = q=1 √skrq χIrq tkrq ∈ λ. Hence, ∞ X q=1

mprq =

√ skp xkp · t → 0.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). Recall that a pair of vector spaces X, X 0 in duality is called a Banach Mackey pair if σ(X, X 0 ) bounded subsets are β(X, X 0 ) bounded. An LCTVS X is a Banach-Mackey space if X, X 0 form a Banach-Mackey pair ([Wi] 10.4.3). The scalar version of Theorem 8.13 has the following corollary. Corollary 8.14. Let λ have 0-GHP. (i) λ, λβ is a Banach-Mackey pair. (ii) If λ0 ⊂ λβ , then λ, λ0 is a Banach-Mackey pair. (iii) If λ0 = λβ and λ is quasi-barrelled, then λ is barrelled. Conditions which guarantee that the hypotheses in (i) and (ii) are satisfied are given in Proposition 2.5. We consider another uniform boundedness result which requires another type of gliding hump property (Appendix B.30 and B.31). Let µ be a sequence space containing c00 . Definition 8.15. The K-space λ has the strong µ gliding hump property (strong µ-GHP) if whenever {Ik } is an increasing sequence of intervals and {tk } is a bounded sequence in λ, then for every s ∈ µ the coordinate sum P∞ of the series j=1 sj χIj tj ∈ λ.

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Definition 8.16. The K-space λ has the weak µ gliding hump property (weak µ-GHP) if whenever {Ik } is an increasing sequence of intervals and {tk } is a bounded sequence in λ, there is a subsequence {nk } such that for P∞ every s ∈ µ the coordinate sum j=1 sj χInj tnj ∈ λ.

The elements s ∈ µ are called multipliers since their coordinates multiply the blocks {χIj tj } determined by the {Ik } and the {tk }. The signedWGHP and signed -SGHP are somewhat similar in that the ”humps” are multiplied by ±1 in these cases. Examples of spaces with the strong µ-GHP and weak µ-GHP are given in Appendix B. For example, any locally complete LCTVS has strong l 1 GHP, l∞ and c0 have strong c0 -GHP and (l2 , σ(l2 , l2 )) has strong l1 -GHP but not 0-GHP. We next establish a basic lemma. If A ⊂ λ and B ⊂ λβ , we write |B · A| = sup{|s · t| : s ∈ B, t ∈ A}. Lemma 8.17. Suppose A ⊂ λ is coordinate bounded and B ⊂ λβ is coordinate bounded. If |B · A| = ∞, then there exist an increasing sequence of intervals {Ik }, {tk } ⊂ A and {sk } ⊂ B such that k s · χ I t k > k 2 k for all k.

Proof: There exist sk ∈ B, tk ∈ A such that sk · tk > k 2 + k. Set Pn1 k1 k1 k1 = 1 and pick n1 such that j=1 sj tj > k12 + 1. By hypothesis for

every j, {skj : k ∈ N} and {tkj : k ∈ N} are bounded so there exists k2 > k1 Pn1 k2 k2 such that k12 j=1 sj tj < 1. Hence, n1 ∞ ∞ X k k X k k X k2 k2 2 2 ≥ s t s t sj 2 tj 2 > k22 . j j − j j j=1 j=1 j=n1 +1 P k2 k2 2 Pick n2 > n1 such that nj=n s t > k22 . Set I2 = [n1 + 1, n2 ] so j j +1 1 k s 2 · χI2 tk2 > k22 .

Now just continue and relabel.

Theorem 8.18. Let λ have weak µ-GHP. Assume (N) {ek : k ∈ N} is β(λ, λβ ) bounded in λ.

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If A ⊂ λ is bounded and B ⊂ λβ is σ(λβ , λ) bounded, then |B · A| < ∞. Proof: Suppose the conclusion fails and let the notation be as in Lemma 8.17. Let {nj } be as in Definition 8.16. Define a linear map T : µ → λ P∞ by T s = j=1 sj χInj tnj [coordinate sum]. We claim that T is σ(µ, µβ ) − σ(λ, λβ ) continuous. Let s ∈ µ, t ∈ λβ . Then (∗) t · T s =

∞ X j=1

sj (t · χInj tnj ).

Equation (∗) implies that {t · χInj tnj } ∈ λβ and t · T s = s · {t · χInj tnj } so T is σ(µ, µβ ) − σ(λ, λβ ) continuous. Hence, T is β(µ, µβ ) − β(λ, λβ ) continuous ([Wi] 11.2.6, [Sw2] 26.15). Thus, by (N), {T ej } = {χInj tnj } is β(λ, λβ ) bounded. But, this contradicts the conclusion of Lemma 8.17. Corollary 8.19. Under the hypothesis of Theorem 8.18, if λ0 ⊂ λβ , then λ is a Banach-Mackey space. Conditions for λ0 ⊂ λβ are given in Proposition 2.5. We next consider some results of the Spanish school in Cadiz which use series to characterize completeness and barrelledness of normed linear spaces ([AP2], [PBA]). For the remainder of this chapter let X be a normed space. Let {xj } ⊂ P X and let M ∞ ( xj )be the space of bounded multipliers for the series P xj : ∞ X X M ∞( xj ) = {tj } ∈ l∞ : tj xj converges . j=1

∞

We equip M ( xj )with the sup-norm from l ∞ . When X is a Banach space, we give necessary and sufficient conditions P P for the space M ∞ ( xj ) to be complete. Recall a series j xj in a normed P∞ 0 0 0 space is wuc iff j=1 |hx , xj i| < ∞ for all x ∈ X (see Definition 3.7). P Theorem 8.20. Let X be a Banach space and j xj be a series in X. P P Then M ∞ ( xj ) is complete iff x is wuc. j j Pn P Proof: Suppose j xj is wuc. Then E = { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded (Proposition 3.8) so let M > 0 be such that kxk ≤ M for all x ∈ E. P Let {tk } be a Cauchy sequence in M ∞ ( xj ) and let t ∈ l∞ be such that

tk − t → 0. Let > 0 and pick n such that ktn − tk < /2M . Since ∞ P∞ n ∞ j=1 tj xj is convergent, there exists N > n such that q > p ≥ N implies P

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P

P Pq

q

q n n (t − t )x ∈ E, (t − t )x

<

j=p tnj xj < /2. Since 2M j j j j j=p j j=p j

P

P

q

/2 so j=p tj xj < for q > p ≥ N . Hence, j tj xj converges since X P is a Banach space and t ∈ M ∞ ( xj ). P Assume that M ∞ ( xj ) is complete. Let t ∈ c0 . It suffices to show that P P P t ∈ M ∞ ( xj ) (Proposition 3.8). For each n, tn = nj=1 tj ej ∈ M ∞ ( xj ) P and ktn − tk∞ → 0 so t ∈ M ∞ ( xj ).

We can use Theorem 8.20 and the multiplier space M ∞ ( acterize completeness of X.

P

xj ) to char-

Theorem 8.21. The normed space X is complete iff for every wuc series P P ∞ xj ) is complete. j xj in X the space M (

Proof: Suppose that X is not complete. Then there exists a nonP∞ P = jxj . Then convergent series j xj with j=1 j kxj k < ∞. Set zjP P 0 0 0 0 does z is wuc since |hx , z i| ≤ kx k kz k for x ∈ X but j j (1/j)zjP j j Pj not converge. That is, {i/j} ∈ / M ∞ ( xj ). Since {i/j} ∈ c0 , M ∞ ( xj ) is not complete. The converse follows from Theorem 8.20. We can also characterize wuc series in terms of the ”summing operator” P P∞ P T : M ∞ ( xj ) → X defined by T t = T {tj } = j=1 tj xj , t ∈ M ∞ ( xj ) (Theorem 2.2). P Theorem 8.22. The summing operator T is continuous iff j xj is wuc. In this case,

n

X

: |tj | ≤ 1, n ∈ N . t x kT k = sup j j

j=1 P ∞ Proof: Suppose that T is continuous. Now c00 ⊂

PM ( x j ) and if

t ∈ c00 , ktk∞ ≤ 1 and ti = 0 for i ≥ n, then kT tk = nj=1 tj xj ≤ kT k .

P P

n Hence, { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded so j xj is wuc by Proposition 3.8.

P P

n Suppose that j xj is wuc and set M = sup{ j=1 tj xj : |tj | ≤ 1, n ∈ P N}. Let t ∈ M ∞ ( xj ), ktk∞ ≤ 1. For every n,

n

n X

X

j

T (

t e ) = t x j j j ≤ M

j=1

j=1

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P

∞ so j=1 tj xj = kT tk ≤ M and T is continuous with kT k ≤ M . The last statement follows from the computations above. We next consider weakly convergent series and the associated multiplier P spaces. Let j xj be a series in X. Define ∞ X X tj xj is weakly convergent . Mw∞ ( xj ) = {tj } ∈ l∞ : j=1

Again we supply Mw∞ ( xj ) with the sup-norm topology from l ∞ . From Proposition 3.8, we have P Lemma 8.23. Let X be a Banach space and j xj a series in X. Then P P ∞ xj ). j xj is wuc iff c0 ⊂ Mw ( P

We have the analogue of Theorem 8.20.

P Theorem 8.24. Let X be a Banach space and j xj a series in X. The P P space Mw∞ ( xj ) is complete iff j xj is wuc. P P Proof: Suppose that j xj is wuc. Let {tk } be a sequence in Mw∞ ( xj ) P∞ which converges to t ∈ l ∞ . Let zk = j=1 tkj xj , where this is the σ(X, X 0 ) Pn n ∈ N} and sup{kxk : sum of the series. Let E = { j=1 aj xj : |aj | ≤ 1, x ∈ E} < M . Let > 0. There exists n such that tk − t ∞ < /3M for k ≥ n. Then

m

X k

(∗) (t − t )x j j ≤ /3 for k ≥ n, m ∈ N j

j=1

so

X

m k

(t − tl )xj ≤ 2/3 for k, l ≥ n, m ∈ N. j j

j=1

Thus, kzk − zl k ≤ for k, l ≥ n. Let z = k·k − lim zk . P We claim that j tj xj is weakly convergent to z. There exists N > n 0 0 0 such P that kzk − zk < /3 for k ≥ N . If x ∈ X , kx k ≤ 1, then by (∗) m k j=1 (tj − tj ) hx0 , xj i ≤ /3 for k ≥ N, m ∈ N. If m ∈ N, then * * + + m m 0 X 0 X x, tj xj − z ≤ 2/3 + x , tN j xj − z N . j=1 j=1

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D P E m − z There exists N1 such that x0 , j=1 tN < /3 for m ≥ N1 . Hence, N j D P E 0 m m ≥ N1 implies that x , j=1 tj xj − z ≤ and the claim is established. P Suppose that Mw∞ ( xj ) is complete. Let t ∈ c0 . For every n, tn = Pn P P j ∞ xj ) so t = k·k − lim tn ∈ Mw∞ ( xj ) by Lemma 8.23. j=1 tj e ∈ Mw (

The analogue of Theorem 8.22 also holds. P Theorem 8.25. Define T : Mw∞ ( xj ) → X by T t = σ(X, X 0 ) − Pn P lim j=1 tj xj . Then T is continuous iff j xj is wuc. In this case,

n

X

kT k = sup t j xj : |tj | ≤ 1, n ∈ N .

j=1

P Proof: Suppose that T is continuous. Now c00 ⊂ Mw∞ ( x j ) and if

P

t ∈ c00 , ktk∞ ≤ 1 and ti = 0 for i ≥ n, then kT tk = nj=1 tj xj ≤ kT k.

P

P

n

Hence, { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded so j xj is wuc by Proposition 3.8.

P

P

n

Suppose that j xj is wuc and set M = sup{ j=1 tj xj : |tj | ≤ 1, n ∈

P

Pn N}. If t ∈ Mw∞ ( xj ), ktk∞ ≤ 1, then for every n we have T ( j=1 tj ej =

P

n

j=1 tj xj ≤ M . Therefore, if x0 ∈ X 0 , kx0 k ≤ 1, then * + ∞ ∞ X X 0 0 0 |hx , T ti| = x , t j xj = hx , tj xj i ≤ M. j=1 j=1 Hence, kT tk ≤ M and T is continuous with kT k ≤ M . The last statement follows from the computations above.

P We next consider the analogue of Theorem 8.21 for Mw∞ ( xj ). We say P P that a series j xj is bounded multiplier Cauchy if the series j tj xj is Cauchy for every t ∈ l ∞ . P Then Lemma 8.26. Suppose j xj is bounded multiplier Cauchy. P P ∞ ∞ Mw ( xj ) = M ( xj ). P P Proof: Let t ∈ Mw∞ ( xj ) and let x = ∞ j=1 tj xj [weak sum]. Since the P∞ partial sums of the series j=1 tj xj form a Cauchy sequence in X, there P∞ exists x00 ∈ X 00 such that x00 = j=1 tj xj [norm limit in X 00 ]. If x0 ∈ X 0 , D P E P ∞ then x0 , j=1 tj xj = hx0 , xi = hx00 , x0 i so x00 = x and t ∈ M ∞ ( xj ).

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Theorem 8.27. X is a Banach space iff for every wuc series P space Mw∞ ( xj ) is complete.

P

j

xj the

Proof: Suppose that X is not complete. By the proof of Theorem P 8.21, there is an absolutely convergent series j xj which is wuc but for P P which M ∞ ( xj ) is not complete. The series j xj is bounded multiplier P P Cauchy since it is absolutely convergent so Mw∞ ( xj ) = M ∞ ( xj ) by P Lemma 8.26 and Mw∞ ( xj ) is not complete. The converse follows from Theorem 8.24.

We next consider series with values in the dual space X 0 and use the P series to characterize barrelled spaces. If j x0j is a series in X 0 , define X ∞ Mw∗ ( x0j ) = {{tj } ∈ l∞ :

∞ We supply Mw∗ (

∞ X

tj x0j converges in X 0 with respect to σ(X 0 , X)}.

j=1

x0j ) with the sup-norm topology from l ∞ . P Theorem 8.28. Let j x0j be a series in X 0 . Consider the following conditions: P 0 is wuc (i) j xjP ∞ l∞ (ii) Mw∗ ( x0j ) = P∞ 0 < ∞ for every x ∈ X. (iii) j=1 xj , x P

Then (i) implies (ii) implies (iii). The conditions (i), (ii) and (iii) are equivalent iff X is barrelled. P P Proof: (i) implies (ii): If j x0j is wuc and t ∈ l∞ , then j tj x0j is also Pn wuc. Hence, { j=1 tj x0j : n ∈ N} is a bounded (equicontinuous) sequence in X 0 that is also σ(X 0 , X) Cauchy and, therefore, σ(X 0 , X) convergent. P ∞ Hence, t ∈ Mw∗ ( xj ).

0 P∞ ∞ (ii) implies

0 (iii): 1For every t ∈ l and x ∈ X, j=1 tj xj , x converges. Hence, { xj , x } ∈ l and (iii) holds. P Suppose that X is barrelled and (iii) is satisfied for the series j x0j . P The set { j∈σ x0j : σ finite} is pointwise bounded on X and is, therefore, P norm bounded since X is barrelled. Thus, j x0j is wuc by Proposition 3.8. Suppose (i),(ii) and (iii) are equivalent and X is not barrelled. Then there exists a subset F ⊂ X 0 which bounded but not norm

0 is weak* 0

y > 22j and set x0 = y 0 /2j . Then bounded. Pick y ∈ F such that j j j

j0 P∞ 0

x > 2j so x , x < ∞ for every x ∈ X so (iii) holds. But, j j Pj=1 P 0 ∞ 0 j j xj is not wuc and (i) fails j=1 xj /2 does not converge in norm and to hold.

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Chapter 9

The Antosik Interchange Theorem

A problem often encountered in analysis is the interchange of two limiting processes. For example, the Lebesgue Dominated Convergence Theorem gives sufficient conditions to interchange the pointwise limit of a sequence of integrable functions with the Lebesgue integral, i.e., to take the ”limit under the integral sign”. In this chapter we consider sufficient conditions for the equality of two iterated series. For real valued series one of the most useP∞ P∞ ful criterion for interchanging the limit of an iterated series i=1 j=1 tij is the absolute convergence of the iterated series. However, absolute convergence for series with values in an LCTVS is a very strong condition and is, therefore, not appropriate. Antosik has given a sufficient condition involving subseries convergence of an iterated series with values in a topological group which has proven to be useful in a number of applications ([A]). We begin this chapter with a presentation of Antosik’s result for series with values in a TVS. We then give generalizations of Antosik’s result to multiplier convergent series. Throughout this chapter let X be a TVS and let λ be a sequence space P containing c00 . Let xij ∈ X for i, j ∈ N. The double series i,j xij converges to x ∈ X if for every neighborhood, U, of 0 in X, there exists Pp Pq N such that i=1 j=1 xij − x ∈ U for p, q ≥ N . We have the following familiar properties of double series. P Proposition 9.1. Let i,j xij be a double series.

P (i) If the double series if the series i,j xij converges to x ∈ X and P∞ P∞ P∞ j=1 xij i=1 j=1 xij converge for each i, then the iterated series converges to x. Pm P∞ (ii) If the series { i=1 j=1 xij : m ∈ N} converge uniformly and if the 145

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P∞ P∞ iterated series i=1 j=1 xij converges to x, then the double series P x converges to x. i,j ij

Proof: (i): Let U be a neighborhood of 0 in X and let V be a symmetric neighborhood such that V + V ⊂ U . There exists N1 such that p, q ≥ N1 Pp Pq implies that i=1 j=1 xij − x ∈ V . For each p there exists N2 (p) such Pp P∞ Pp Pq that i=1 j=1 xij − i=1 j=1 xij ∈ V for q ≥ N2 (p). Let p ≥ N1 and fix q ≥ max{N1 , N2 (p)}. Then p X ∞ X i=1 j=1

xij − x =

p X ∞ X i=1 j=1

xij −

q p X X

xij +

q p X X i=1 j=1

i=1 j=1

xij − x ∈ V + V ⊂ U.

P P (ii): There exists N such that pi=1 ∞ x ∈ V for q > N and for j=q+1 Pp ij P∞ every p ∈ N. There exists M > N such that i=1 j=1 xij − x ∈ V for p ≥ M . If p, q ≥ M , then q p X X i=1 j=1

xij − x =

p X ∞ X i=1 j=1

xij − x −

p ∞ X X

i=1 j=q+1

xij ∈ V + V ⊂ U.

9.2. (Antosik) Let {xij } ⊂ X. Suppose the series converges for every increasing sequence {mj }. Then the i=1 j=1 xim Pj double series i,j xij converges and

Theorem P∞ P∞

(∗)

X i,j

xij =

∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X

xij .

j=1 i=1

P∞ Proof: Note that the series x converges for every k [consider i=1 P∞ P∞ P∞ik P∞ the difference between the two series i=1 j=1 xinj and i=1 j=1 ximj , where nj = j for every j and {mj } is the sequence {1, ..., k−1, k+1, ...}]. Set P P Pm P zmj = m i=1 xij . Then for σ ⊂ N, j∈σ zmj = i=1 j∈σ xij converges P∞ P to i=1 j∈σ xij as m → ∞ by hypothesis. By the Hahn-Schur Theorem P P∞ 7.17, the series ∞ j=1 ( i=1 xij ) is subseries convergent and lim m

m X X i=1 j∈σ

xij =

∞ XX

xij

j∈σ i=1

P∞ P∞ P∞ P∞ uniformly for σ ⊂ N. In particular, i=1 xij . j=1 j=1 xij = i=1 By Proposition 9.1 the uniform convergence implies that the double series P i,j xij converges and (∗) holds. We give applications of Theorem 9.2 later in the chapter.

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Although Antosik’s Theorem is easy to apply in many concrete situations, it is only a necessary condition for the equality of the two iterated series. For example, suppose that aj , bj ∈ R and xij = ai bj . If both series P P j bj converge, then j aj and ∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X j=1 i=1

xij =

∞ X

ai

i=1

∞ X

bj .

j=1

P However, if the ”inner” series, ∞ j=1 bj is conditionally convergent, the hypothesis in Theorem 9.2 is not satisfied. Stuart has given a result which covers this case ([St3]). The result uses Stuart’s weaker form of the HahnSchur Theorem 7.24. Theorem 9.3. Let F be an IQσ family which contains N (Appendix B.23). P P If the series ∞ i=1 j∈σ xij converges for every σ ∈ F, then ∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X

xij .

j=1 i=1

P∞ Proof: As before in Theorem 9.2, the series i=1 xij converges for evPm ery j. As in the proof of Theorem 9.2, set zmj = i=1 xij . For any P P∞ P σ ∈ F, limm j∈σ zmj exists and equals x i=1 j∈σ ij . By the version of the Hahn-Schur Theorem for IQσ families given in Theorem 7.24, P∞ P∞ P∞ P∞ j=1 xij . i=1 i=1 xij ) converges and equals j=1 (

Note that by employing the weaker version of the Hahn-Schur Theorem in Theorem 7.24, we cannot assert the uniform convergence of the limit, P limm j∈σ zmj , and the existence of the double series. However, Theorem 9.3 is sufficiently strong to cover the case mentioned prior to Theorem 9.3 P since if j bj is a conditionally convergent series, the set F={σ ⊂ N : P j∈σ bj converges} is an IQσ family containing N (Appendix B.24). As has been done before, Antosik’s Interchange Theorem can be viewed as a result concerning m0 multiplier convergent series and this suggests generalizations to more general λ multiplier convergent series. The hypothesis P P∞ in Theorem 9.2 that the series ∞ every increasi=1 j=1 ximj converges for P P∞ ing sequence {mj } can be restated to assert that the series ∞ i=1 j=1 tj xij converges for every t = {tj } ∈ m0 . This suggests that we might generalize Antosik’s theorem by replacing m0 by other sequence spaces λ. We now give such a generalization. Recall that λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence

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P {nj } such that the coordinate sum of the series ∞ j=1 sj χInj t belongs to λ; if the signs can all be chosen equal to 1, then λ has the weak gliding hump property (WGHP) [see Appendix B for examples]. Theorem 9.4. Let λ have signed-WGHP. Let {xij } ⊂ X. Suppose that the P series j xij is λ multiplier convergent for every i and the iterated series P∞ P∞ t = {t } ∈ λ. Then for every t ∈ λ, i=1 j=1 tj xij converges for every Pm P∞ j the sequence of iterated series { i=1 j=1 tj xij } converge uniformly for m ∈ N. Proof: If the conclusion fails to hold, then there exists a neighborhood of 0, U , in X such that for every k there exist jk > k and mk such that mk X ∞ X tj xij ∈ / U. i=1 j=jk

Pick a balanced neighborhood of 0, V , such that V + V ⊂ U . There exists lk > jk such that mk X ∞ X tj xij ∈ V i=1 j=lk +1

so

(∗)

lk mk X X

i=1 j=jk

tj xij ∈ / V.

By the condition in (∗) for k = 1, there exist j1 < l1 and m1 such that P m1 P l 1 / V. There exists J1 > j1 such that j=j1 tj xij ∈ i=1 (∗∗)

m n+p X X i=1 j=n

tj xij ∈ V for 1 ≤ m ≤ m1 , n > J1 and p > 0.

P m2 P l 2 By (∗) there exist l2 > j2 > J1 and m2 such that / V. i=1 j=j2 tj xij ∈ By (∗∗), m2 > m1 . We can continue this construction to produce increasing sequences mk , lk , jk with lk−1 < jk < lk and (∗ ∗ ∗)

lk mk X X

i=1 j=jk

tj xij ∈ / V.

Put Ik = [jk , lk ] so {Ik } is an increasing sequence of intervals. Define a matrix mp X X M = [mpq ] = [ tj xij ]. i=1 j∈Iq

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We claim that M is a signed K-matrix (Appendix D.3). First, note P∞ that the series i=1 tj xij converges for every j by setting t = ej in the hypothesis. Thus, each column of M converges. Next, given an increasing sequence of positive integers there is a subsequence {nq } and a sequence of P∞ P signs {sq } such that the coordinate sum u = q=1 sq j∈Inq tj ∈ λ. Then ∞ X q=1

sq mpnq =

∞ X

sq

q=1

mp X X

tj xij =

i=1 j∈Inq

mp ∞ X X i=1 q=1

sq

X

tj xij =

j∈Inq

mp ∞ X X

uj xij

i=1 j=1

P P∞ P∞ so limp ∞ q=1 sq mpnq = i=1 j=1 uj xij exists. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem, the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗ ∗ ∗). From Proposition 9.1 and Theorem 9.4, we obtain Corollary 9.5. Under the hypothesis of Theorem 9.4, for every t ∈ λ the P double series i,j tj xij converges and X i,j

tj xij =

∞ X ∞ X i=1 j=1

tj xij =

∞ X ∞ X

tj xij .

j=1 i=1

By strengthening the hypothesis on the multiplier space λ, we can also strengthen the conclusions of Theorem 9.4 and Corollary 9.5 to uniform convergence over bounded sets in the multiplier space. Recall that the K-space λ has the signed strong gliding hump property (signed-SGHP) if whenever {tj } is a bounded sequence in λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence {nj } P nj such that the coordinate sum of the series ∞ belongs to λ; j=1 sj χInj t if all of the signs can be chosen equal to 1, then λ has the strong gliding hump property (SGHP) [see Appendix B for examples]. Theorem 9.6. Let λ have signed-SGHP. Let {xij } ⊂ X. Suppose that the P series j xij is λ multiplier convergent for every i and the iterated series P∞ P∞ i=1 j=1 tj xij converges for every t = {tj } ∈ λ. Then the family of iterated series m X ∞ X tj xij : m ∈ N, t ∈ B i=1 j=1

converge uniformly for every bounded subset B ⊂ λ.

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Proof: The proof is similar to the proof of Theorem 9.4 which we now sketch. If the conclusion fails, there exists a neighborhood U of 0 in X such that for every k there exist jk > k, tk ∈ B and mk such that mk X ∞ X

i=1 j=jk

tkj xij ∈ / U.

Pick a balanced neighborhood V such that V + V ⊂ U . Then there exists lk > jk such that mk X ∞ X

i=1 j=lk +1

so

(∗)

tkj xij ∈ V

lk mk X X

i=1 j=jk

tkj xij ∈ / V.

By the condition in (∗) for k = 1, there exist j1 < l1 , t1 ∈ B and m1 such that (∗∗)

m1 X l1 X

i=1 j=j1

t1j xij ∈ / V.

By Theorem 2.16, there exists J1 > j1 such that (∗ ∗ ∗)

m n+p X X i=1 j=n

tj xij ∈ V when 1 ≤ m ≤ m1 , n > J1 , p > 0 and t ∈ B

(this is where signed-SGHP is used to guarantee the uniform convergence over B). By (∗∗) there exist l2 > j2 > J1 , m2 and t2 ∈ B such that P m2 P l 2 2 / V . By (∗ ∗ ∗) m2 > m1 . Continuing this construction j=j2 tj xij ∈ i=1 produces increasing sequences mk , lk , jk with lk−1 < jk < lk and tk ∈ B such that (∗ ∗ ∗∗)

lk mk X X

i=1 j=jk

tkj xij ∈ / V.

Put Ik = [jk , lk ] so {Ik } is an increasing sequence of intervals. Define a matrix M = [mpq ] = [

mp X X

i=1 j∈Iq

tqj xij ].

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Using the signed-SGHP, the proof of Theorem 9.4 can now be employed to show that M is a signed K-matrix. By the signed version of the AntosikMikusinski Matrix Theorem, the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗ ∗ ∗∗). We can now use Theorem 9.6 to give a generalization of Corollary 9.5. P Let xij (a) ∈ X for i, j ∈ N and a ∈ A. The double series i,j xij (a) converge uniformly for a ∈ A if for every neighborhood, U , of 0 in X, there P∞ P∞ exists N such that i=p j=q xij (a) ∈ U for p, q ≥ N and a ∈ A. The Pm P∞ proof of Proposition 9.1 shows that if the series i=1 j=1 xij (a) converge P uniformly for m ∈ N and a ∈ A, then the double series i,j xij (a) converge uniformly for a ∈ A. Thus, from Theorem 9.6, we have Corollary 9.7. Under the hypothesis of Theorem 9.6, the double series P i,j tj xij converge uniformly for t ∈ B.

Using the proof of Theorem 9.6 and Theorem 2.22 on the uniform convergence over null sequences for multiplier spaces with 0-GHP in place of Theorem 2.16 on the uniform convergence of multiplier convergent series over bounded sets with signed-SGHP, we can obtain the following results. Theorem 9.8. Let λ have 0-GHP. Let {xij } ⊂ X. Suppose that the seP ries j xij is λ multiplier convergent for every i and the iterated series P∞ P ∞ i=1 j=1 tj xij converges for every t ∈ λ. Then the family of iterated series m X ∞ X tkj xij : m, k ∈ N i=1 j=1

converge uniformly whenever {tk } is a null sequence in λ.

Corollary 9.9. Under the hypothesis of Theorem 9.8, the double series P k k i,j tj xij converge uniformly for any null sequence {t } in λ.

As an application of Theorem 9.4, we establish a multiplier convergent version of an Orlicz-Pettis Theorem for non-locally convex TVS with a Schauder basis due to Stiles ([Sti]) which was considered for LCTVS in Chapter 4.74. Stiles’ result seems to be the first version of an OrliczPettis Theorem for non-locally convex spaces. Indeed, Kalton remarks that Stiles’ result motivated his far-reaching generalization of the OrliczPettis Theorem for series with values in a topological group ([Ka2]). Stiles’ version of the Orlicz-Pettis Theorem is for subseries convergent series with

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values in an F-space with a Schauder basis and his proof uses the metric properties of the space. Other proofs of Stiles’ result have been given in [Bs] and [Sw5]. We will establish a version of Stiles’ result for multiplier convergent series which requires no metrizability assumption. Let X be a TVS with a Schauder basis {bj } and associated coordinate functionals {fj }. That is, every x ∈ X has a unique series representation P∞ x = j=1 tj bj and fj : X → R is defined by hfj , xi = tj . We do not assume that the coordinate functionals {fj } are continuous although this is the case if X is a complete metric linear space ([Sw2] 10.1.13). Let F = {fj : j ∈ N}. We consider the weak topology σ(X, F ) on X and use the interchange theorem to establish a multiplier convergent version of the Orlicz-Pettis Theorem for σ(X, F ) and the original topology of X. P Theorem 9.10. Let λ have signed-WGHP. If j xj is λ multiplier conP vergent with respect to σ(X, F ), then j xj is λ multiplier convergent with respect to the original topology of X. P∞ P∞ Proof: Let t ∈ λ. Consider the iterated series i=1 j=1 tj hfi , xj i bi . P∞ σ(X, F ) sum of the series. Then for each i, Let x = j=1 tj xj be theP P∞ ∞ t hf , xi = hf , xi so j i i j=1 tj hfi , xi bi = hfi , xi bi , where the converj=1 gence is in the original topology of X. But, ∞ X ∞ X i=1 j=1

tj hfi , xj i bi =

∞ X i=1

hfi , xi bi = x

with convergence in the original topology of X since {bj } is a Schauder basis. By Corollary 9.5, ∞ X ∞ X i=1 j=1

tj hfi , xj i bi =

∞ X ∞ X j=1 i=1

tj hfi , xj i bi =

∞ X

t j xj

j=1

with convergence in the original topology of X.

The use of Corollary 9.5 in the proof of Theorem 9.10 removes the metrizability and completeness assumptions in Stiles’ Theorem. Note that we did not use the continuity of the coordinate functional {fj } in the proof so the topology of X and the topology σ(X, F ) may not even be compatible! Remark 9.11. Theorem 9.10 covers the non-locally convex case when λ = lp , 0 < p < 1. The case when λ = bs gives a generalization of Stiles’ result for subseries convergent series. If X is an AK-space, then {ej }

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is a Schauder basis and the weak topology σ(X, {ej }) is just the topology of coordinatewise convergence on X. Thus, from Theorem 9.10 a series in X which is λ multiplier convergent with respect to the topology of coordinatewise convergence is λ multiplier convergent with respect to the original topology of X. In particular, these remarks apply to the classical sequence spaces X = lp , 0 < p < ∞, c0 , c00 . We next consider a generalization of Theorem 9.10 for multiplier convergent series; see also Theorem 4.75 for a locally convex version. Let (X, τ ) be a TVS and assume that there exists a sequence of linear operP∞ ators Pj : X → X such that x = j=1 Pj x [convergence in X] for every x ∈ X. When each Pj is a continuous projection, {Pj } is called a Schauder decomposition ([LT]). If X has a Schauder basis {bj } with continuous coordinate functionals {fj } and Pj x = hfj , xi bj for x ∈ X, then {Pj } is a Schauder decomposition for X. Theorem 9.12. Let λ have signed-WGHP. Let (X, τ ) be a TVS and σ a Hausdorff topology on X. Assume that each Pj : X → X is σ − τ P continuous. If j xj is λ multiplier convergent with respect to σ, then P x is λ multiplier convergent with respect to τ . j j P∞ Proof: Let t ∈ λ and j=1 tj xj be the σ sum of the series. For each P∞ P∞ i, j=1 tj Pi xj = Pi ( j=1 tj xj ), where the series is τ convergent by the continuity of Pi . Hence, ∞ X ∞ X

Pi (tj xj ) =

i=1 j=1

∞ X

Pi (

i=1

∞ X

t j xj )

j=1

converges with respect to τ . By Antosik’s Interchange Theorem 9.4 (Corollary 9.5), ∞ X ∞ X i=1 j=1

Pi (tj xj ) =

∞ X

t j xj ,

j=1

where the convergence is with respect to τ . Note that Theorem 9.10 is a corollary of Theorem 9.12. For suppose that {bj } is a Schauder basis for X with coordinate functionals {fj } and define Pj : X → X by Pj x = hfj , xi bj . If F = {fj : j ∈ N} and σ = σ(X, F ), then each Pj is σ − τ continuous and Theorem 9.12 applies. We indicate applications to vector valued sequence spaces; see also Example 4.76.

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Example 9.13. Let E be a vector space of X valued sequences equipped with a vector topology τ . E is a K-space if the coordinate maps fj : E → X, fj ({xj }) = xj are continuous for every j. If E is a K-space and every P∞ x = {xj } ∈ E has a representation x = j=1 ej ⊗ xj , then E is an AKspace [Appendix C; here ej ⊗ x is the sequence with x in the j th coordinate and 0 in the other coordinates]. If E is an AK-space, define Pj : E → E by Pj ({xj }) = ej ⊗ xj . If E is an AK-space and σ is the topology of coordinatewise convergence on E, then each Pj is σ − τ continuous, where τ is the original topology of E. Thus, Theorem 9.12 is applicable and P any series j xj in E which is coordinatewise λ multiplier convergent is λ multiplier convergent in E. For situations where Example 9.13 is applicable, let X be a metric linear space whose topology is generated by the quasi-norm |·|. For 0 < p < ∞, P∞ p let lp (X) be all X-valued sequences such that j=1 |xj | < ∞. If 1 ≤ p 0 such that (∗) s · Atk > δ for every k. Set k1 = 1. Pick m1 , n1 such that X n1 m1 X k1 (∗∗) si aij tj = Pm1 s · APn1 tk1 > δ. i=1 j=1 From (b) and (c),

(∗ ∗ ∗) s · APn1 tk = AT s · Pn1 tk = limk tkj

n1 X

tkj (AT s)j for every k.

j=1

= 0 for every j so it follows from (∗ ∗ ∗) that Since λ is a K-space, s · APn1 tk → 0 as k → ∞. Therefore, there exists k2 > k1 such that (♣) s · APn1 tk2 < δ/2.

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From (∗) and (♣), s · A(tk2 − Pn1 tk2 > δ/2.

Pick m2 > m1 , n2 > n1 such that X n2 X m2 k2 si aij tj = Pm2 s · A(Pn2 tk2 − Pn1 tk2 ) > δ/2. i=1 j=n1 +1

Continuing this construction produces increasing sequences {kp }, {mp }, and {np } such that Pmp s · A(Pnp tkp − Pnp−1 tkp ) > δ/2. Let Ip = {j ∈ N : np−1 < j ≤ np }. So {Ip } is an increasing sequence of intervals with (♠) Pmp s · AχIp tkp > δ/2. Define a matrix

M = [mpq ] = [Pmp s · AχIq tkq ] = [AT Pmp s · χIq tkq ].

We claim that M is a K-matrix (Appendix D.2). First, by (a), the columns of M converge to s · AχIq tkq . Next, given any increasing sequence {rq }, by 0-GHP there is a further subsequence, still denoted by {rq }, such that P∞ t = q=1 χIrq tkrq ∈ λ. Therefore, the sequence {

∞ X q=1

mprq } = {AT Pmp s · t} = {Pmp s · At}

converges with limit s · At by (a) and (c). Hence, M is a K-matrix and the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (♠). We give several corollaries of Theorem 10.10. Corollary 10.11. Let (λ, τ ) be a K-space with 0-GHP. Let η be a vector topology on µ such that (µ, η)0 = µ0 ⊂ µβ . If A : λ → µ, then A is τ − η bounded. Proof: The hypothesis implies that σ(µ, µβ ) bounded sets are η bounded so the result follows from Theorem 10.10. For example, the hypothesis in Corollary 10.11 is satisfied if µ is an AK-space (Proposition 2.5). Corollary 10.11 is not applicable to l ∞ or its subspace m0 . We give a result which is applicable to these spaces.

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Corollary 10.12. Let (λ, τ ) be a K-space with 0-GHP. Let η be a vector topology on µ such that η and σ(µ, µβ ) have the same bounded sets. If A : λ → µ, then A is τ − η bounded. Since (l∞ )β = (m0 )β = cβ = l1 , Corollary 10.12 is applicable if µ = c, m0 or l∞ with the sup-norm. Even though bs, the space of bounded series, is not an AK-space, Corollary 10.12 is applicable to bs with its natural topology since bs = (bv0 )0 and (bs)β = bv0 (Appendix B or [KG] p. 69). We next establish another automatic continuity result which requires assumptions on the range space µ. Theorem 10.13. Let (λ, τ ) be a K-space with 0-GHP. Assume that (µ, η) is a separable K-space such that µ0 = (µ, η)0 ⊂ µβ and (1) the sectional projections {Pn } are η equicontinuous. If A : λ → µ, then A is τ − η sequentially continuous. Proof: Let xk → 0 in τ . It suffices to show that y k · Axk → 0 when {y k } is an equicontinuous subset of µ0 . If this fails we may assume that there exists δ > 0 such that k y · Axk > δ for every k. Set k1 = 1 and pick m1 , n1 such that Pm1 y k1 · Pn1 xk1 > δ.

By (b), AT y k ∈ s, the space of all sequences, and since AT y k · ej = y k · Aej = (AT y k )j

and {y k } ⊂ µ0 is σ(µ0 , µ) bounded, {(AT y k )j : k ∈ N} is bounded for every j. Since limk xkj = 0 for every j, by Lemma 7.7, limk (AT y k )j xkj = 0 for every j. Therefore, there exists k2 > k1 such that X n1 T k (A y 2 )j xk2 = y k2 · APn1 xk2 < δ/2. j j=1 Hence,

k y 2 · A(xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that Pm2 y k2 · A(Pn2 xk2 − Pn1 xk2 ) > δ/2.

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Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (z) Pmp y kp · AχIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Since {Pn } is equicontinuous, {Pn0 = Pn } will carry equicontinuous subsets of µ0 into equicontinuous subsets of µ0 . Therefore, {Pn y k : n, k ∈ N} ⊂ µ0 is equicontinuous. Since η is separable, {Pmp y kp } has a subsequence, still denoted by {Pmp y kp }, which is σ(µ0 , µ) convergent to some y ∈ µ0 ⊂ µβ ([Wi] 9.5.3, [Sw] 18.9). Define a matrix M = [mpq ] = [Pmp y kp · AχIq xkq ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to y ·AχIq xkq . Next, given any subsequence {rq } there is a further P∞ subsequence, still denoted by {rq }, such that t = q=1 χIrq xkrq ∈ λ. Then ∞ X q=1

mprq = Pmp y kp · At → y · At.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (z). Proposition 2.5 gives sufficient conditions for µ0 = (µ, η)0 = µβ to be satisfied. If µ is a barrelled AB-space (Appendix B.3), then condition (1) is satisfied. Thus, Theorem 10.13 in particular applies to the spaces l p , 1 ≤ p < ∞. Without some assumptions on the domain space λ the continuity conclusions in Theorems 10.10 and 10.13 may fail. Example 10.14. Let A = [aij ] be the matrix aij = 1 if i ≤ j and aij = 0 otherwise. Then A : c00 → c is not norm continuous with respect to the P sup-norm and c is separable (kA( nj=1 ej )k∞ = n).

The method of proof of Theorem 10.10 can also be used to establish another boundedness result with respect to the strong topology β(µ, µβ ). In contrast to Theorem 10.10 this result requires an assumption on the range space µ. Actually we are able to establish a Uniform Boundedness result for families of pointwise bounded matrix mappings. Theorem 10.15. Let (λ, τ ) be a K-space with 0-GHP. Let Ak = [akij ]:λ → µ for every k ∈ N. Assume that µ satisfies

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(2) The sectional projections Pn : µβ → µβ are uniformly bounded on σ(µβ , µ) bounded subsets with respect to σ(µβ , µ) (i.e., if B ⊂ µβ is σ(µβ , µ) bounded and t ∈ µ, then {Pn s · t : s ∈ B, n ∈ N} is bounded). If {Ak } is pointwise bounded on λ with respect to β(µ, µβ ), then {Ak } is uniformly bounded on bounded subsets of λ with respect to β(µ, µβ ). Proof: If the conclusion fails, we may assume, by passing to a subsequence if necessary, that there exist xk → 0 in λ, {y k } ⊂ µβ which is σ(µβ , µ) bounded, tk → 0 and δ > 0 such that (∗) tk y k · Ak xk > δ for all k. P P n1 m1 Ak1 xkj 1 > δ. Now Set k1 = 1. Pick m1 , n1 such that i=1 tk1 yik1 j=1 tk y k · Ak Pn1 xk = tk (Ak )T y k · Pn1 xk = limk tk xkj

n1 X

tk xkj ((Ak )T y k )j .

j=1

Since λ is a K-space, = 0 for each j, and the pointwise boundedness assumption implies that {((Ak )T y k )j : k ∈ N} is bounded for each j. Hence, limk tk y k · Ak Pn1 xk = 0 so there exist k2 > k1 such that tk2 y k2 · Ak2 Pn1 xk2 < δ/2. Therefore, from (∗), tk2 y k2 · Ak2 (xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that tk2 Pm2 y k2 · Ak2 (Pn2 xk2 − Pn1 xk2 ) > δ/2.

Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (♥) tkp Pmp y kp · Akp χIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Define a matrix

M = [mpq ] = [tkp Pmp y kp · Akp χIq xkq ].

We claim that M is a K-matrix (Appendix D.2). Since {Pn y k : n, k ∈ N} is σ(µβ , µ) bounded by (2) and {Akp χIq xkq : p ∈ N} is β(µ, µβ ) bounded, the columns of M converge to 0. Since xk → 0 in λ, by the 0-GHP, given a subsequence {rq } there is a further subsequence, still denoted by {rq }, P∞ such that x = q=1 χIrq xkrq ∈ λ. Therefore, ∞ X q=1

mprq = tkp (Akp )T Pmp y kp · x = tkp Pmp y kp · Akp x → 0

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as above. Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.2). But, this contradicts (♥). Of course, Theorem 10.15 is applicable to a single matrix A and gives a boundedness result. We now give sufficient conditions for (2) to hold. Proposition 10.16. Consider the following conditions: (α) If (µ, µβ ) is a Banach-Mackey pair [i.e., if σ(µβ , µ) bounded sets are β(µβ , µ) bounded], then (2) holds. (β) If (µ, β(µ, µβ )) is an AB-space (Appendix B.3), then (2) holds. Proof: Let B ⊂ µβ be σ(µβ , µ) bounded and x ∈ µ. (α) : By (d), {Pn x} is σ( µ, µβ ) bounded so

sup{|Pn y · x| : y ∈ B, n ∈ N} = sup{|y · Pn x| : y ∈ B, n ∈ N} < ∞

and (2) holds. (β) : {Pn x} is β(µ, µβ ) so (2) holds by the computation in part (α). We can use the methods of Theorem 10.15 to establish a BanachSteinhaus type result. Theorem 10.17. Let (λ, τ ) be a K-space with 0-GHP. Let Ak = [akij ]:λ → µ for every k ∈ N. Assume that µ satisfies (2) The sectional projections Pn : µβ → µβ are uniformly bounded on σ(µβ , µ) bounded subsets with respect to σ(µβ , µ) (i.e., if B ⊂ µβ is σ(µβ , µ) bounded and t ∈ µ, then {Pn s · t : s ∈ B, n ∈ N} is bounded). If limk Ak x exists with respect to the strong topology β(µ, µβ ), then {Ak } is τ −β(µ, µβ ) sequentially equicontinuous. Proof: If the conclusion fails to hold, we may assume, by passing to a subsequence if necessary, that there exist δ > 0, xj → 0 in λ and {y j } ⊂ µβ which is σ(µβ , µ) bounded such that j y · Aj xj > δ. Set k1 = 1 and pick m1 , n1 such that Pm1 y k1 · Ak1 Pn1 xk1 > δ. Since {Ak ej : k ∈ N} is strong bounded for every j, {(Ak )T y k · ej = y k · Ak ej = ((Ak )T y k )j : k ∈ N}

is bounded for every j. Since limk xkj = 0 for every j, by Lemma 7.7, lim((Ak )T y k )j xkj = 0 k

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for every j. Therefore, there exist k2 > k1 such that k y 2 · Ak2 Pn1 xk2 < δ/2. Therefore,

k y 2 · Ak2 (xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that Pm2 y k2 · Ak2 (Pn2 xk2 − Pn1 xk2 ) > δ/2.

Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (∗) Pmp y kp · Akp χIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Define a matrix

M = [mpq ] = [Pmp y kp · Akp χIq xkq ]. We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge since {Ak } converges pointwise with respect to the strong topology and (2) implies that {Pmp y kp } is σ(µβ , µ) bounded. Next, given an increasing sequence {rq } there exists a further subsequence, still denoted P∞ by {rq }, such that t = q=1 χIrq xkrq ∈ λ. Then ∞ X q=1

mprq = Pmp y kp · Akp t

and the sequence {Pmp y kp · Akp t} converges by the same argument that the columns converge. Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗). The original version of the Hellinger-Toeplitz Theorem asserts that if P P∞ 2 the series a(x, y) = ∞ i=1 j=1 aij xj yi converges for every x, y ∈ l , then there exists M such that |a(x, y)| ≤ M for kxk2 ≤ 1, kyk2 ≤ 1 ([HT]). That is, the bilinear form a induced by the matrix A is a continuous bilinear form a : l2 × l2 → R. We can use the method of proof in Theorem 10.10 to establish the sequential continuity of bilinear forms between products of sequence spaces which are induced by matrices. If the series a(x, y) = P∞ P∞ i=1 j=1 aij xj yi = y · Ax converges for every x ∈ λ, y ∈ µ, where λ and µ are sequence spaces, then a is a bilinear form on λ × µ. We use the

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method of proof in Theorem 10.10 to establish a sequential continuity result for such bilinear forms. Theorem 10.18. Let (λ, τ ) be a K-space with 0-GHP and let (µ, η) be a K-space such that µβ ⊂ µ0 = (µ, η)0 . Assume (3) the sectional projections Pn : µ → µ are sequentially equicontinuous with respect to η. P P∞ If a(x, y) = ∞ i=1 j=1 aij xj yi = y · Ax converges for every x ∈ λ, y ∈ µ, then a : λ × µ → R is sequentially τ × η continuous.

Proof: If the conclusion fails, there exist sequences xk → 0 in τ , y k → 0 in η and δ > 0 such that k k k a(x , y ) = y · Axk > δ

for all k. Set k1 = 1 and pick m1 , n1 such that X n1 m1 X k1 k1 aij xj yj > δ. i=1 j=1

Note that Ax ∈ µβ ⊂ µ0 for each x ∈ λ by the convergence of the series P∞ P∞ i=1 j=1 aij xj yi for each y ∈ µ. For each j the series a(ej , y k ) =

∞ X i=1

aij yik = y k · Aej = (AT y k )j

converges, and since {y k } is η convergent to 0 and Aej ∈ µ0 by the observation above, for each j, {(AT y k )j : k ∈ N } is bounded. Since λ is a K-space, limk xkj = 0 for for each j. Therefore, by Lemma 7.7, lim k

n1 X j=1

xkj (AT y k )j = lim k

n1 X ∞ X

aij xkj yik = 0

j=1 i=1

so there exists k2 > k1 such that n1 ∞ X X k k2 k2 2 · APn1 xk2 < δ/2. a x y ij j i = y j=1 i=1 Hence,

k y 2 · A(xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that m2 n 2 X X k2 k2 a x y ij j j > δ/2. i=1 j=n1 +1

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Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that Pmp y kp · A(Pnp − Pnp−1 )xkp > δ/2. Set Ip = {j ∈ N : np−1 < j ≤ np } so (♦) Pmp y kp · AχIp xkp > δ/2. Define a matrix

M = [mpq ] = [Pmp y kp · AχIq xkq ]. We claim that M is a K-matrix (Appendix D.2). First, since y k → 0 in η, from (3) Pk y k → 0 in η and AχIq xkq ∈ µ0 , the columns of M converge to 0. Next, given any subsequence {rq }, there is a further subsequence, P∞ kr q still denoted by {rq }, such that t = ∈ λ by the 0-GHP q=1 χIrq x assumption. Then ∞ X q=1

0

mprq = Pmp y kp · At → 0

since At ∈ µ by the observation above. Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (♦). Proposition 2.5 gives sufficient conditions for the condition µβ ⊂ µ0 = (µ, η)0 to be satisfied. Note that this condition is satisfied by the topology σ(µ, µβ ). If µ is a barrelled AB-space, then condition (3) is satisfied. Thus, Theorem 10.18 is applicable to a wide range of sequence spaces including lp , 0 < p < ∞, and includes the original Hellinger-Toeplitz result.

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Chapter 11

Operator Valued Series and Vector Valued Multipliers

In this chapter we will consider operator valued series but allow the space of multipliers to be vector valued with values in the domain space of the operators. Throughout this chapter let X, Y be LCTVS and L(X, Y ) the space of all continuous linear operators from X into Y . Let E be a vector valued sequence space with values in X which contains c00 (X), the space of all X valued sequences which are eventually 0. P Definition 11.1. A series j Tj in L(X, Y ) is E multiplier convergent if P∞ the series j=1 Tj xj converges in Y for every x = {xj } ∈ E. The series P P∞ j Tj is E multiplier Cauchy if the series j=1 Tj xj is Cauchy for every x = {xj } ∈ E. The elements of E are called multipliers. P If E = l∞ (X), a series j Tj which is l∞ (X) multiplier convergent is said to be bounded multiplier convergent. If E = m0 (X), then a series P j Tj which is m0 (X) multiplier convergent is subseries convergent in the strong operator topology. We now consider the basic properties of E multiplier convergent series. Many of these properties are the same as those for scalar valued multiplier convergent series and the proofs are essentially identical. When this phenomena occurs we will make references to the appropriate scalar results. However, due to the presence of continuous linear operators, the results for vector valued multipliers and operator valued series often require additional technical assumptions which we will indicate. P Let j Tj be E multiplier convergent. The summing operator S : E → P Y (with respect to j Tj and E) is defined by Sx =

∞ X j=1

Tj xj , x = {xj } ∈ E. 169

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Recall the β-dual of E (with respect to the scalar field) is defined to be E β = {{x0j } : x0j ∈ X 0 ,

∞ X

x0j , xj

j=1

converges for every {xj } ∈ E}

P∞ 0 and E, E β form a dual pair under the pairing x0 · x = j=1 xj , xj (Appendix C). Also, recall that a locally convex topology w defined for dual pairs is a Hellinger-Toeplitz topology if whenever X, X 0 and Y, Y 0 are dual pairs and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is a continuous linear operator, then T : (X, w(X, X 0 )) → (Y, w(Y, Y 0 )) is continuous (Appendix A.1). As in Theorem 2.2, we have Theorem 11.2. The summing operator S : E → Y is σ(E, E β ) − σ(Y, Y 0 ) continuous. Therefore, S is w(E, E β ) − w(Y, Y 0 ) continuous for any Hellinger-Toeplitz topology w. As in Corollaries 2.3 and 2.4, we have Corollary 11.3. If B is σ(E, E β ) bounded, then SB = { B} is bounded in Y.

P∞

j=1

T j xj : x ∈

Corollary 11.4. Let E be a K-space. If E β ⊂ E 0 and B ⊂ E is bounded, P∞ then SB = { j=1 Tj xj : x ∈ B} is bounded in Y.

For conditions which guarantee that E β ⊂ E 0 , we have the analogue of Proposition 2.5. For this we need the following property of vector valued sequence spaces. If z ∈ X and j ∈ N, recall that ej ⊗ z is the sequence with z in the j th coordinate and 0 in the other coordinates. The space E has the property (I) if the maps z → ej ⊗ z are continuous from X into E for every j. Proposition 11.5. We have the following conditions:

(i) If E is a barrelled K-space, then E β ⊂ E 0 . (ii) If E is an AK-space with property (I), then E 0 ⊂ E β . (iii) If E is a barrelled AK-space with property (I), then E 0 = E β .

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Proof: (i): Let y ∈ E β . For each n define Fn : E → R by Fn (x) = Pn j=1 hyj , xj i. Since E is a K-space, each Fn is continuous and linear. Since Fn (x) → y · x for every x ∈ E, the map x → y · x is continuous by the barrelledness assumption. Hence, y ∈ E 0 .

(ii): Let F ∈ E 0 . Define yj : X → R by hyj , xi = F, ej ⊗ x . Since E has property (I), yj ∈ X 0 . Set y = {yj }. If x ∈ E, then * ∞ + ∞ ∞ X X

X j hF, xi = F, e ⊗ xj = F, ej ⊗ xj = hyj , xj i . j=1

j=1

j=1

Therefore, y ∈ E β and hF, xi = y · x. (iii) follows from (i) and (ii). Corollary 11.6. Assume that E is a barrelled AB-space (Appendix C.3) P T is E multiplier convergent. If B ⊂ E is bounded, then and Pn j j { j=1 Tj xj : n ∈ N, x ∈ B} is β(Y, Y 0 ) bounded. P Proof: Let Pn : E → E be the section operator Pn (x) = nj=1 ej ⊗ xj . By the AB assumption {Pn : n} is pointwise bounded on E and, therefore, equicontinuous since E is barrelled. Since E is barrelled, E has the strong topology β(E, E 0 ) so {Pn x : n ∈ N, x ∈ B} is β(E, E 0 ) bounded. By Proposition 11.5 , E β ⊂ E 0 so {Pn x : n ∈ N, x ∈ B} is β(E, E β ) bounded. The result now follows from Theorem 11.2 since the strong topology is a Hellinger-Toeplitz topology. Recall the condition that E β ⊂ E 0 is important even in the scalar case (Example 2.9). From Corollary C.7 in Appendix C if X is a Frechet space, then l∞ (X) is a Frechet space with property AB so we have P Corollary 11.7. Let j Tj be bounded multiplier convergent. If X is a P∞ Frechet space and B ⊂ l ∞ (X) is bounded, then { j=1 Tj xj : x ∈ B} is β(Y, Y 0 ) bounded. We have the analogue of Theorem 2.11. Recall that E has 0-GHP if whenever xj → 0 in E and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that the coordinatewise sum of the series P∞ nj ∈ E (Appendix C). j=1 χInj x P Theorem 11.8. Let E be a K-space with 0-GHP. If j Tj is E multiplier convergent, then the summing operator S : E → Y is sequentially continuous and, therefore, bounded.

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We next consider the uniform convergence of operator valued series over bounded subsets of the multiplier space E. Recall that E has the signedSGHP if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence P∞ {nj } such that the coordinatewise sum of the series j=1 sj χInj xnj ∈ E (Appendix C). We have the analogue of Theorem 2.16. P Theorem 11.9. Let E have signed-SGHP. If j Tj is E multiplier converP∞ gent, then the series j=1 Tj xj converge uniformly for x = {xj } belonging to bounded subsets of E. Without the signed-SGHP assumption, the conclusion of Theorem 11.9 may fail. Example 11.10. Let 1 ≤ p < ∞ and define Qk : lp → lp by Qk t = tk ek . P P P For t ∈ lp and σ ⊂ N, k∈σ Qk t = k∈σ tk ek converges in l p so k Qk is m0 (lp ) multiplier convergent since m0 (lp ) = span{χσ t : σ ⊂ N, t ∈ P lp }. However, the series k Qk tk do not converge uniformly for t = {tk } belonging to bounded subsets of m0 (lp ). For let tk be the constant sequence P P∞ k k k in m0 (lp ) with ek in each coordinate. Then ∞ j=n Qj tj = j=n Qj e = e if k ≥ n. As a corollary of Theorem 11.9 we have an important property of bounded multiplier convergent series which was established for Banach spaces by Batt ([Bt]). P Corollary 11.11. Let j Tj be bounded multiplier convergent. Then the P∞ series j=1 Tj xj converge uniformly for x = {xj } belonging to bounded subsets of l∞ (X). Remark 11.12. We can give another interesting proof of Corollary 11.11 above by employing the lemma of Li (Lemma 3.29). Suppose that B ⊂ l∞ (X) is bounded and let p be a continuous semi-norm on X. There exists M > 0 such that sup{p(xj ) : x = {xj } ∈ B} ≤ M . Put Ej = {x ∈ X : P p(x) ≤ M } and define fj : Ej → X by fj (x) = Tj x. Then ∞ j=1 fj (xj ) = P∞ ∞ j xj converges for every x = {xj } ∈ Πj=1 Ej . By Lemma 3.29, the j=1 T P ∞ series j=1 Tj xj converge uniformly for x = {xj } ∈ Π∞ j=1 Ej so the series converge uniformly for {xj } ∈ B. The analogue of Theorem 2.22 also holds.

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P Theorem 11.13. Let E be a K-space with 0-GHP. If j Tj is E multiplier P ∞ convergent and xk → 0 in E, then the series j=1 Tj xkj converge uniformly for k ∈ N. We next consider uniform convergence results for families of E multiplier convergent series. The β-dual of E with respect to Y is E βY = {{Tj } : Tj ∈ L(X, Y ),

∞ X j=1

Tj xj converges for every {xj } ∈ E}.

The topology w(E βY , E) on E βY is defined to be the weakest topology such P βY that the mappings T = {Tj } → T · x = ∞ into Y are j=1 Tj xj from E continuous for every x ∈ E. If X is the scalar field, these notations agree with those employed in Chapter 2. The analogue of Theorem 2.26 holds. Recall E has signed-WGHP if for every x ∈ E and every increasing sequence {Ij }, there exist a sequence of signs {sj } and a subsequence {nj } such that P∞ the coordinatewise sum of the series j=1 sj χInj x ∈ E (Appendix C).

Theorem 11.14. Assume that E has signed-WGHP. If {T k } ⊂ E βY is such that limk T k · x exists for every x ∈ E, then for every x ∈ E the series P∞ k j=1 Tj xj converge uniformly for k ∈ N. We next consider the analogue of Stuart’s weak completeness theorem (Corollary 2.28). This is the point where the significant differences between the scalar and vector cases appear.

Definition 11.15. The pair (X, Y ) has the Banach-Steinhaus property if whenever {Tj } ⊂ L(X, Y ) is pointwise convergent, lim Tj x = T x exists for every x ∈ X, then T ∈ L(X, Y ). For example, if X is barrelled, then (X, Y ) has the Banach-Steinhaus property for every LCTVS Y ([Sw2] 24.12, [Wi] 9.3.7). Lemma 11.16. Let {T k } ⊂ E βY be such that there for every j, there exists Tj ∈ L(X, Y ) with limk Tjk x = Tj x for every x ∈ X. If for every x ∈ E, P k limk T k · x exists and the series ∞ j=1 Tj xj converge uniformly for k ∈ N, βY k then T = {Tj } ∈ E and T → T in w(E βY , E). Proof: Let x ∈ E and set u = limk T k · x. It suffices to show that P∞ u = j=1 Tj xj . Let U be a balanced neighborhood of 0 in Y and pick a balanced neighborhood of 0, V , such that V + V + V ⊂ U . There exists p

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P∞ such that j=n Tjk xj ∈ V for n ≥ p, k ∈ N. Fix n ≥ p. There exists k = kn P∞ Pn such that j=1 Tjk xj − u ∈ V and j=1 (Tjk − Tj )xj ∈ V . Then n X j=1

Tj xj −u = (

∞ X j=1

Tjk xj −u)−

and the result follows.

n ∞ X X (Tjk −Tj )xj − Tjk xj ∈ V +V +V ⊂ U j=1

j=n+1

Lemma 11.17. Let {T k } ⊂ E βY and let (X, Y ) have the BanachSteinhaus property. If for every x ∈ E, limk T k · x exists and the series P∞ k βY such j=1 Tj xj converge uniformly for k ∈ N, then there exists T ∈ E k βY that T → T in w(E , E). Proof: For each j define a linear map Tj : X → Y by Tj z = lim T k · (ej ⊗ z) = lim Tjk z. k

k

By the Banach-Steinhaus assumption, Tj ∈ L(X, Y ). The result now follows from Lemma 11.16. From Theorem 11.14 and Lemma 11.17, we can now obtain Stuart’s completeness result for vector valued sequence spaces. Corollary 11.18. (Stuart) Let E have signed-WGHP, Y be sequentially complete and (X, Y ) have the Banach-Steinhaus property. If {T k } is Cauchy in w(E βY , E), then there exists T ∈ E βY such that T k → T in w(E βY , E). That is, w(E βY , E) is sequentially complete. Proof: For each z ∈ X and j, the sequence {Tjk z}k = {T k · (ej ⊗ z)}k is Cauchy in Y . By the sequential completeness assumption, limk Tjk z = Tj z exists. The result now follows from Theorem 11.14 and Lemma 11.17. The assumption that the pair (X, Y ) has the Banach-Steinhaus property is necessary for the conclusion of Corollary 11.18 to hold. For suppose that Tk ∈ L(X, Y ) and lim Tk x = T x exists for every x ∈ X. Define T k ∈ E βY by T k = (Tk , 0, 0, ...). Then lim T k · x = lim Tk x1 exists for every {xj } ∈ E so {T k } is w(E βY , E) Cauchy. If w(E βY , E) is sequentially complete and T 0 = w(E βY , E) − lim T k , then T10 = T ∈ L(X, Y ) so (X, Y ) has the Banach-Steinhaus property. We state another corollary of Theorem 11.14. A subset F of E βY is conditionally w(E βY , E) sequentially compact if every sequence {T k } ⊂ F has a subsequence {T nk } which is such that lim T nk · x exists for every x ∈ E. From Theorem 11.14, we have

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Corollary 11.19. Let E have signed-WGHP and (X, Y ) have the BanachSteinhaus property. If F ⊂ E βY is conditionally w(E βY , E) sequentially P∞ compact and x ∈ E, then the series j=1 Tj xj converge uniformly for T ∈ F. We next consider a uniform convergence result for the strong topology of Y . This requires the ∞-GHP assumption. The space E has the ∞-GHP if whenever x ∈ E and {Ij } is an increasing sequence of intervals there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinatewise sum of P the series ∞ j=1 apj χIpj x ∈ E (Appendix C).

Theorem 11.20. Assume that E has ∞-GHP. If F ⊂ E βY is pointwise bounded on E with respect to β(Y, Y 0 ) , then for every x ∈ E the series P∞ 0 j=1 Tj xj converge uniformly in β(Y, Y ) for T ∈ F .

Proof: If the conclusion fails, there exist > 0, {T k } ⊂ F, {yk0 } ⊂ Y 0 σ(Y 0 , Y ) bounded and an increasing sequence of intervals {Ik } such that * + 0 X k Tl xl > for all k. (∗) yk , l∈Ik

By ∞-GHP, there exist {nk }, apk > 0, apk → 0 such that every subsequence of {nk } has a further subsequence {pk } such that the coordinatewise P∞ sum of the series j=1 apj χIpj x ∈ E. Define a matrix * + X M = [mij ] = yn0 i /ani , Tlni (apj xl ) . l∈Ij

We claim that M is a K-matrix (Appendix D.2). First, since F is pointwise β(Y 0 , Y ) bounded on E , {yi0 } is σ(Y 0 , Y ) bounded and 1/api → 0, the columns of M converge to 0. Next, given any subsequence there is a further P∞ subsequence {pj } such that u = j=1 apj χIpj x ∈ E. Therefore, * + ∞ ∞ X X pi 0 T uj → 0 mipj = yni /ani , j=1

j=1

by the same argument as above. Hence, M is a K-matrix and the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). From Theorem 11.20 and Lemma 11.17, we have the following weak sequential completeness result.

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Corollary 11.21. Let E have ∞-GHP, Y be a sequentially complete, barrelled space and (X, Y ) have the Banach-Steinhaus property. Then w(E βY , E) is sequentially complete. Proof: Since Y is barrelled the original topology of Y is just β(Y, Y 0 ). Thus, the result follows from Theorem 11.20 and Lemma 11.17 since any w(E βY , E) Cauchy sequence {T k } is pointwise bounded on E and lim T k · x exists for every x ∈ E by the sequential completeness of Y . As noted in Remark 2.34, the ∞-GHP and signed-WGHP are independent so the results in Corollaries 11.18 and 11.21 cover different spaces. We have the vector analogues of Theorems 2.35 and 2.39. Theorem 11.22. Assume that E has signed-SGHP. If {T k } ⊂ E βY is such that lim T k · x exists for every x ∈ E and B ⊂ E is bounded, then the P k series ∞ j=1 Tj xj converge uniformly for k ∈ N, x ∈ B.

Theorem 11.23. Assume that E has 0-GHP. If {T k } ⊂ E βY is such that lim T k · x exists for every x ∈ E and xk → 0 in E, then the series P∞ k l j=1 Tj xj converge uniformly for k, l ∈ N.

Examples 2.30 and 2.40 show the gliding hump assumptions above are important. We next discuss the relationships between absolute convergence, bounded multiplier convergence and subseries convergence for operator valued series. For the topologies on the space L(X, Y ) which are employed, see Appendix C. P Theorem 11.24. Let Y be sequentially complete. If the series j Tj is absolutely convergent in Lb (X, Y ), then the series is bounded multiplier convergent. Proof: Let {xj } ⊂ X be bounded and set A = {xj : j ∈ N}. Let p be a continuous semi-norm on Y . If n > m, then n n n X X X p( T j xj ) ≤ p(Tj xj ) ≤ pA (Tj ) j=m

j=m

j=m

so the result follows from the completeness of Y .

The converse of the result above does not hold. Example 11.25. Define Tj : R → c0 by Tj t = (t/j)ej . Then Tj is continuous, linear with kTj k = 1/j. If {tj } is bounded in R, then the series

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P∞

P P j = ∞ j=1 (tj /j)e converges in c0 so the series j Tj is bounded multiplier convergent but not absolutely convergent. P Theorem 11.26. If the series j Tj is bounded multiplier convergent, then the series is subseries convergent in Lb (X, Y ). P Proof: We first claim that the series j Tj is convergent in Lb (X, Y ). If this fails to hold, there exist a continuous semi-norm p on Y , a bounded subset A ⊂ X, an increasing sequence of intervals {Ij } and > 0 such that P P pA ( i∈Ij Ti ) > for all j. Pick xj ∈ A such that p(( i∈Ij Ti )xj ) > for all j. Define z ∈ l∞ (X) by zi = xj if i ∈ Ij and zi = 0 otherwise. Then the P∞ series j=1 Tj zj is not convergent since the series fails the Cauchy criterion. Since the same argument can be applied to any subseries of the series P j Tj , the result follows. j=1 tj Tj

The converse of the result in Theorem 11.26 fails to hold.

Example 11.27. Let X = Y = l 1 and define Tj : l1 → l1 by Tj t = P

P

(tj /j)ej . If σ ⊂ N, then j∈σ Tj ≤ supj∈σ |1/j| so the series j Tj is subseries convergent in the uniform operator topology. However, the series P∞ P T is not bounded multiplier convergent since the series j=1 Tj ej = Pj∞ j j 1 j=1 (1/j)e is not convergent in l .

Obviously, a series which is subseries convergent in Lb (X, Y ) is subseries convergent in the strong operator topology, Ls (X, Y ), but the converse is false. Example 11.28. Let X = Y = l 1 and define Tj : l1 → l1 by Tj t = tj ej so Tj is continuous, linear with kTj k = 1. If σ ⊂ N and t ∈ l 1 , then

X

X

|tj | T t j j =

j∈σ

j∈σ P so the series j Tj is subseries convergent in the strong operator topology. However, since kTj k = 1 the series is not subseries convergent in the uniform operator topology. We establish partial converses to the statements above. Proposition 11.29. Let X be a normed space and Y a Banach space. Every l∞ (X) multiplier convergent series is absolutely convergent iff Y is finite dimensional.

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Proof: Suppose that Y is infinite dimensional. By the DvoretskyRogers Theorem ([Sw2] 30.1.1, [Day]), there is a subseries convergent series P Pick x0 ∈ X 0 , x0 6= 0. Define j yj in Y which is not absolutely convergent. P Tj ∈ L(X, Y ) by Tj x = hx0 , xi yj . Then j Tj is l∞ (X) multiplier conP vergent since the series j yj is also bounded multiplier convergent (2.54). P However, j Tj is not absolutely convergent since kTj k = kx0 k kyj k. P Suppose Y = Rn and j Tj is l∞ (X) multiplier convergent in L(X, Y ). P Since l∞ (X) is monotone, the series ∞ j=1 Tj xj is subseries convergent in P ∞ For each j, pick Y for every {xj } ∈ l (X). Therefore, ∞ j=1 kTj xj k < ∞.P ∞ xj ∈ X, kxj k ≤ 1, such that kTj k ≤ kTj xj k + 1/2j . Then j=1 kTj k < ∞. Proposition 11.30. If X = Rn and Y is a Banach space, then every P series j Tj which is subseries convergent in the strong operator topology P of L(X, Y ) is l∞ (X) multiplier convergent. In particular, the series j Tj is subseries convergent in Lb (X, Y ).

Proof: Let {ei }ni=1 be the canonical base in Rn . For i = 1, ..., n and P i i x ∈ l∞ (X), the series ∞ j=1 (e · xj )Tj e is bounded multiplier convergent since Y is complete (2.54). Therefore, the series n X ∞ X i=1 j=1

i

i

(e · xj )Tj e =

∞ X n X j=1 i=1

i

i

(e · xj )Tj e =

∞ X

T j xj

j=1

converges in Y . Proposition 11.31. Suppose X is a normed space and Y is a Banach space. Then every series in L(X, Y ) which is subseries convergent in the uniform operator topology is l ∞ (X) multiplier convergent iff X is finite dimensional. Proof: Suppose X is infinite dimensional. Since X 0 is infinite dimenP sional, there exists a series j x0j in X 0 which is subseries convergent in X 0 but ∞ X

0

x = ∞ j j=1

(Dvoretsky-Rogers Theorem ([Day], Pick y ∈ Y, y 6= 0. [Sw2] 30.1.1)). P Define Tj : X → Y by Tj x = x0j , x y. Then j Tj is subseries convergent in the uniform operator topology l ∞(X) multiplier convergent (pick

0 but not 0

xj ∈ X, kxj k ≤ 1, such that xj ≤ xj , xj + 1/2j ). The converse follows from Theorem 11.30.

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We next establish several results relating vector and scalar multiplier convergent series. For the definition of the spaces ν{X}, see Appendix C. P Proposition 11.32. If the series j Tj is ν{X} multiplier convergent, then the series is ν multiplier convergent in the strong operator topology. Proof: Let t ∈ ν and x ∈ X so tx ∈ ν{X} and P∞ j=1 tj Tj x converges.

P∞

j=1

Tj (tj x) =

The converse of the result above fails to hold.

P Example 11.33. Define Tj ∈ L(l2 , l2 ) by Tj t = tj ej . The series j Tj is bounded multiplier convergent in the strong operator topology since if P∞ P j 2 = ∞ s ∈ l∞ and t ∈ l2 , the series j=1 sj tj e converges in l . j=1 sj Tj tP P ∞ ∞ j ∞ 2 j j However, {e } ∈ l (l ) and j=1 Tj e = j=1 e does not converge in l 2 P so the series j Tj is not l∞ (l2 ) multiplier convergent. We do have a partial converse to Proposition 11.32.

P Proposition 11.34. Let X = Rn be finite dimensional. If the series j Tj is ν multiplier convergent in the strong operator topology of L(X, Y ), then the series is ν{X} multiplier convergent. Proof: Let {ei }ni=1 be the canonical basis for R n . If x ∈ ν{X}, then {kxj k} ∈ ν and since ν is normal and kxj k ≥ ei · xj for every i, {ei ·xj }j ∈ ν. Therefore, lim m

m X

Tj xj = lim m

j=1

= lim m

so the series

P∞

j=1

m X j=1

Tj (

n X

n X m X i=1 j=1

i=1

(ei · xj )ei )

(ei · xj )Tj ei =

n X ∞ X i=1 j=1

(ei · xj )Tj ei

Tj xj converges.

The results in Propositions 11.32 and 11.34 hold if ν = c0 or ν = lp (0 < p ≤ ∞). We now consider some applications to operator valued set functions. For this we assume that X and Y are normed spaces. Let A be an algebra of subsets of a set S and µ : A → L(X, Y ) be finitely additive. The (operator) semi-variation of µ is defined to be

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) ( n

X

µ(Ai )xi : {Ai } ⊂ A a partition of A, kxi k ≤ 1 . µ ˆ (A) = sup

i=1

It is clear that the semi-variation is subadditive and monotone. The operator semi-variation is employed when integrating X valued measurable functions with respect to operator valued measures with values in L(X, Y ) ([Bar]). Since µ ˆ (A) ≥ kµk(A) for any A ∈ A (as computed with respect to the norm in L(X, Y )), if µ ˆ (S) < ∞, then µ has bounded semi-variation and is, therefore, bounded (Proposition 3.45). However, µ can be bounded and have infinite operator semi-variation. P Example 11.35. Define µ : 2N → L(l2 , R) = l2 by µ(A) = j∈A Tj where Tj = (1/j)ej ∈ L(l2 , R) = l2 . Then µ is countably additive and P n Pn bounded. However, Tj ej = (1/j)ej · ej = 1/j so j=1 Tj ej = j=1 1/j Pn and µ ˆ ({1, ...n}) = j=1 1/j → ∞ so the operator semi-variation of µ is infinite. We have a multiplier convergent characterization of operator valued set functions with finite semi-variation. Theorem 11.36. The following conditions are equivalent: (i) µ has finite (operator) semi-variation. P (ii) For every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) is c0 (X) multiplier Cauchy. Proof: Assume (i). Let {Aj } ⊂ A be pairwise

let x ∈ c0 (X).

P disjoint and

ˆ (S). Hence, Put zj = xj / kxj k. For any finite σ ⊂ N, j∈σ µ(Aj )zj ≤ µ P j µ(Aj )zj is c0 multiplier Cauchy (Proposition 3.8). Therefore, X j

kxj k µ(Aj )zj =

X

µ(Aj )xj

j

is Cauchy so (ii) holds. Assume (ii). Suppose (i) fails so µ ˆ (S) = ∞. First, note that µ is bounded. For, if tj → 0 and {xj } ⊂ X is bounded, then {tj xj } ∈ c0 (X) so P j µ(Aj )tj xj is Cauchy and µ(Aj )tj xj → 0. Since {tj } ∈ c0 is arbitrary, {µ(Aj )xj } is bounded for any bounded sequence {xj }. Hence, {kµ(Aj )k} is bounded, and µ is bounded by Theorem 3.28. Put M = sup{kµ(A)k

: A ∈ A}. There exist a partition {A11 , ..., A1n , A1n+1 } and x11 , ..., x1n+1 , x1j ≤ 1,

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P

n+1 with j=1 µ(A1j )x1j > 1 + M , where some {A1j }, say A1n+1 , satisfies

µ ˆ (A1n+1 ) = ∞. Then

X

n 1 1

µ(Aj )xj ≥ 1 + M − µ(A1n+1 )x1n+1 ≥ 1.

j=1

Now treat A1n+1 as S above to obtain a partition {A21 , ..., A2m , A2m+1 }

P

2 2 of A1n+1 and x21 , ..., x2m+1 , x2j ≤ 1, with m j=1 µ(Aj )xj ≥ 2 and

µ ˆ (A2m+1 ) = ∞. Continuing this construction produces a pairwise disjoint sequence {Bj } = {A11 , ..., A1n , A21 , ..., A2m , ...} and a null sequence

{zj } = {x11 , ..., x1n , x21 /2, ..., x2m /2, ...} P such that the series j µ(Bj ) corresponding to the {Bj } is not c0 (X) multiplier Cauchy. For sequentially complete spaces, we can use Li’s Lemma 3.29 to give an improvement to Theorem 11.36. Corollary 11.37. Let Y be sequentially complete. The following are equivalent: (i) µ has finite (operator) semi-variation. P (ii) For every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) is c0 (X) multiplier convergent. (iii) For every pairwise disjoint sequence {Aj } ⊂ A and x = {xj } ∈ c0 (X), P the series j µ(Bj )zj converge uniformly for Bj ∈ A, Bj ⊂ Aj , kzj k ≤ kxj k. Proof: (i) and (ii) are equivalent by Theorem 11.36. Clearly, (iii) implies (ii). Assume (ii). We use Li’s Lemma 3.29. For this, set Ej = {(B, z) : B ∈ A, B ⊂ Aj , z ∈ X, kzk ≤ kxj k} and define fj : Ej → Y by fj (B, z) = µ(B)z. The condition (iii) now follows from Li’s Lemma.

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Another important property of operator valued set functions is that the semi-variation is continuous from above in the sense that if {Aj } ⊂ A and Aj ↓ ∅, then µ ˆ (Aj ) → 0. For example, this property was utilized by Bartle in developing properties of bilinear vector integrals ([Bar]). Note that if µ ˆ is continuous from above, then µ is countably additive (kµ(E)k ≤ µ ˆ (E) for E ∈ Σ). However, µ may be countably additive and µ ˆ may fail to be continuous from above. P Example 11.38. Let X be a Banach space and j Tj a series in L(X) which is subseries convergent in L(X) with respect to the operator norm but not bounded multiplier convergent (Example 11.27). Define µ : 2N → P L(X) by µ(E) = countably additive with respect j∈E Tj . Then µ is P to the operator norm. Since the series j Tj is not bounded multiplier convergent, there exist > 0, an increasing

sequence of intervals {Ij } and

P xj ∈ X, kxj k ≤ 1 such that k∈Ij Tk xk > . Put Ej = ∪k≥j Ik . Then Ej ↓ ∅ and µ ˆ (Ej ) ≥ µ ˆ (Ij ) ≥ so µ ˆ is not continuous from above. We give a series characterization for countably additive measures whose operator semi-variation is continuous from above. Theorem 11.39. Let Σ be a σ-algebra of subsets of a set S and µ : Σ → L(X, Y ) be countably additive in Ls (X, Y ). The following are equivalent: (i) µ ˆ is continuous from above. P (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series j µ(Aj ) is l∞ (X) multiplier Cauchy. Proof: Assume (i). Let {Aj } ⊂ Σ be pairwise disjoint and kxj k ≤ 1. Then

n+p

X n+p

ˆ (∪j=n Aj ) ≤ µ ˆ (∪∞ µ(Aj )xj j=n Aj ) → 0

≤µ

j=n

P since ∪∞ j=n Aj ↓ ∅. Therefore, j µ(Aj )xj is Cauchy and (ii) holds. Assume (ii). If (i) fails, there exist > 0, Aj ↓ ∅, Aj ∈ Σ, such that µ ˆ (Aj ) > . There exist n1 , {Ej : j = 1, ..., n1 } ⊂ Σ pairwise

disjoint and

P n 1 {xj : j = 1, ..., n1 }, kxj k ≤ 1, such that j=1 µ(A1 ∩ Ej )xj > . Since µ is countably additive in Ls (X, Y ), limi µ(Ai ∩ Ej )xj = 0 for j = 1, ..., n1 .

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Put k1 = 1. There exists k2 > k1 such that

n1

n1

X

X

(µ(Ak1 ∩ Ej )xj − µ(Ak2 ∩ Ej )xj ) =

µ((A A ) ∩ E )x k1 k2 j j > .

j=1

j=1

Treat Ak2 as A1 above to obtain a partition {Ej : j = n1 + 1, ..., n2 } of Ak2 ,

P 2 kxj k ≤ 1 for j = n1 + 1, ..., n2 , such that nj=n µ(Ak2 ∩ Ej )xj > 1 +1

P

n2

µ((Ak2 Ak3 ) ∩ Ej )xj > . Continuing and k3 > k2 such that j=n 1 +1 this construction produces a disjoint sequence {Ej } ⊂ Σ, kxj k ≤ 1, and increasing sequences {nj }, {kj } with

nj+1

X

µ((A A ) ∩ E )x kj+1 kj i i >

i=nj +1

for every j. The series

j+1 ∞ n X X

j=1 i=nj +1

µ((Akj+1 Akj ) ∩ Ei )xi

is not Cauchy so (ii) fails. Again if Y is sequentially complete, we can obtain an improvement of Theorem 11.39. Corollary 11.40. Let Y be sequentially complete and µ as in the theorem above. The following are equivalent: (i) µ ˆ is continuous from above. P (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series j µ(Aj ) is l∞ (X) multiplier convergent. (iii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series P∞ j=1 µ(Bj )xj converge uniformly for Bj ⊂ Aj , Bj ∈ Σ and xj ∈ X, kxj k ≤ 1. Proof: (i) and (ii) are equivalent by Theorem 11.39. Clearly, (iii) implies (ii). Assume (ii). We use Li’s Lemma 3.29. Set Ej = {(B, x) : B ∈ Σ, B ⊂ Aj , x ∈ X, kxk ≤ 1} and define fj : Ej → Y by fj (B, x) = µ(B)x. Then (iii) follows from Li’s Lemma.

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A sequence of finitely additive set functions µi : Σ → L(X, Y ) is uniformly continuous from above if Ej ↓ ∅ implies that limj µ ˆ i (Ej ) = 0 uniformly for i ∈ N. We have a characterization of uniform continuity from above in terms of multiplier convergent series. Theorem 11.41. Let µi : Σ → L(X, Y ) be countably additive in Ls (X, Y ) for every i ∈ N. The following are equivalent: (i) {µi } is uniformly continuous from above. (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ the series P∞ { j=1 µi (Aj ) : i ∈ N} are uniformly l ∞ (X) multiplier Cauchy.

Proof: That (i) implies (ii) follows as in Theorem 11.39. Suppose that (i) fails to hold. Then there exist > 0, Aj ↓ ∅ such that for every j there exist nj > j and mj with µ ˆ mj (Anj ) > . For j = 1 there exist n1 , m1 such that µ ˆ m1 (An1 ) > . There exist pairwise disjoint {El : 1 ≤ l ≤ N1 } and xl ∈ X, kxl k ≤ 1 such that

N

1

X

µm1 (An1 ∩ El )xl > .

l=1

Since each µi is countably additive in Ls (X, Y ), limj µm1 (Aj ∩ El )xl = 0 for 1 ≤ l ≤ N1 . There exist k1 > n1 such that

N 1

X

(µm1 (An1 ∩ El )xl − µm1 (Ak1 ∩ El )xl )

l=1

N 1

X

= µm1 ((An1 Ak1 ) ∩ El )xl > .

l=1

There exists n2 > k1 , m2 such that µ ˆ m2 (An2 ) > . By the construction above there exist N2 , pairwise disjoint {El : N1 + 1 ≤ l ≤ N2 } ⊂ Σ, xl ∈ X, kxl k ≤ 1,N1 + 1 ≤ l ≤ N2 and k2 > n2 such that

N 2

X

µm2 (An2 ∩ El )xl >

l=N1 +1

and

N 2

X

µm2 ((An2 Ak2 ) ∩ El )xl > .

l=N1 +1

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Continuing this construction produces a disjoint sequence {El }, xl ∈ X, kxl k ≤ 1, a sequence {mi } and increasing sequences {nj }, {kj }, kj > nj satisfying

NX

j+1

µmi ((Anj Akj ) ∩ El )xl

> .

l=Nj +1

The series

j+1 ∞ N X X

j=1 l=Nj +1

µmi ((Anj Akj ) ∩ El )xl : i ∈ N

are not uniformly l ∞ (X) multiplier Cauchy so (ii) fails to hold.

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Chapter 12

Orlicz-Pettis Theorems for Operator Valued Series

In this chapter we consider Orlicz-Pettis type theorems for operator valued series and vector valued multipliers. Throughout this chapter let X and Y be LCTVS and L(X, Y ) the space of all continuous linear operators from X into Y . Let E be a vector space of X valued sequences which contains P the subspace c00 (X) of all sequences which are eventually 0. If j Tj is a P∞ series in L(X, Y ) and x = {xj } is a multiplier, then the series j=1 Tj xj has values in Y so any Orlicz-Pettis Theorem must focus on the topology of Y . We first have a straightforward result. P Theorem 12.1. Assume that E is monotone. If the series j Tj is E multiplier convergent with respect to the weak topology σ(Y, Y 0 ) of Y , then P the series j Tj is E multiplier convergent with respect to the topologies γ(Y, Y 0 ) and λ(Y, Y 0 ) (see Appendix A for these topologies). In particular, P the series j Tj is E multiplier convergent in the original topology of Y .

P Proof: Let x ∈ E. Since the space E is monotone, the series ∞ j=1 Tj xj 0 is subseries convergent in the weak topology σ(Y, Y ). By the Orlicz-Pettis result in Corollary 4.11, the result follows immediately. Examples of vector valued sequence spaces which are monotone are given in Appendix C. We also have the vector analogue of Theorem 4.5. Recall the β-dual of E (with respect to the scalar field) is ∞ X

x0j , xj converges for every x = {xj } ∈ E E β = {x0j } ⊂ X 0 : j=1

187

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and that E, E β form a dual pair under the bilinear pairing 0

x ·x=

{x0j }

· {xj } =

∞ X

x0j , xj

j=1

(Appendix C). Also, a locally convex topology w defined for dual pairs X, X 0 is a Hellinger-Toeplitz topology if whenever X, X 0 and Y, Y 0 are dual pairs and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is linear and continuous, then T : (X, w(X, X 0 )) → (Y, w(Y, Y 0 )) is continuous (Appendix A.1). Theorem 12.2. Let w be a Hellinger-Toeplitz topology for dual pairs. If P (E, w(E, E β )) is an AK-space and j Tj is E multiplier convergent with P respect to σ(Y, Y 0 ), then j Tj is E multiplier convergent with respect to w(Y, Y 0 ). Proof: Let S : E → Y be the summing operator with respect to the P P 0 series j Tj and the topology σ(Y, Y 0 ), Sx = ∞ j=1 Tj xj (σ(Y, Y ) sum; β 0 Chapter 11). By Theorem 11.2, S is w(E, E ) − w(Y, Y ) continuous. If P x ∈ E, x = w(E, E β ) − limn nj=1 ej ⊗ xj so Sx = w(Y, Y 0 ) − lim n

n X j=1

S(ej ⊗ xj ) = w(Y, Y 0 ) − lim n

n X j=1

T j xj =

∞ X

T j xj .

j=1

Examples of vector valued AK-spaces are given in Appendix C. We next consider the strong topology on Y . The following example shows that without some condition on the multiplier space E the series P j Tj xj will not, in general, converge in the strong topology of Y .

Example 12.3. Equip X = l ∞ with the weak topology σ(l ∞ , l1 ) and let E = l∞ (X). Define Qk : l∞ → l∞ by Qk t = tk ek . If x = {xk } ∈ E, then P∞ P∞ k k Q k xk = x e is σ(l∞ , l1 ) convergent, but if x = {ek } ∈ E, k=1P k=1 P∞k k ∞ k then k=1 Qk e = k=1 e is not β(l∞ , l1 ) = k·k∞ convergent.

We now establish the analogue of Theorem 5.7. The space E has ∞GHP if whenever x ∈ E and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every

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subsequence of {nj } has a further subsequence {pj } such that the coordiP∞ natewise sum of the series j=1 apj χIpj x ∈ E (Appendix C). P Theorem 12.4. Let E have ∞-GHP. If j Tj is E multiplier convergent P with respect to σ(Y, Y 0 ), then j Tj is E multiplier convergent with respect to β(Y, Y 0 ). Proof: If the conclusion fails to hold, there exist x ∈ E, {yk0 } σ(Y 0 , Y ) bounded, > 0 and an increasing sequence of intervals {Ik } such that X 0 (∗) hyk , Tj xj i > for all k. j∈Ik

Since E has ∞-GHP, there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such that the coordiP natewise sum of the series ∞ k=1 aqk χIqk x ∈ E. Define a matrix X

yp0 i /api , Tl (apj xl ) ]. M = [mij ] = [ l∈Ipj

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since {yi0 } is σ(Y 0 , Y ) bounded and api → ∞. Next, given a subsequence there is a further subsequence {qj } such that y = P∞ j=1 aqj χIqj x ∈ E. Let z=

∞ X l=1

Tl y l =

∞ X X

Tl (aqj xl )

j=1 l∈Iqj

be the σ(Y, Y 0 ) sum of this series. Then ∞ X j=1

miqj = yp0 i /api , z → 0

so M is a K-matrix. By the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗). Examples of vector valued sequence spaces with ∞-GHP are given in Appendix C.

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Chapter 13

Hahn-Schur Theorems for Operator Valued Series

In this chapter we consider versions of the Hahn-Schur Theorem for series with values in the space of continuous linear operators and with multipliers which are vector valued with values in the domain of the operators. Throughout this chapter let X and Y be LCTVS, L(X, Y ) the space of all continuous linear operators from X into Y , and E a vector space of X valued sequences which contains c00 (X), the space of all X valued sequences which are eventually 0. The analogue of the hypothesis (H) for the Hahn-Schur Theorem in Chapter 7 for E multiplier convergent series is straightforward. P (H) Let j TijPbe E multiplier convergent in L(X, Y ) for every i. ∞ Assume that limi j=1 Tij xj exists for every x = {xj } ∈ E and that Tj x = limi Tij x exists for every x ∈ X. We, of course, need to have that Tj ∈ L(X, Y ) for every j for reasonable conclusions to hold. We will either assume this or impose conditions which guarantee that this holds. Similarly, the analogues of the conclusions of the Hahn-Schur Theorems of Chapter 7 are also straightforward. P∞ P (C1) The series j Tj is E multiplier convergent and limi j=1 Tij xj = P∞ every {xj } ∈ E. j=1 Tj xj for P∞ P∞ (C2) limi j=1 Tij xj = j=1 Tj xj uniformly for x = {xj } belonging to bounded subsets of E. P∞ (C3) The series j=1 Tij xj converge uniformly for x = {xj } belonging to bounded subsets of E. From Theorem 11.14 and Lemma 11.16, we have the following connection between (H) and conclusion (C1). Theorem 13.1. Assume (H), that E has signed-WGHP, and that there 191

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exist Tj ∈ L(X, Y ) such that limi Tij x = Tj x for every x ∈ E. Then (C1) holds. Hypothesis (H) implies that limi Tij x = Tj x exists for every x ∈ X so if the pair (X, Y ) has the Banach-Steinhaus property, it follows that Tj ∈ L(X, Y ) for every j and the hypothesis in Theorem 13.1 is satisfied. The hypothesis in (H) implies in particular, that if Tj ∈ L(X, Y ), then (i) For each j ∈ N, limi Tij = Tj in the strong operator topology of L(X, Y ). To see this, take x = ej ⊗ z for j ∈ N and z ∈ X. The following example shows that even in the presence of the BanachSteinhaus property, property SGHP, and condition (i), the hypothesis (H) does not imply conclusion (C2). Example 13.2. Let X = l 1 and Y = R so L(X, Y ) = (l 1 )0 = l∞ . For i, j ∈ N, let Tij = ei /2j so Tij t = (ei /2j ) · t = ti /2j for t ∈ l1 . Thus, for each j, limi Tij = 0 in the strong operator topology of L(X, Y ) = l ∞ which in this case is just σ(l ∞ , l1 ). Note, however, that {Tij }i does not converge to 0 in the norm topology k·k∞ of L(X, Y ) = l∞ . Let E = l∞ (l1 ), where j l 1 has

the norm topology k·k1 and E has the sup norm. If {t } ∈ E with

tj ≤ 1 for every j, we have 1 ∞ X

Tij tj =

j=1

∞ X j=1

ei · tj /2j =

∞ X

tji /2j

j=1

P∞ and since tji ≤ 1, the series j=1 Tij tj converges (absolutely). That is, P for each i, the series j Tij is E multiplier convergent. Since ∞ ∞ ∞ ∞ X ∞ ∞ X ∞ X X j j X X j j X Tij tj = 1/2j < ∞, ti /2 ≤ ti /2 = i=1 j=1

P∞

j=1 i=1

i=1 j=1

j

j=1

limi j=1 Tij t = 0. Thus, conditions (H) and (i) hold. However, condition (C2) does not hold. Indeed, for any i, let ti ∈ E be the constant sequence ti = {ei }. Then ∞ ∞ ∞ X X X Tij ti = ei · ei /2j = 1/2j = 1 j=1

j=1

j=1

while

∞ X j=1

Tkj ti = 0

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P∞ for k 6= i. Thus, limi j=1 Tij tk = 0 does not hold uniformly for {tk } belonging to bounded subsets of E and conclusion (C2) fails. Note that E = l ∞ (l1 ) has SGHP so a straightforward analogue of the Hahn-Schur Theorem given in Theorem 7.10 does not hold. In order to obtain versions of the Hahn-Schur Theorem in which conclusions (C2) and (C3) hold, it is necessary to strengthen condition (i). In particular, we need to replace the assumption that the sequence {Tij }i converges to Tj in the strong operator topology with convergence in Lb (X, Y ) in order to obtain the conclusion (C2). Actually, we consider the more general situation where the sequences {Tij }i converge in LA (X, Y ) (see Appendix A). We first consider the conclusion (C3). P Theorem 13.3. Let in L(X, Y ) for j Tij be E multiplier convergent P∞ every i. Assume that E has signed-SGHP. If limi j=1 Tij xj exists for P∞ every x = {xj } ∈ E, then for every bounded set A the series j=1 Tij xj converge uniformly for x ∈ A, i ∈ N, i.e., conclusion (C3) holds. Proof: If the conclusion fails to hold there exist a closed neighborhood of 0,U , such that for every i there exist ki > i, a finite interval Ii with min Ii > i, xi ∈ A with X Tki k xik ∈ / U. k∈Ii

Put i1 = 1. By the condition above there exist k1 > 1, an interval I1 with min I1 > i1 , x1 ∈ A with X Tk1 k x1k ∈ / U. k∈I1

By Theorem 11.9 (this uses signed-SGHP), there exists j1 such that ∞ X k=j

Tik xk ∈ U

for every x ∈ A, j ≥ j1 , 1 ≤ i ≤ k1 . Set i2 = max[I1 + 1, j1 ]. By the condition above, there exist k2 > i2 , an interval I2 with min I2 > i2 , x2 ∈ P 2 / U . Note that k2 > k1 . Continuing this A such that k∈I2 Tk2 k xk ∈ construction produces an increasing sequence {ki }, an increasing sequence of intervals {Ii }, xi ∈ A such that X (∗) Tki xik ∈ / U. k∈Ii

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Define an infinite matrix X Tki k xjk ]. M = [mij ] = [ k∈Ij

We claim that M is a signed K-matrix (Appendix D). First the columns of M converge by hypothesis. Next, given any subsequence there exist a further subsequence {pj } and a sequence of signs {sj } such that x = P∞ pj ∈ E. Then j=1 sj χIpj x ∞ ∞ ∞ X X X X p T k i k xk Tki k xkj = sj mipj = sj j=1

j=1

P∞

k=1

k∈Ipj

so limi j=1 sj mipj exists by hypothesis. Hence, M is a signed K-matrix so the diagonal of M converges to 0 by the signed version of the AntosikMikusinski Matrix Theorem (Appendix D.3). But, this contradicts (∗).

We next consider the converse of the Hahn-Schur result above. In what follows A will denote a family of bounded subsets of X whose union is X(see Appendix A). P Theorem 13.4. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ) and for every A ∈ A the series j=1 Tij xj converge uniformly P∞ for x ∈ A, i ∈ N. Then the sequences { j=1 Tij xj }i satisfy a Cauchy condition uniformly for x ∈ A. Proof: Let U be a neighborhood of 0 in Y and pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists n such P that ∞ Tij xj ∈ V for every x ∈ A, i ∈ N. There exists m such that Pn−1 j=n (T − Tkj )xj ∈ V for x ∈ A, and i, k ≥ m. If x ∈ A and i, k ≥ m, ij j=1 then ∞ ∞ X X Tij xj − Tkj xj j=1

j=1

=

n−1 X j=1

(Tij − Tkj )xj +

and the conclusion holds.

∞ X

j=n

Tij xj −

∞ X

j=n

Tkj xj ∈ V + V + V ⊂ U

From Theorem 13.4, we have condition (H). P Corollary 13.5. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in

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P Ls (X, Y ). If for each x ∈ E the series ∞ j=1 Tij xj converge uniformly for P∞ i ∈ N, then { j=1 Tij xj }i is a Cauchy sequence. Hence, if Y is sequentially complete, hypothesis (H) is satisfied. We can also obtain a boundedness result from uniform convergence of series. P Proposition 13.6. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ) and for every A ∈ A the series j=1 Tij xj converge uniformly P∞ for x ∈ A, i ∈ N. Then B = { j=1 Tij xj : i ∈ N, x ∈ A} is bounded for every A ∈ A . Proof: Let U be a neighborhood of 0 in Y and pick a balanced neighborhood, V , such that V + V ⊂ U . There exists n such that P∞ j=n Tij xj ∈ V for i ∈ N, x ∈ A. For every j, {Tij xj : i ∈ N, xj for x = {xj } ∈ A} is bounded by the hypothesis so there exists t > 1 Pn−1 P∞ such that { j=1 Tij xj : i ∈ N, x ∈ A} ⊂ tV . Therefore, j=1 Tij xj = Pn−1 P∞ j=1 Tij xj + j=n Tij xj ∈ tV + V ⊂ tU for i ∈ N, x ∈ A.

Theorem 13.3 gives sufficient conditions for the uniform convergence hypothesis in Theorem 13.4, Corollary 13.5 and Proposition 13.6 to hold. We next consider the conclusions (C1) and (C2). P Theorem 13.7. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ). If limi j=1 Tij xj exists for every x = {xj } ∈ E and for every P∞ A ∈ A the series for x ∈ A, i ∈ N, then j=1 Tij xj converge uniformly P P∞ P∞ T is E multiplier convergent and lim T j i j j=1 ij xj = j=1 Tj xj uniformly for x ∈ A. That is, conclusions (C1) and (C2) hold for the family A. P Proof: That j Tj is E multiplier convergent and lim i

∞ X j=1

Tij xj =

∞ X

T j xj

j=1

for x ∈ E follows from Lemma 11.16. Let U be a neighborhood of 0 in Y and pick a closed, symmetric neighborhood of 0, V , such that V + V + V ⊂ U . Pn There exists N such that j=m Tij xj ∈ V for n > m ≥ N, i ∈ N and Pn x ∈ A. Hence, j=m Tj xj ∈ V for n > m ≥ N, i ∈ N and x ∈ A and,

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P therefore, ∞ j=m Tj xj ∈ V for m ≥ N, x ∈ A. There exists M such that PN −1 j=1 (Tij − Tj )xj ∈ V for i ≥ M, x ∈ A. If x ∈ A and i ≥ M , we have ∞ X j=1

Tij xj −

∞ X j=1

T j xj =

∞ X

j=N

and the result follows.

Tij xj −

∞ X

j=N

T j xj +

N −1 X j=1

(Tij −Tj )xj ∈ V +V +V ⊂ U

From Theorems 13.3 and 13.7, we have a Hahn-Schur Theorem for LA (X, Y ). P Corollary 13.8. Assume that E has signed-SGHP. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist P∞ Tj ∈ L(X, Y ) such that limi Tij = Tj in LA (X, Y ). If limi j=1 Tij xj P exists for every x = {xj } ∈ E, then j Tj is E multiplier converP P∞ gent, limi ∞ T x = T x uniformly for x ∈ A and the series j=1 ij j j=1 j j P∞ j=1 Tij xj converge uniformly for x ∈ A, i ∈ N. That is, conclusions (C2) and (C3) hold for the family A. A sufficient condition for there to exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every x ∈ X is that the pair (X, Y ) has the Banach-Steinhaus property. However, as Example 13.2 shows we must have convergence in LA (X, Y ) in order to obtain the conclusion in Theorem 13.7. Corollary 13.9. Assume that E has signed-SGHP and that Y is sequenP tially complete. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in LA (X, Y ). The following are equivalent: (i) (H). P∞ (ii) For each x ∈ E the series j=1 Tij xj converge uniformly for i ∈ N. P∞ (iii) For each A ∈ A the series j=1 Tij xj converge uniformly for i ∈ N, x ∈ A. P (iv) For each A ∈ A, the series j Tj is E multiplier convergent and P∞ P∞ limi j=1 Tij xj = j=1 Tj xj uniformly for x ∈ A. Proof: That (i) implies (iii) follows from Theorem 13.3; (iii) implies (iv) from Theorem 13.7; (ii) implies (i) from Corollary 13.5; clearly (iii) implies (ii) and (iv) implies (i).

We give a statement of the results above for the case of bounded (l ∞ (X)) multiplier convergent series. Note that l ∞ (X) has SGHP so these results apply.

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P Corollary 13.10. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj P∞ in Lb (X, Y ). If limi j=1 Tij xj exists for every x = {xj } ∈ l∞ (X), P∞ P then the series j Tj is bounded multiplier convergent, limi j=1 Tij xj = P∞ for x = {xj } belonging to bounded subsets A ⊂ l ∞ (X) j=1 Tj xj uniformly P∞ and the series j=1 Tij xj converge uniformly for x = {xj } belonging to bounded subsets A ⊂ l ∞ (X). P Corollary 13.11. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in Ls (X, Y ). If Y is sequentially complete and for every x = {xj } ∈ l∞ (X), P∞ P∞ the series j=1 Tij xj converge uniformly for i ∈ N, then limi j=1 Tij xj exists for every x = {xj } ∈ l∞ (X). From Proposition 13.6, we have a boundedness result. P Corollary 13.12. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ Lb (X, Y ). If for each bounded subset A of l ∞ (X) the series j=1 Tij xj P∞ converge uniformly for i ∈ N, x ∈ A, then B = { j=1 Tij xj : i ∈ N, x ∈ A} is bounded. Unlike the case of scalar multipliers, the space m0 (X) (or the subset {χσ x : σ ⊂ N, x ∈ X}) does not have SGHP so the results above do not apply to m0 (X) multiplier convergent series. However, m0 (X) does have WGHP so Theorem 13.1 applies and gives the following result. P Theorem 13.13. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij x = Tj x for P∞ every x ∈ X. If limi j=1 Tij xj exists for every x = {xj } ∈ m0 (X), then P∞ P∞ P j=1 Tj xj j=1 Tij xj = j Tj is m0 (X) multiplier convergent and limi for every x = {xj } ∈ m0 (X). The following example shows that the uniform convergence conclusions in (C2) and (C3) do not hold for m0 (X) multiplier convergent series. Example 13.14. Let 1 ≤ p < ∞ and define Qj : lp → lp by Qj t = tj ej as P in Example 11.10. As noted in Example 11.10, the series j Qj is m0 (lp ) multiplier convergent. Now define Tij = Qj if j ≤ i and Tij = 0 if j > i. We have that ∞ ∞ X X Q j tj Tij tj = (∗) lim i

j=1

j=1

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for any t = {tj } ∈ m0 (lp ) and limi Tij = Qj in Lb (lp ) for every j. However, the limit in (∗) is not uniform for t belonging to bounded subsets of m0 (lp ) [take t to be the constant sequence {ek } in m0 (lp ) so P∞ Pi k k j=1 Tij tj = j=1 Qj e = e if i ≥ k]. Similarly, the uniform convergence condition in (C3) does not hold. From Theorems and 7.17 and Corollary 7.19, we do have the following results for m0 (X) multiplier convergent series. P Theorem 13.15. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every P x ∈ X. If limi ∞ j=1 Tij xj exists for every {xj } ∈ m0 (X), then

P∞ P Tj is m0 (X) multiplier convergent and limi j=1 Tij xj = j P∞ } ∈ m0 (X), j=1 Tj xj for every {x P Pj (ii) for every x ∈ X, limi j∈σ Tij x = j∈σ Tj x uniformly for σ ⊂ N, P (iii) for every x ∈ X, the series j∈σ Tij x converge uniformly for i ∈ N, σ ⊂ N. (i)

P Theorem 13.16. Let Y be sequentially complete. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every x ∈ X. The following are equivalent: P∞ (1) limi j=1 Tij xj exists for every {xj } ∈ m0 (X). P (2) The series j Tj is m0 (X) multiplier convergent and for every x ∈ X, P P limi j∈σ Tij x = j∈σ Tj x uniformly for σ ⊂ N. P (3) For every x ∈ X the series j∈σ Tij x converge uniformly for i ∈ N, σ ⊂ N. P (4) For every x ∈ X and σ ⊂ N the series j∈σ Tij x converge uniformly for i ∈ N. P In the scalar case, if X is sequentially complete, j xij is m0 multiplier P convergent for every i, limi xij = xj exists for every j and limi j∈σ xij P exists for every σ ⊂ N, then the series j xj is l∞ multiplier convergent and P∞ P∞ limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k1 ≤ 1 (Theorem 7.29). As the following example shows, the analogue of this statement does not hold for m0 (X) and l∞ (X) multiplier convergent series. P Example 13.17. Let X = l 1 and let j Tj be a series in L(X) which is subseries convergent but not l ∞ (X) multiplier convergent (Example 11.27). Set Tij = Tj if j ≤ i and Tij = 0 if j > i. If σ ⊂ N and x ∈ X,

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P P∞ P P then limi j∈σ Tij x = j∈σ Tj x so limi ∞ j=1 Tij xj = j=1 Tj xj for any {xj } ∈ m0 (X). However, if x = {xj } ∈ l∞ (X), k{xj }k ≤ 1, is such that P∞ P∞ j=1 Tj xj does not converge, then lim i j=1 Tij xj does not exist.

In Chapter 7 it was noted that a scalar matrix [aij ] which maps m0 into c also maps l∞ into c (condition (S) following Theorem 7.2). It was also shown that a vector valued matrix [xij ] which maps m0 into c(X), the space of convergent X valued sequences, also maps l ∞ into c(X) (see condition (S’) following Theorem 7.29). The example above shows that an operator valued matrix [Tij ] may map m0 (X) into c(Y ) but fail to map l ∞ (X) into c(Y ). Finally, we give an application of the operator version of the HahnSchur Theorem to obtain a version of the Nikodym Convergence Theorem for operator valued measures. Let Σ be a σ-algebra of subsets of a set S and let µi : Σ → LA (X, Y ) be countably additive. The Nikodym Convergence Theorem given in Theorem 7.47 implies that if limi µi (E) = µ(E) exists in LA (X, Y ) for every E ∈ Σ, then µ : Σ → LA (X, Y ) is countably additive and {µi } is uniformly countably additive in LA (X, Y ). We seek to obtain a version of the Nikodym Convergence Theorem for operator valued measures whose operator semi-variation is continuous from above (see the definition following Corollary 11.37). Let X, Y be normed spaces and µ : Σ → L(X, Y ) be finitely additive. Recall the operator semi-variation of µ, µ ˆ , is defined to be

n

X

: {Aj }n a partition of E, kxj k ≤ 1 µ ˆ (E) = sup µ(A )x j j j=1

j=1

(see the definition preceding Theorem 11.34). The semi-variation of µ, µ ˆ , is continuous from above if Ej ↓ ∅ implies that µ ˆ (Ej ) ↓ 0. Theorem 11.39 gives necessary and sufficient conditions for µ ˆ to be continuous from above. We derive a Nikodym Convergence Theorem for measures whose variation is continuous from above. First, the following example shows that a straightforward analogue of the Nikodym Convergence Theorem fails for such measures. P Example 13.18. Let X be a Banach space and j Tj be a series in Lb (X) which is subseries convergent in Lb (X) with respect to the operator norm but not bounded multiplier convergent (Example 11.27). Define µi : 2N → P Lb (X) by µi (E) = j∈E∩[1,i] Tj . Then each µi is countably additive and its semi-variation is continuous from above. However, limi µi (E) = µ(E) =

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j∈E Tj exists in Lb (X) but the semi-variation of µ is not continuous from above (Example 11.38).

We obtain a version of the Nikodym Convergence Theorem for measures whose semi-variation is continuous from above by strengthening the condition that limi µi (E) = µ(E) exists in Lb (X, Y ) for every E ∈ Σ to the condition that P∞ (#) limi j=1 µi (Ej )xj exists in Y for every pairwise disjoint sequence {Ej } ⊂ Σ and kxj k ≤ 1. Recall that a sequence of finitely additive set functions {µi } with finite semi-variation is uniformly continuous from above if Ej ↓ ∅ implies that limj µi (Ej ) = 0 uniformly for i ∈ N. Theorem 11.41 gives necessary and sufficient conditions for a sequence of set functions to be uniformly continuous from above. Theorem 13.19. Let Y be sequentially complete and let µi : Σ → Ls (X, Y ) be countably additive with µ ˆ i continuous from above. Assume that condition (#) above holds and for every E ∈ Σ there exists µ(E) ∈ L(X, Y ) with limi µi (E) = µ(E) in Lb (X, Y ). Then the semi-variation of µ, µ ˆ , is continuous from above and the semi-variations {ˆ µi } are uniformly continuous from above. Proof: Let {Ej } ⊂ Σ be pairwise disjoint and xj ∈ X, kxj k ≤ 1. P Then condition (#) and Corollary 13.10 imply that the series j µ(Ej ) P is bounded multiplier convergent and the series { j µi (Ej )}i∈N are uniformly bounded multiplier convergent. Theorems 11.39 and 11.41 now give the desired conclusion.

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Chapter 14

Automatic Continuity for Operator Valued Matrices

In this chapter we will establish several automatic continuity results for operator valued matrices analogous to those for scalar valued matrices established in Chapter 10. We first consider the relationship between an operator matrix and its transpose which require some technical assumptions. Let X, Y be TVS with L(X, Y ) the space of all continuous linear operators from X into Y . Let E [F ] be a vector space of X [Y ] valued sequences which contains c00 (X) [c00 (Y )], the space of X [Y ] valued sequences which are eventually 0 and let A = [Aij ] be an infinite matrix with Aij ∈ L(X, Y ). P∞ We say that A maps E into F if the series j=1 Aij xj converges for P∞ every i ∈ N and x = {xj } ∈ E and Ax = { j=1 Aij xj }i ∈ F for every x = {xj } ∈ E; we write A : E → F if A maps E into F . Note that if A : E → F , then the rows of A must be E multiplier convergent. We begin by considering the analogue of Theorem 10.3. Recall that the (scalar) β-dual of E is defined to be ∞ X E β = {yj } : yj ∈ X 0 , hyj , xj i = y · x converges for every x = {xj } ∈ E j=1

and E, E β form a dual pair under the pairing y · x (Appendix C).

Theorem 14.1. Let A : E → F and assume that σ(E β , E) is sequentially complete. Then A is σ(E, E β ) − σ(F, F β ) continuous and, therefore, w(E, E β ) − w(F, F β ) continuous for every Hellinger-Toeplitz topology w (Appendix A.1). 201

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Proof: Let y ∈ F β , x ∈ E. Then + + * * m ∞ ∞ ∞ X X X X yi , Aij xj y · Ax = yi , Aij xj = lim i=1

= lim m

j=1 m ∞ X X j=1 i=1

m

i=1

j=1

A0ij yi , xj = lim z m · x, m

Pm 0 m where zjm = = {zjm }j ∈ E β . Then {z m } is σ(E β , E) i=1 Aij yi , z Cauchy and, therefore, there exists z ∈ E β such that z m → z with respect to σ(E β , E) with y · Ax = z · x for every x ∈ E. This implies that A is σ(E, E β )−σ(F, F β ) continuous. The last statement follows from Appendix A.1. A sufficient condition for σ(E β , E) to be sequentially complete is that E have the signed-WGHP and X 0 be σ(X 0 , X) sequentially complete (Corollary 11.18). In the scalar case Theorem 14.1 also contains a statement concerning the transpose matrix of A; however, the operator case which we now discuss is more complicated. The transpose of the matrix A is defined to be AT = [A0ji ]. In order for the transpose AT to be defined on F β it is necessary P∞ that the series i=1 A0ij yi converge in X 0 for each y ∈ F β with respect to some locally convex topology. In order for this to be the case we henceforth assume (*) (X 0 , σ(X 0 , X)) is sequentially complete. Under the assumption in (*) the transpose matrix AT will map F β into s(X 0 ), the space of all X 0 valued sequences (Appendix C). Theorem 14.1 gives sufficient conditions for AT to map F β into E β . Corollary 14.2. Let A : E → F and assume that σ(E β , E) is sequentially complete. Under assumption (*), AT : F β → E β and y · Ax = AT y · x

for x ∈ E, y ∈ F β . Hence, AT is σ(F β , F ) − σ(E β , E) continuous and, therefore, w(F β , F )− w(E β , E) continuous with respect to any HellingerToeplitz topology w (Appendix A.1). Proof: Let the notation be as in the proof of Theorem 14.1. Then y · Ax = lim z m · x = z · x = AT y · x m P∞ 0 m since limm zj = zj = i=1 Aij yi for each j. The last statements are immediate from the equation y · Ax = AT y · x.

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Recall that even in the scalar case the matrix A may map E into F but the transpose matrix AT may fail to map F β into E β (Example 10.6). It can also be the case that A : E → F and AT : F β → E β but the condition y · Ax = AT y · x may fail to hold (Example 10.9). We next consider the problem of when AT : F β → E β will imply that A : E → F . For this we always assume that dual spaces carry the strong topology and we write E ββ = (E β )β and AT T = (AT )T . Then E ββ is a space of X 00 valued sequences and AT T consists of linear operators A00ij : X 00 → Y 00 whose restriction to X is just Aij . In order that the transpose matrix AT T map E ββ into s(Y 00 ), we assume (**) (Y 00 , σ(Y 00 , Y 0 )) is sequentially complete. From Theorem 14.1 and Corollary 14.2, we have Corollary 14.3. Assume that σ(F ββ , F β ) is sequentially complete. If AT : F β → E β , then AT T : E ββ → F ββ and z · AT y = AT T z · y for all z ∈ E ββ , y ∈ F β . Moreover, AT is σ(F β , F ββ ) − σ(E β , E ββ ) continuous and AT T is σ(E ββ , E β ) − σ(F ββ , F β ) continuous. Proof: Note that the assumption that σ(F ββ , F β ) is sequentially complete implies condition (**) so the transpose matrix AT T is defined on E ββ and the result follows from Theorem 14.1 and Corollary 14.2. To consider the problem of when AT : F β → E β will imply that A : E → F we use the second transpose AT T . For this we establish a lemma. Lemma 14.4. Let Y be semi-reflexive. If σ(F, F β ) is sequentially complete, then F = F ββ . Proof: Let z ∈ F ββ . Since Y is semi-reflexive and F ⊃ c00 (Y ), z n = (z1 , ...zn , 0, 0, ...) ∈ F , and since z ∈ F ββ , {z n } is σ(F, F β ) Cauchy. Since σ(F, F β ) is sequentially complete, {z n } converges to an element of F which must be z. Note that semi-reflexivity of Y is a necessary condition for the identity F = F ββ to hold. From Corollary 14.3 and Lemma 14.4, we obtain Corollary 14.5. Assume that σ(F, F β ) is sequentially complete and Y is semi-reflexive. If AT : F β → E β , then AT T : E ββ → F and z · AT y = AT T z · y for z ∈ E ββ , y ∈ F β so AT T is σ(E ββ , E β ) − σ(F, F β ) continuous and AT is σ(F β , F ) − σ(E β , E) continuous. In particular, A : E → F and y · Ax = AT y · x holds for x ∈ E, y ∈ F β and A is σ(E, E β ) − σ(F, F β )

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continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Corollary 14.6. Assume that Y is semi-reflexive and that σ(F, F β ) and σ(E β , E) are sequentially complete. The following are equivalent: (i) A : E → F (ii) AT : F β → E β (iii) AT T : E ββ → F. We now give applications of the automatic continuity results to concrete sequences spaces. Recall that l 1 (X) is the space of all absolutely convergent X valued series and if X is quasi-barrelled, the β-dual of l 1 (X) is l∞ (Xb0 ), the space of all strongly bounded sequences with values in X 0 (Appendix C.25). Theorem 14.7. Let X be barrelled and Y be quasi-barrelled. If A : l1 (X) → l1 (Y ), then AT : l∞ (Yb0 ) → l∞ (Xb0 ) and A is σ(l1 (X), l∞ (Xb0 )) − σ(l1 (Y ), l∞ (Yb0 )) continuous and AT is ∞ 0 1 ∞ 0 1 σ(l (Yb ), l (Y )) − σ(l (Xb ), l (X)) continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Proof: Since X is barrelled, condition (*) is satisfied so the transpose map AT is defined on l∞ (Yb0 ). The space l1 (X) is monotone and (X 0 , σ(X 0 , X)) is sequentially complete so Corollary 11.18 implies that (l∞ (Xb0 ), σ(l∞ (Xb0 ), l1 (X))) is sequentially complete. Theorem 14.1 and Corollary 14.2 give the result. We next consider matrices acting between l ∞ spaces. The β-dual of l∞ (X) is l1 (Xb0 ) (Appendix C.23). Theorem 14.8. Let X be barrelled. If A : l ∞ (X) → l∞ (Y ), then AT : l1 (Yb0 ) → l1 (Xb0 ) and A is σ(l∞ (X), l1 (Xb0 )) − σ(l∞ (Y ), l1 (Yb0 )) continuous and AT is σ(l1 (Yb0 ), l∞ (Y )) − σ(l1 (Xb0 ), l∞ (X)) continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Proof: The space l ∞ (X) is monotone and (X 0 , σ(X 0 , X)) is sequentially complete so Corollary 11.18 implies that (l 1 (Xb0 ), σ(l1 (Xb0 ), l∞ (X))) is sequentially complete. Theorem 14.1 and Corollary 14.2 give the result. We give an additional automatic continuity result. The space E has the property I if the injections x → ej ⊗ x from X into E are continuous for every j.

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Theorem 14.9. Assume that every T ∈ E βY induces a continuous linear operator belonging to L(E, Y ), (E, F ) has the Banach-Steinhaus property and F is an AK-space with property I. If A : E → F , then A is continuous. Proof: Let Ri be the ith row of A so that Ri ∈ L(E, Y ). By hypothesis x → Rj · x → (Rj · x)ej is continuous from E into F so the operator P An : E → F defined by An x = nj=1 (Rj · x)ej is continuous. By the AK assumption, An → A pointwise so A is continuous by the Banach-Steinhaus assumption. Sufficient conditions for each T ∈ E βY to induce a continuous linear operator are given in Theorem 11.2. The analogue of Theorem 10.10 also holds with essentially the same proof. Theorem 14.10. Let (E, τ ) be a K-space with 0-GHP and assume that condition (*) is satisfied. If A : E → F , then A is τ − σ(F, F β ) continuous.

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Appendix A Topological Vector Spaces

In this appendix we will record some of the results pertaining to topological vector spaces (TVS) which will be used throughout the text. For convenience we assume that all vector spaces are real. A topological vector space (TVS) is a vector space X supplied with a topology τ such that the operations of addition and scalar multiplication are continuous with respect to τ . A subset U of a TVS X is symmetric (balanced) if x ∈ U implies −x ∈ U (x ∈ U implies tx ∈ U for |t| ≤ 1). Any TVS has a neighborhood base at 0 which consists of symmetric (balanced, closed) sets. See [Sch], [Sw2] or [Wi] for discussions of TVS. One other result pertaining to TVS will be used. A quasi-norm on a vector space X is a map |·| : X → [0, ∞) satisfying |0| = 0, |x + y| ≤ |x|+|y| and |x| = |−x| for x, y ∈ X, and if tk → t in R and xk , x ∈ X with |xk − x| → 0, then |tk xk − tx| → 0. If the quasi-norm satisfies |x| = 0 iff x = 0, then the quasi-norm is said to be total. If |·| is a quasi-norm on X, then d (x, y) = |x − y| defines a semi-metric on X which is a metric iff |·| is total. The semi-metric d is translation invariant in the sense that d(x + z, y + z) = d(x, y) for x, y, z ∈ X. The space X is a TVS under the semi-metric d. A useful fact which we will use is that the topology of any TVS is generated by a family of quasi-norms ([BM ]). That is, if τ is the vector topology of X, then there exists a family δ of quasi-norms {|·|a : a ∈ A} which generate δ τ in the sense that a net {x } in X converges to 0 with respect to τ iff x − x a → 0 for every a ∈ A. A TVS X is locally convex (LCT V S) if X has a neighborhood base at 0 consisting of convex sets. Any LCTVS also has a base at 0 consisting of closed, absolutely convex sets. The topology τ of any LCTVS is generated by a family of semi-norms {pa : a ∈ A} as above. See [Sch], [Sw2] or [Wi] 207

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for the basic properties of LCTVS. We now give a description of polar topologies which will play an important role when we discuss Orlicz-Pettis Theorems. A pair of vector spaces X, X 0 are said to be in duality if there is a bilinear map h·, ·i : X 0 × X → R such that (i) {h·, xi : x ∈ X, x 6= 0} separates the points of X and (ii) {hx0 , ·i : x0 ∈ X 0 , x0 6= 0} separates the points of X 0 . If X, X 0 are in duality, the weak topology of X (X 0 ), σ(X, X 0 ) (σ(X 0 X)), is the locally convex vector topology generated by the semi-norms p(x) = |< x0 , x >| , x0 ∈ X 0 (p (x0 ) = |< x0 , x >|) , x ∈ X). A subset A ⊂ X is σ(X, X 0 ) bounded iff sup {|< x0 , x >| : x ∈ A} < ∞ for every x0 ∈ X 0 . Let A be a family of σ(X 0 X) bounded subsets of X 0 . For A ∈ A, set pA (x) = sup{|hx, x0 i| : x0 ∈ A}. The semi-norms {pA : A ∈ A} generate a locally convex topology τA on X called the polar topology of uniform convergence on A (for the reason the topology is called a polar topology, see [Sw2] 17). Thus, a net {xδ } converges to 0 in τA iff x0 , xδ → 0 uniformly for x0 ∈ A for every A ∈ A. We will use the following polar topologies in the text. (1) The weak topology σ(X, X 0 ) is generated by the family A of all finite subsets of X 0 . (2) The strong topology of X, denoted by β(X, X 0 ), is generated by the family of all σ(X 0 , X) bounded subsets of X 0 . (3) The Mackey topology, denoted by τ (X, X 0 ), is generated by the family of all absolutely convex, σ(X 0 , X) compact subsets of X 0 . (4) The polar topology generated by the family of all σ(X 0 , X) compact subsets of X 0 is denoted by λ(X, X 0 ). (5) A subset A ⊂ X 0 is said to be conditionally σ(X 0 , n X) sequentially o compact if every sequence x0j ⊂ A has a subsequence x0nj which is E D σ(X 0 , X) Cauchy, i.e., lim x0nj , x exists for every x ∈ X. The polar topology generated by the family of conditionally σ(X 0 , X) sequentially compact sets is denoted by γ(X, X 0 ). The topology λ(X, X 0 ) was introduced by G. Bennett and Kalton ([BK]) and is obviously stronger than the Mackey topology τ (X, X 0 ); it can be strictly stronger ([K1] 21.4).

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Let w(X, X 0 ) be a polar topology defined for all dual pairs X, X 0 . We have the following useful notion introduced by Wilansky ([Wi]). Definition A.1. The topology w(·, ·) is a Hellinger-Toeplitz topology if whenever T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then T : (X, w (X, X 0 )) → (Y, w (Y, Y 0 )) is continuous. Wilansky has given a very useful criterion for Hellinger-Toeplitz topologies ([Wi] 11.2.2). If T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then the adjoint (transpose) operator of T is the linear operator T 0 : Y 0 → X 0 defined by hT 0 y 0 , xi = hy 0 , T xi for x ∈ X, y 0 ∈ Y 0 . The adjoint T 0 is σ (Y 0 , Y ) − σ(X 0 , X) continuous. Let A(X 0 , X) be a family of σ(X 0 , X) bounded subsets which is defined for all dual pairs X, X 0 . Let w(X, X 0 ) be the polar topology generated by the elements of A(X 0 , X). We have Theorem A.2. The topology w(X, X 0 ) is a Hellinger-Toeplitz topology if whenever T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then T 0 A (Y 0 , Y ) ⊂ A(X 0 , X). 0 Proof: Let {xδ } be a net in X which converges to 0 in

0w(X,δ X ). 0 0 0 0 0 = Let , X) so y , T x

0 0 Aδ ∈ A(Y , Y ). Then {T0 y : y ∈ A} ∈ A(X T y , x → 0 uniformly for y ∈ A. That is, T xδ → 0 in w(Y, Y 0 ).

Theorem A.2 clearly implies that the polar topologies given in (1) - (5) are all Hellinger-Toeplitz topologies. We next consider another notion due to Wilansky which is useful in treating Orlicz-Pettis results.

Definition A.3. Let X be a vector space and σ and τ two vector topologies on X. We say that τ is linked to σ if τ has a neighborhood base at 0 consisting of σ closed sets. [The terminology is that of Wilansky ([Wi] 6.1.9).]

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For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are linked to the weak topology σ(X, X 0 ). Lemma A.4. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. (i) If {xj } ⊂ X is τ Cauchy and if σ-lim xj = x, then τ -lim xj = x. (ii) If (X, σ) is sequentially complete and σ ⊂ τ, then (X, τ ) is sequentially complete. Proof: (i): Let U be a τ neighborhood of 0 which is σ closed. There exists N such that j, k ≥ N implies xj − xk ∈ U . Since U is τ closed, xj − x ∈ U for j ≥ N . (ii) follows from (i). Remark A.5. It is important that the topologies σ and τ are linked in the Lemma. For example, consider the space c with its weak topology P σ(c, l1 ) and the topology of pointwise convergence p. The series j ej is p convergent, the partial sums of the series are σ(c, l 1 ) Cauchy, but the series is not σ(c, l1 ) convergent [here we are using the pairing between c and l 1 where l1 is the topological dual of c ([Sw2] 5.12)]. We now establish a basic lemma. Lemma A.6. Let X be a vector space and σ and τ two vector topologies P on X such that τ is linked to σ. If every series j xj which is σ subseries convergent satisfies τ − lim xj = 0, then every series in X which is σ subseries convergent is τ subseries convergent. Proof: By the previous lemma it suffices to show that every σ subseries P Pn convergent series j xj is such that its partial sums sn = j=1 xj form P a τ Cauchy sequence. If j xj is σ subseries convergent but {sn } is not τ Cauchy, there exists a τ neighborhood of 0, U , and a pairwise disjoint sequence of finite subsets, {Ik }, of N such that max Ik < min Ik+1 and P P ∈ / U . The series k zk is σ subseries convergent, being zk = j∈Ik xj P a subseries of / U and j xj , so τ − lim zk = 0. This contradicts zk ∈ establishes the result. We next consider topologies for spaces of continuous linear operators. Let X, Y be LCTVS and L (X, Y ) the space of all continuous linear operators T : X → Y. Let A be a family of bounded (σ (X, X 0 )) subsets of X

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and let Q be the family of all continuous semi-norms on Y. Then A and Q generate a locally convex topology on L (X, Y ) defined by (∗) pqA (T ) = sup {q(T x) : x ∈ A} , q ∈ Q, A ∈ A. We denote by LA (X, Y ), L(X, Y ) with the locally convex topology generated by the semi-norms in (∗). A net {T δ } in L (X, Y ) converges to 0 in LA (X, Y ) iff T δ x → 0 uniformly for x ∈ A for every A ∈ A and for this reason, the topology is called the topology of uniform convergence on A or A -uniform convergence. We have the following examples which will be considered. (i) If A is the family of all finite subsets of X, we denote the topology generated by A by Ls (X, Y ). This is just the topology of pointwise convergence on X and is called the strong operator topology. Thus, a net {T δ } converges to 0 in Ls (X, Y ) iff T δ x → 0 in Y for every x ∈ X. (ii) Let A again be the family of all finite subsets of X but equip Y with the weak topology σ(Y, Y 0 ). This topology is called the weak operator topology. Thus, a net {T δ } in L(X, Y ) converges to 0 in the weak operator iff for every x ∈ X , T δ x → 0 in σ(Y, Y 0 ) or, equivalently,

0 topology δ y , T x → 0 for every x ∈ X, y 0 ∈ Y 0 . (iii) Let A be the family of all bounded subsets of X. Then the topology generated by A is denoted by Lb (X, Y ). If X and Y are normed spaces, the topology Lb (X, Y ) is generated by the operator norm kT k = sup{kT xk : kxk ≤ 1} and is called the uniform operator topology. (iv) Let A be the family of all precompact subsets of X. Then the topology generated by A is denoted by Lpc (X, Y ). (v) Let A be the family of all compact subsets of X. Then the topology generated by A is denoted by Lc (X, Y ). (vi) Let A be the family of all sequences {xj } ⊂ X which converge to 0. Then the topology generated by A is denoted by L→0 (X, Y ).

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Appendix B Scalar Sequence Spaces

In this appendix we will list the sequence spaces and their properties which will be used in the text. If λ is a vector space of (real) sequences containing c00 , the space of all sequences which are eventually 0, the β-dual of λ is defined to be λβ ( =

s = {sj } :

∞ X

)

sj tj = {sj } · {tj } = s · t converges for every t = {tj } ∈ λ .

j=1

Since λ ⊃ c00 , the pair λ, λβ are in duality with respect to the pairing s · t = {sj } · {tj } for s ∈ λβ , t ∈ λ. We now list some of the scalar valued sequence spaces which will be encountered in the text. • • • • • •

c00 = {{tj } : tj = 0 eventually} c0 = {{tj } : lim tj = 0} cc = {{tj } : tj is eventually constant} c= {{tj } : lim tj exists} m0 = {{tj } : the range of {tj } is finite} = span{χσ : σ ⊂ N} l∞ = {{tj } : supj {|tj |} = k{tj }k∞ < ∞}

All of the sequence spaces above are usually equipped with the supnorm, k·k∞ , defined above. For 0 < p < 1, • lp = {{tj } :

P∞

j=1

p

|tj | = |{tj }|p < ∞} 213

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The space lp (0 < p < 1) is usually equipped with the quasi-norm |·|p which generates the metric dp (s, t) = |s − t|p under which it is complete. For 1 ≤ p < ∞, • lp = {{tj } : (

P∞

j=1

p

|tj | )1/p = k{tj }kp < ∞}

The space lp is usually equipped with the norm k·kp under which it is a Banach space. P n • bs= {{tj } : supn { j=1 tj = ktkbs < ∞}

The space bs is called the space of bounded series and is usually equipped with the norm k·kbs under which it is a Banach space. • cs= {{tj } :

P∞

converges}

P∞

|tj+1 − tj | < ∞}

j=1 tj

The space cs is a subspace of bs and is called the space of convergent series; cs is a closed subspace of bs under the norm k·kbs . • bv= {{tj } :

j=1

The space bv is called the space of sequences of bounded variation and P∞ is a Banach space under the norm k{tj }kbv = j=1 |tj+1 − tj | + |t1 | . • bv0 = bv ∩ c0

The space bv0 is a closed subspace of bv. • s= the space of all real valued sequences. The space s is a Frechet space under the metric d(s, t) =

∞ X j=1

|sj − tj | /2j (1 + |sj − tj |)

of coordinatewise convergence. We give a list of the β-duals and topological duals of the spaces above. For these, see [HK] and [Bo].

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space c00 c0 cc c m0 l∞ lp (0 < p < 1) lp (1 ≤ p < ∞) bs cs bv0 s

β-dual s l1 cs l1 l1 l1 l∞ lq ( p1 + bv0 bv bs c00

1 q

= 1)

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topological dual l1 l1 l1 l1 ba ba l∞ lq ( p1 + q1 = 1) bv bv bs c00

We now list some of the properties of sequence spaces which will be encountered in the sequel. Throughout the remainder of this appendix λ will denote a sequence space containing c00 . Suppose that λ is equipped with a Hausdorff vector topology. Definition B.1. The space λ is a K-space if the coordinate functionals t = {tj } → tj are continuous from λ into R for every j. If the K-space λ is a Banach (Frechet) space, λ is called a BK-space (FK-space). All of the spaces listed above are K-spaces under their natural topologies. Let ej be the sequence with a 1 in the j th coordinate and 0 in the other coordinates. Definition B.2. The K-space λ is an AK-space if the {ej } form a Schauder Pn basis for λ, i.e., if t = {tj } ∈ λ, then t = limn j=1 tj ej , where the convergence is in λ. The spaces c00 , c0 , lp (0 < p < ∞), cs, bv and bv0 are AK-spaces. The spaces m0 and l∞ are not AK-spaces. For each n let Pn : λ → λ be the sectional projection (operator) defined Pn by Pn t = j=1 tj ej = (t1 , ..., tn , 0, 0, ...).

Definition B.3. The K-space λ is an AB-space if {Pn t : n ∈ N} is bounded for each t ∈ λ, i.e., if the {Pn } are pointwise bounded on λ.

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Definition B.4. The K-space λ has the sections uniformly bounded property (SUB) if {Pn t : n ∈ N, t ∈ B} is bounded for every bounded subset B of λ, i.e., if the {Pn } are uniformly bounded on bounded subsets of λ. Definition B.5. The K-space λ has the property SE (sections equicontinuous) if the sectional operators {Pn } from λ into λ are equicontinuous. Obviously, property SUB implies that λ is an AB-space and property SE implies property SUB. If λ is a barrelled AB-space, then λ has property SE and, therefore, SUB ([Sw2], [Wi]). If λ is a metric linear space whose topology is generated by the quasi-norm |·| which satisfies |Pn t| ≤ M |t| for some M and all t ∈ λ, then λ has property SE; e.g., s, l ∞ and its subspaces, lp (0 ≤ p < ∞), and bs and its subspace cs. Throughout this text numerous gliding hump properties are employed. We now list these gliding hump properties and give examples of sequence spaces which satisfy the various gliding hump properties. If σ ⊂ N, χσ will denote the characteristic function of σ and if t = {tj } is any sequence (scalar or vector), χσ t will denote the coordinatewise product of χσ and t. A sequence space λ is monotone if χσ t ∈ λ for every σ ⊂ N and t ∈ λ. A sequence space λ is normal (solid ) if t ∈ λ and |sj | ≤ |tj | implies that s = {sj } ∈ λ. Obviously, a normal space is monotone; the space m0 is monotone but not normal. The spaces c00 , c0 , lp (0 < p ≤ ∞) and s are normal whereas cc , c, bs, cs, bv and bv0 are not monotone. An interval in N is a subset of the form [m, n] = {j ∈ N : m ≤ j ≤ n}, where m, n ∈ N with m ≤ n. A sequence of intervals {Ij } is increasing if max Ij < min Ij+1 for every j. A sequence of signs is a sequence {sj } with sj = ±1 for every j. We begin with 2 gliding hump properties which are algebraic and require no topology on the sequence space λ. Definition B.6. Let Λ ⊂ λ. Then Λ has the signed weak gliding hump property (signed-WGHP) if for every t ∈ Λ and every increasing sequence of intervals {Ij }, there is a subsequence {nj } and a sequence of signs {sj } P∞ such that the coordinatewise sum of the series j=1 sj χInj t belongs to Λ. If the signs sj can all be chosen to be equal to 1 for every t ∈ Λ, then Λ has the weak gliding hump property (WGHP). The weak gliding hump property was introduced by Noll ([No]) and the signed weak gliding hump property was introduced by Stuart ([St1],

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[St2]). We now give some examples of sequence spaces with WGHP and signed-WGHP. Example B.7. Any monotone space has WGHP. We now show that the space cs of convergent series is not monotone but has WGHP. Example B.8. cs has WGHP but is not monotone. Let t ∈ cs and {Ij } P be an increasing sequence of intervals. Since the series j tj converges, X tk → 0 χIj ∩J · t = k∈Ij ∩J

for any interval J. Pick a subsequence {nj } such that χInj ∩J · t < 1/2j P∞ for every interval J. Then j=1 χInj t ∈ cs since this series satisfies the Cauchy criterion.

If t = {(−1)j /j}, then t ∈ cs, but if σ = {1, 3, 5, ...}, then χσ t ∈ / cs so cs is not monotone. Example B.9. The space c of convergent sequences does not have WGHP. P∞ For example, t = {1, 1, ...} ∈ c and if Ij = {2j}, then j=1 χInj t ∈ / c for any subsequence {nj }. We show later in Example B.26 that the space bs of bounded series has signed-WGHP but not WGHP so the inclusion of signs in Definition B.6 is important. We next establish a general criterion for a space to have WGHP. A TVS X is a K-space if whenever {xj } is a null sequence in X and {nj } is a subsequence, then there is a further subsequence {mj } of {nj } such that P∞ the series j=1 xmj converges in X. For example, any complete metric linear space is a K-space ([Sw1] 3.2.3, there are further examples in this text). Theorem B.10. If λ is a K-space and an AK-space, then λ has WGHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Since P P∞ t = j=1 tj ej converges in λ, limk j∈Ik tj ej = 0. Since λ is a K-space, P∞ P j there exists {nk } such that k=1 j∈Ink tj e converges to some s ∈ λ. P ∞ P Since λ is a K-space, the series k=1 j∈In tj ej also converges to s pointk wise. Theorem B.10 applies to spaces such as c0 , lp (0 < p < ∞) and cs.

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We next consider results related to ideas due to Garling ([Ga]). For this we need the following observation. Example B.11. Let B0 be the closed unit ball of bv0 . We show that B0 has WGHP. Let t ∈ B0 and {Ik } be an increasing sequence of intervals. P∞ Then k=1 |tk+1 − tk | ≤ 1 and lim tk = 0. Choose {nk } such that ∞ X X

k=1 j∈Ink

|tj+1 − tj | < 1/2

and ∞ X k=1

2 max{ tmin Ink , tmax Ink } < 1/2

(this is possible since t ∈ c0 , so we can extract a subsequence which belongs to l1 ). Then χ∪∞ I t has total variation less than or equal to k=1 nk ∞ X X

k=1 j∈Ink

so χ

∪∞ k=1 Ink

|tj+1 − tj | +

t ∈ B0 .

∞ X k=1

2 max{ tmin Ink , tmax Ink } < 1

Definition B.12. (Garling) Let S be any subset of sequences. The space λ is S invariant if Sλ = λ. For example, λ is monotone iff λ is m0 invariant. Proposition B.13. If S has signed-WGHP (WGHP) and λ is S invariant, then λ has signed-WGHP (WGHP). Proof: For t ∈ λ there exists u = {uk } ∈ S,v ∈ λ such that t = uv. Let {Ik } be an increasing sequence of intervals. By hypothesis there exist a subsequence {Ink } and a sequence of signs {sk } such that u0 = P∞ sk χInk u ∈ S (coordinate sum). Then {u0k vk } ∈ λ and since {u0k vk } = Pk=1 ∞ k=1 sk χInk t, λ has signed-WGHP. From Example B.11 and Proposition B.14, we have

Corollary B.14. If λ is B0 invariant, then λ has WGHP. We next consider a notion introduced by Noll ([No]). Definition B.15. The multiplier space of λ, M (λ), is defined to be {s : st ∈ λ for all t ∈ λ}.

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Proposition B.16. If M (λ) has signed-WGHP (WGHP), then λ has signed-WGHP (WGHP). Proof: Consider the case of WGHP. The constant sequence e with 1 in each coordinate belongs to M (λ). Let t ∈ λ and {Ik } be an increasing P sequence of intervals. There exist {nk } such that ∞ k=1 χInk e ∈ M (λ) so P∞ P∞ ( k=1 χInk e)t = k=1 χInk t ∈ λ.

The results above in Propositions B.14 and B.16 are used in Chapter 2 to establish weak sequential completeness for β-duals. We next consider gliding hump properties which depend on the topology of the sequence space λ. In what follows we assume that λ is a K-space.

Definition B.17. Let Λ ⊂ λ. Then Λ has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {tj } ⊂ Λ and every increasing sequence of intervals {Ij }, there is a subsequence {nj } and a sequence of signs {sj } such that the coordinatewise sum of the series P∞ nj ∈ Λ. If the signs sj can be chosen equal to 1 for every j=1 sj χInj t t ∈ Λ, then Λ is said to have the strong gliding hump property (SGHP). The strong gliding hump property was introduced by Noll ([No]) and the signed strong gliding hump property was introduced in [Sw4]. Example B.18. The space l ∞ has SGHP. The spaces l p (0 < p < ∞) and c0 do not have SGHP (consider {ej } and Ij = {j}). Example B.19. The subset M0 = {χσ : σ ⊂ N} ⊂ m0 has SGHP while the space m0 = spanM0 does not have SGHP. We consider smaller subsets of 2N whose characteristic functions have SGHP. Definition B.20. A family F of subsets of N is an FQσ family if F contains the finite sets and if whenever {Ij } is a pairwise disjoint sequence of finite subsets, there is a subsequence {Inj } such that ∪∞ j=1 Inj ∈ F. This notion is due to Sember and Samaratanga ([SaSe]). We give an example of an FQσ family which is a proper subset of 2N . Example B.21. Haydon has given an example of an algebra H of subsets of N such that H is an FQσ family but for no infinite A ⊂ N do we have 2A = {A ∩ B : B ∈ H} ([Hay]).

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Example B.22. If F is an FQσ family and Λ = {χσ : σ ∈ F} ⊂m0 , then Λ has SGHP (if {χσj } ⊂ Λ and {Ij } is an increasing sequence of intervals, then there exists a subsequence {nj } such that I = ∪∞ j=1 σnj ∩ Inj ∈ F so P∞ χI = j=1 χInj χσnj ∈ Λ). Definition B.23. A family of subsets F of N is an IQσ family if F contains the finite sets and if whenever {Ij } is an increasing sequence of intervals, there is a subsequence {Inj } such that ∪∞ j=1 Inj ∈ F.

This notion is also due to Sember and Samaratanga ([SaSe]). We give an example of an IQσ family containing N. P∞ Example B.24. Let j=1 tj be a conditionally convergent scalar series. P Put F = {σ : j∈σ tj converges}. Then F is an IQσ family containing N. For suppose that {Ij } is an increasing sequence of intervals. There exists P a subsequence {Inj } such that i∈In ∩J ti < 1/2j for every j and every j P the interval J. Then I = ∪∞ j=1 Inj ∈ F since the series j∈I tj satisfies P N Cauchy condition. Note that N ∈ F but F 6=2 since the series j tj is conditionally convergent. As in Example B.22, we have Example B.25. Let F be an IQσ family and Λ = {χσ : σ ∈ F} ⊂m0 . Then Λ has SGHP. We next show that the space bs of bounded series has signed-SGHP but not SGHP. Example B.26. The space bs has signed-SGHP but not SGHP. Let {tj } ⊂ bs be bounded and {Ij } be an increasing sequence of intervals. Put ( ) X j ti : j ∈ N, I an interval in N < ∞. M = sup i∈I

Define signs inductively by setting s1 = signχI1 · t1 and sn+1 = −[sign

n X

k=1

sk χIk · tk ][signχIn+1 · tn+1 ].

P∞ k Put y = k=1 Psk χIk t . We show kykbs ≤ 2M . We first Pshow by in max In max I duction that j=1 yj ≤ M for every n. For n = 1, j=1 1 yj =

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P j∈I1 s1 t1j ≤ M . Suppose the inequality holds for n. Then max max In+1 X XIn X yj + yj ≤ M yj = j=1 j=1 j∈In+1 P P P max I since j∈In+1 yj = j∈In+1 sn+1 yj ≤ M and j=1 n yj ≤ M and both of these terms have opposite signs. Now for arbitrary n, let k = kn be the largest integer such that max Ik ≤ n. Then n max Ik n n XIk X X max X X k+1 yj = yj + yj ≤ yj + sk+1 tj ≤ 2M j=1 j=1 j=1 j=min Ik+1 j=min Ik+1

so kykbs ≤ 2M as desired.

Note that bs does not have WGHP (consider t = {1, −1, 1, −1, ...} and Ij = {2j − 1}). The proof above is essentially that of Stuart who showed that bs has signed-WGHP but not WGHP ([St1], [St2]). Further examples of spaces with SGHP (WGHP) and signed-SGHP (signed-WGHP) are constructed later in this appendix. Note that in the proof of Example B.26 it was not necessary to pass to a subsequence in the definition of signed-SGHP. Definition B.27. The K-space λ has the zero gliding hump property (0GHP) if whenever tj → 0 in λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that the coordinate sum of the series P∞ nj belongs to λ. j=1 χInj t

The 0-GHP was essentially introduced by Lee Peng Yee ([LPY]); see also [LPYS]. We give some examples of spaces with 0-GHP. Recall the section operPn ators Pn : λ → λ are defined by Pn t = j=1 tj ej .

Proposition B.28. Let λ be a K-space with property SE. Then λ has 0GHP.

Proof: Let tj → 0 in λ and let {Ij } be an increasing sequence of intervals. Then χIj tj → 0 in λ by property SE. Since λ is a K-space, there is P a subsequence {nj } such that the subseries j χInj tnj converges to some P t ∈ λ. Since λ is a K-space, the series j χInj tnj converges coordinatewise to t.

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From Proposition B.28, it follows that the spaces l p (0 < p ≤ ∞), s, c, cs, bs and c0 have 0-GHP. The space c00 does not have 0-GHP. Further examples are given later in this appendix and can be found in [Sw1] 12.5. The gliding hump properties WGHP and 0-GHP are independent; the space c has 0-GHP but not WGHP while the space c00 has WGHP but not 0-GHP. We give a simple proposition which relates the two conditions. Proposition B.29. If λ is an AK-space with 0-GHP, then λ has WGHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Since λ has AK, χIj t → 0. By 0-GHP, there is a subsequence {nj } such that P∞ j=1 χInj t ∈ λ, where the series is coordinatewise convergent since λ is a K-space. We next define a gliding hump property which is used to establish uniform boundedness principles. Let µ be sequence space containing c00 . Definition B.30. The K-space λ has the strong µ gliding hump property (strong µ-GHP) if whenever {tj } is a bounded sequence in λ and {I } is an increasing sequence of intervals, the coordinate sum of the series Pj∞ j j=1 uj χIj t belongs to λ for every u = {uj } ∈ µ.

Definition B.31. The K-space λ has the weak µ gliding hump property (weak µ-GHP) if whenever {tj } is a bounded sequence in λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that P∞ nj belongs to λ for every the coordinate sum of the series j=1 uj χInj t u = {uj } ∈ µ. Of course, the difference in the strong µ-GHP and the weak µ-GHP is the necessity to pass to a subsequence in Definition B.31. We refer to the elements u = {uj } ∈ µ as multipliers since the coordinates of u multiply the blocks or ”humps”, {χIj tj }, determined by the {tj } and {Ij }. This is analogous to the situation in the signed-WGHP or signed-SGHP where the humps are multiplied by {±1}. We give examples of spaces with µ-GHP. Proposition B.32. If λ is a locally complete K-space with property SUB, then λ has strong l 1 -GHP. Proof: Let {tj } ⊂ λ be bounded and {Ij } be an increasing sequence of intervals. By SUB, {χIj tj } is bounded in λ so if u ∈ l1 , the series P∞ j is absolutely convergent in λ and, therefore, converges to j=1 uj χIj t

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an element t ∈ λ by local completeness. Since λ is a K-space, the series P∞ j is coordinatewise convergent to t. j=1 uj χIj t

Example B.33. From Proposition B.32, it follows that the spaces l p (0 < p ≤ ∞), s, c, cs, bs and c0 have strong l 1 -GHP. We also have Example B.34. The spaces l ∞ and c0 have strong c0 -GHP; lp (0 < p ≤ ∞) has strong l∞ -GHP. We next give examples of non-complete spaces with weak l p -GHP. This example requires some properties of integration with respect to finitely additive set functions. We refer the reader to [RR] for a discussion of the integrals. Example B.35. Let 1 ≤ p < ∞. Let P be the power set of N and let α : P → [0, ∞) be a finitely additive set function with α({j}) > 0 for every j. Let lp (α) = Lp (α) beR the space of all pth power α-integrable functions p with the norm kf kp = ( N |f | dα)1/p [see [RR] for details; the assumption p that α({j}) > 0 makes l (α) a K-space]. We show that l p (α) has weak lp -GHP. Let {fj } ⊂ lp (α) be bounded with kfj kp ≤ 1 and {Ij } be an increasing sequence of intervals. By Drewnowski’s Lemma (Appendix E.2), there is a subsequence {nj } such that α is countably additive on the σP∞ algebra generated by {Inj }. Suppose that t ∈ l p . Put f = j=1 tj χInj fnj P∞ [coordinate sum]. We claim that f ∈ l p (α) and the series j=1 tj χInj fnj p converges to f in l (α) by using Theorem 4.6.10 of [RR]; this will establish Pn the result. Put sn = j=1 tj χInj fnj and note that sn → f α-hazily [i.e., in α measure] since if > 0, α({j : |sn (j) − f (j)| ≥ }) ≤ α(∪∞ j=n+1 Inj ) =

∞ X

j=n+1

α(Inj ) → 0

by the countable additivity. Next, {sn } is Cauchy in lp (α) since

p

n n X X

p p

|tj | → 0. ksn − sm kp = t j χ In j f n j ≤

j=m+1 j=m+1 p

R

p

It follows that { |sn | dα} is uniformly α-continuous and using [RR] this justifies the claim.

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We next consider a gliding hump property called the infinite gliding hump property which has been proven to be useful when considering strong convergence. This property is algebraic and requires no topology on λ. Definition B.36. The sequence space λ has the infinite gliding hump property (∞-GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that P the coordinate sum of the series ∞ j=1 apj χIpj t ∈ λ. The term ”infinite” gliding hump property is used to suggest that the humps {χIj t} are multiplied by a sequence of scalars which tend to infinity. We now give examples of sequence spaces with ∞-GHP. For this, we introduce another property for a sequence space.

Definition B.37. The space λ is c0 -factorable (called c0 -invariant by Garling ([Ga])) if t ∈ λ implies that there exist s ∈ c0 , u ∈ λ with t = su [coordinate product]. Proposition B.38. If λ is normal and c0 -factorable, then λ has ∞-GHP. Proof: Let t ∈ λ with t = su, s ∈ c0 , u ∈ λ and let {Ij } be an increasing sequence of intervals. Pick an increasing sequence {nj } such that sup{|si | : i ∈ Inj } = bnj > 0 (if this choice is not possible there is nothing to do). Note that bnj → 0 so anj = 1/bnj → ∞. Define vj = sj ank if j ∈ Ink and vj = 0 otherwise; then v ∈ l ∞ so vu ∈ λ since λ is normal. We P∞ P j have ∞ k=1 ank χInk t ∈ λ. Since the same argument can be j=1 (vu)j e = applied to any subsequence, λ has ∞-GHP. We now give some examples of spaces which satisfy the conditions of Proposition B.38. Example B.39. The space c0 is normal and c0 -factorable so has ∞-GHP. Example B.40. Let 0 < p < ∞. Let t ∈ l p . Pick an increasing sequence Pnj+1 p {nj } with n0 = 0 such that i=n |tj | < 1/2j(p+1) and set Ij = [nj + Pj +1 P∞ −j ∞ j p 1, nj+1 ], s = j=1 2 χIj t. Then t = su so l is c0 j=1 2 χIj , u = factorable and is obviously normal so Proposition B.38 applies. Example B.41. As in Example B.40, it can be shown that the spaces d = {t : sup{|tj |1/j < ∞} and δ = {t : lim |tj |1/j = 0} are c0 -factorable and are both clearly normal so Proposition B.38 applies [see [KG] for these spaces].

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We give examples of non-normal spaces with ∞-GHP. Proposition B.42. If λ is a Banach AK-space, then λ has ∞-GHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Choose

P

a subsequence {nj } such that i∈In ∩J tj ej < 1/j2j for any interval J. j P Consider s = ∞ jχInj t. If J is any interval contained in the interval j=1

P P

P

P

∞ k j [min Inj , ∞), then j∈J sj ej = ∞ i=j k∈In ∩J itk e ≤ i=j 1/2 = j

2−j+1 so the partial sums of the series generated by s are Cauchy and, therefore, convergent. Hence, s ∈ λ and λ has ∞-GHP since λ is a K-space.

Example B.43. For example, it follows from Proposition B.42 that the non-normal space cs of convergent series has ∞-GHP. Likewise, bv0 has ∞-GHP. Example B.44. The spaces l ∞ , m0 , bs and bv do not have ∞-GHP. To this point the only example of a space with signed-SGHP (signedWGHP) is bs. We will now describe a method which can be used to construct more sequence spaces with these and other gliding hump properties ([BSS]). Let A = [aij ] be an infinite matrix with scalar entries. We use A and λ to generate a further sequence space. The matrix domain of A and λ is defined to be λA = {t = {tj } : At ∈ λ}.

Thus, A is a linear map from λA → λ. Some of the familiar sequence spaces can be generated by this procedure. In particular, cA and c0A = (c0 )A are the spaces of all sequences which are A-summable and A-summable to 0, respectively ([Bo]). Example B.45. Let B = [bij ] be the matrix with bij = 1 for j ≤ i and ∞ bij = 0 otherwise. Then lB = bv, the space of sequences with bounded variation, and cB = cs, the space of convergent series. Example B.46. Let B1 = [bij ] be the matrix with bii = 1, bi+1,i = −1 and 1 = bv. bij = 0 otherwise. Then lB 1 Example B.47. Let C = [cij ] be the Cesaro matrix , cij = 1/i for 1 ≤ j ≤ i ∞ and cij = 0 otherwise. Then lC is the space of sequences {tj } with bounded P j averages, supj i=1 ti /j < ∞.

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Let {dj } be a scalar sequence and let D be the diagonal matrix with the {dj } as the entries down the diagonal. Theorem B.48. If λ has WGHP (signed-WGHP), then λD has WGHP (signed-WGHP). Proof: Let t ∈ λD and {Ij } be an increasing sequence of intervals. Since Dt ∈ λ, there is a subsequence {nj } such that ∞ X v= χInj Dt ∈ λ. j=1

Then

u=

∞ X j=1

χ In j t ∈ λ D

since Du = v. The proof for the signed-WGHP case is similar. Corollary B.49. bsD has signed-WGHP for any diagonal matrix D. If each dj 6= 0, the spaces bs and bsD are algebraically isomorphic, but, in general, the spaces may have very different topological properties depending on the growth of the sequence {dj }. For example, if dj → ∞ and t ∈ bsD , then t ∈ c0 so bsD ⊂ c0 in this case. On the other hand, if dj → 0 and dj 6= 0 for all j, then u = (1/d1 , −1/d2 , 1/d3 , ...) ∈ bsD

and lim uj = ∞ so u ∈ / l∞ in this case. Also, if dj = 0 for some j, then th the j coordinate of any element in bsD can be arbitrarily large so bsD will be very different from bs. Thus, the spaces bsD furnish a large class of sequence spaces with signed-WGHP. We next consider the construction above when the sequence space λ is a K-space. Assume that λ is a K-space whose locally convex topology is generated by a family of semi-norms P. We give the space λA the locally convex topology generated by the semi-norms pA (t) = p(At) f or p ∈ P and pk (t) = |tk | , k ∈ N.

Since only triangular matrices A will be considered below, this agrees with the usual topology defined on λA ([Wi2] 4.3.12). Note that λA is a K-space and A : λA → λ is a linear, continuous operator. Theorem B.50. If λ has SGHP (signed-SGHP), then λD has SGHP (signed-SGHP) for any diagonal matrix D.

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Proof: Let {tj } be bounded in λD and {Ij } be an increasing sequence of intervals. Since D : λD → λ is continuous, {Dtj } is bounded in λ. There exists a subsequence {nj } such that the coordinate sum of the series P∞ nj v = ∈ λ. Then the coordinate sum of the series u = j=1 χInj Dt P∞ nj χ t ∈ λ since Du = v. D j=1 Inj The other case is treated similarly. ∞ Corollary B.51. lD has SGHP and bsD has signed-SGHP for any diagonal matrix D. ∞ As noted earlier the spaces lD furnish examples of spaces of a different nature with SGHP. The only spaces other than l ∞ with SGHP seem to have been constructed by Noll ([No]).

Theorem B.52. If λ has 0-GHP, then λD has 0-GHP. Proof: Let tj → 0 in λD and {Ij } be an increasing sequence of intervals. Since D : λD → λ is continuous, Dtj → 0 in λ. There exists a subsequence P∞ {nj } such that the coordinate sum of the series v = j=1 χInj Dtnj ∈ λ. P∞ Then the coordinate sum of the series u = j=1 χInj tnj ∈ λD since Du = v. Again Theorem B.52 furnishes a large number of examples of spaces with 0-GHP. Let µ be a sequence space containing c00 . As in Theorems B.48, B.50 and B.52, we have Theorem B.53. If λ has strong µ-GHP (weak µ-GHP), then λD has strong µ-GHP (weak µ-GHP). Further examples of sequence spaces λA with various gliding hump properties for matrices which are not diagonal matrices are given in [BSS].

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Appendix C Vector Valued Sequence Spaces

In this appendix we give a list of vector valued sequence spaces, review the gliding hump properties of sequence spaces and give examples of the sequence spaces satisfying various gliding hump properties. Let X be a TVS. We give a list of X valued sequence spaces and their natural topologies. • • • • • •

c00 (X) : all X valued sequences which are eventually 0. c0 (X) : all X valued sequences which converge to 0. cc (X) : all X valued sequences which are eventually constant. c(X) : all X valued sequences which are convergent. m0 (X) : all X valued sequences with finite range. l∞ (X) : all X valued sequences which are bounded.

Let X be an LCTVS whose topology is generated by the family of seminorms X . The natural topology of all the spaces above is generated by the semi-norms q∞ ({xj }) = sup q(xj ), q ∈ X . j

There is a significant difference between the scalar and vector case for spaces of bounded sequences. In the scalar case m0 is dense in l∞ with respect to k·k∞ . However, when X is an infinite dimensional Banach space, m0 (X) is not dense in l∞ (X). For suppose that X is an infinite dimensional Banach space. By Riesz’s Lemma ([Sw2] 7.6), There exists a sequence {xj } such that kxi − xj k ≥ 1 for i 6= j with kxj k = 1 for all j. Then x = {xj } ∈ l∞ (X). However, if m0 (X) is dense in l∞ (X), then {xj : j ∈ N} has a finite -net for every > 0 and is, therefore, precompact which is clearly not the case. Let 0 < p < ∞. Then 229

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• lp (X) : all X valued sequences such that

P∞

j=1

q(xj )p < ∞, q ∈ X .

If 1 ≤ p < ∞, the topology of l p (X) is generated by the semi-norms 1/p ∞ X qp ({xj }) = q(xj )p , q ∈ X . j=1

If 0 < p < 1, the topology of l p (X) is generated by the quasi-norms qp ({xj }) =

∞ X j=1

q(xj )p , q ∈ X .

Pn • BS(X) : all X valued sequences {xj } satisfying {{xj } : { j=1 xj }n is bounded}. P • CS(X) : all X valued sequences {xj } satisfying {{xj } : j xj is Cauchy}. If X is the scalar field, then bs = BS(X) and cs = CS(X). We define a locally convex topology on BS(X) ⊃ CS(X) by the semi-norms X q 0 ({xj }) = sup q xj : I a finite interval , q ∈ X . j∈I

• BV (X) : all X valued sequences satisfying X. • BV0 (X) = BV (X) ∩ c0 (X).

P∞

j=1

q(xj+1 −xj ) < ∞, q ∈

If X is the scalar field, then bv = BV (X) and bv0 = BV0 (X). These spaces are topologized by the semi-norms ∞ X qˆ({xj }) = q(x1 ) + q(xj+1 − xj ), q ∈ X . j=1

• s(X) : all X valued sequences. We now describe a method of constructing vector valued sequence spaces from scalar valued sequence spaces. Let ν be a scalar valued sequence space which is normal and a K-space under a locally convex topology generated by the semi-norms N . If t = {tj } ∈ ν , we write |t| = {|tj |}; note |t| ∈ ν since ν is normal. We will also consider a monotone property for ν. The sequence space ν satisfies condition

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(M ) if there is a family of semi-norms N generating the topology of ν which is such that q(s) ≤ q(t) whenever s, t ∈ ν and |s| ≤ |t| and q ∈ N . Define ν{X} = {{xj } : xj ∈ X, {q(xj )} ∈ ν for all q ∈ X }. Since ν is normal, ν{X} is a vector space. We supply ν{X} with the locally convex topology generated by the semi-norms (#) πq,p (x) = q({p(xj }), p ∈ X , q ∈ N . Thus, we have c00 (X) = c00 {X}, c0 (X) = c0 {X} and lp (X) = lp {X} for 0 < p ≤ ∞. We extend some of the topological properties of scalar sequence spaces to vector valued sequence spaces. Let E be a vector space of X valued sequences equipped with a locally convex Hausdorff topology. Definition C.1. The space E is a K-space if the maps x = {xj } → xj from E into X are continuous for every j ∈ N. Since ν is a K-space, then ν{X} is a K-space. If z ∈ X, then ej ⊗z will denote the sequence with z in the j th coordinate and 0 in the other coordinates. For every n ∈ N, the sectional operator (projection) is defined to be the map Pn : E → E defined by Pn (x) =

n X j=1

ej ⊗ xj = (x1 , ..., xn , 0, ...).

Definition C.2. The K-space E is an AK-space if for every x ∈ E, we P∞ have x = j=1 ej ⊗ xj , with convergence in E.

Definition C.3. The K-space E is an AB-space if {Pn x} is bounded for every x ∈ E.

Definition C.4. The K-space E has the sections uniformly bounded property (SUB) if {Pn x : n ∈ N, x ∈ B} is bounded for every bounded set B ⊂ E. Definition C.5. The K-space E has the property SE (sections equicontinuous) if the sectional operators {Pn } are equicontinuous.

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The spaces ν{X} supply a large number of spaces with the properties defined above. Proposition C.6. Consider the following properties for ν{X}: (i) (ii) (iii) (iv) (v)

If ν is an AK-space, then ν{X} is an AK-space. If ν is an AB-space, then ν{X} is an AB-space. If ν has property SUB, then ν{X} has property SUB. If ν has property SE, then ν{X} has property SE. If X is sequentially complete, ν is sequentially complete, has property (M ) and the topology of ν is linked to the topology of coordinatewise convergence, then ν{X} is sequentially complete.

Proof: We prove (i); the other proofs of (ii)-(iv) are similar. Let x ∈ E and let πq,p be a basic semi-norm for ν{X}. Then ∞ ∞ X X p(xj )ej ) → 0 ej ⊗ xj ) = q( πq,p ( j=n

j=n

since {p(xj )} ∈ ν, so the result follows. (v): Suppose {xk } is Cauchy in ν{X}. Since X is sequentially complete, limk xkj = xj exists for every j. Set x = {xj }. Let p ∈ X and q ∈ N . By (M ), q({p(xkj )}j − {p(xlj )}j ) ≤ q({p(xkj − xlj )}j ) so {p(xkj )}j is a Cauchy sequence in ν. Since ν is sequentially complete, there exists u ∈ ν such that {p(xkj )} → u as k → ∞. Also, limk p(xkj ) = p(xj ) for each j so u = {p(xj )} ∈ ν and, therefore, x = {xj } ∈ ν{X}. Since {xk } is Cauchy in ν and xk → x ∈ ν{X} in the topology of coordinatewise convergence, xk → x in ν since the topology of ν is linked to the topology of coordinatewise convergence (Appendix A.4). From (v) above, we have Corollary C.7. If X is sequentially complete, then l p (X) and c0 (X) are sequentially complete for 1 ≤ p ≤ ∞. We now give statements of various gliding hump properties for vector valued sequence spaces. These statements are straightforward generalizations of the corresponding properties for scalar sequence spaces (see

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Appendix B). If σ ⊂ N and x = {xj } is an X valued sequence, the coordinatewise product of χσ and x is denoted by χσ x. Definition C.8. The space E has the signed weak gliding hump property (signed-WGHP) if for every x ∈ E and every increasing sequence of intervals {Ij }, there exist a subsequence {nj } and a sequence of signs {sj } such that P∞ the coordinatewise sum of the series j=1 sj χInj x ∈ E. If the signs above can be chosen to be equal to 1 for every x ∈ E, then E has the weak gliding hump property (WGHP). The space E is monotone if χσ x ∈ E for every σ ⊂ N and x ∈ E. Any monotone space has WGHP. The spaces ν{X} are all monotone since ν is assumed to be normal so ν{X} always has WGHP. The proof in Example 8 of Appendix B shows that the non-monotone space CS(X) has WGHP (use the semi-norms |·| which generates the topology of X in place of absolute value). Definition C.9. The K-space E has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {xj } from E and every increasing sequence of intervals {Ij }, there exist a subsequence {nj } and a sequence of signs {sj } such that the coordinatewise sum of the series P∞ nj ∈ E. If the signs above can be chosen equal to 1 for every j=1 sj χInj x x ∈ E, then E has the strong gliding hump property (SGHP). The space l∞ (X) has SGHP. Proposition C.10. If ν has SGHP and if X is normed, then ν{X} has SGHP. Proof: Let {xj } be bounded in ν{X} and {Ij } be an increasing sequence

j of intervals. Then {{ xi }i : j ∈ N} is bounded in ν so there exists a

nj P

}i ∈ ν so subsequence {nj } such that the coordinate sum ∞ j=1 χInj { xi P∞ nj ∈ ν{X}. j=1 χInj x

Definition C.11. The K-space E has the zero gliding hump property (0GHP) if for every null sequence {xj } in E and every increasing sequence of intervals {Ij }, there is a subsequence {nj } such that the coordinatewise P∞ sum of the series j=1 χInj xnj ∈ E. As in Proposition C.10, we have

Proposition C.12. If ν has 0-GHP and if X is normed, then ν{X} has 0-GHP.

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Let µ be a scalar sequence space containing c00 . Definition C.13. The K-space E has the strong µ gliding hump property (strong µ-GHP) if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, then for every u = {uj } ∈ µ the P∞ coordinatewise sum of the series j=1 uj χIj xj ∈ E.

Definition C.14. The K-space E has the weak µ gliding hump property (weak µ-GHP) if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, then there is a subsequence {nj } such that for every u = {uj } ∈ µ the coordinatewise sum of the series P∞ nj ∈ E. j=1 uj χInj x Proposition C.15. If ν has strong µ-GHP, then ν{X} has strong µ-GHP.

Proof: Let {xj } be bounded in ν{X} and let {Ij } be an increasing P j sequence of intervals. Let u = {uj } ∈ µ and set x = ∞ j=1 uj χIj x (coordiP∞ j nate sum). Let p ∈ X and note that p(x(·)) = j=1 |uj | χIj p(x (·)), where x(·) is the function j → xj . Now {{p(xji )}i : j ∈ N} is bounded in ν by definition. By strong µ-GHP, {p(xj )} ∈ ν so x ∈ ν{X}. As in Propositions C.10 and C.12, we have Proposition C.16. If ν has weak µ-GHP and X is normed, then ν{X} has weak µ-GHP. We give an example of a non-monotone space with strong l 1 -GHP. Example C.17. CS(X) has strong l 1 -GHP. Suppose that {xj } is bounded in CS(X) and {Ij } is an increasing sequence of intervals. Let u = P∞ j and set {uj } ∈ l1 . Put x = j=1 uj χIj x . Let > 0 and p ∈ X P j M = sup{p( i∈I xi ) : I a finite interval, j ∈ N}. Pick N such that P∞ Suppose I is a finite interval such that min I > N . Then j=N |uj | < . P P∞ p( j∈I xj ) ≤ j=N |uj | M < M so x ∈ CS(X). As in the example above, BS(X) also has strong l 1 -GHP. We next extend the ∞-GHP to vector valued sequence spaces (Appendix B.36).

Definition C.18. The space E has the infinite gliding hump property (∞-GHP) if whenever x ∈ E and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that

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every subsequence of {nj } has a further subsequence {pj } such that the P∞ coordinatewise sum of the series j=1 apj χIpj ∈ E. As in Propositions C.10 and C.12, we have

Proposition C.19. If X is normed and ν has ∞-GHP, then ν{X} has ∞-GHP. In particular, if X is normed, then the spaces c0 (X) and lp (X) (0 < p < ∞) have ∞-GHP (Examples B.39 and B.40 of Appendix B). As in Proposition B.42 of Appendix B, we also have Proposition C.20. If E is a Banach AK-space, then E has ∞-GHP. Let Y be a TVS. The β-dual of E with respect to Y is defined to be ∞ X E βY = {Tj } : Tj ∈ L(X, Y ), Tj xj converges for every {xj } ∈ E . j=1

Thus, E βY consists of all operator valued series in L(X, Y ) which are E multiplier convergent. If Y is the scalar field, we write E βR = E β ; in this case, E, E β form a dual pair under the pairing ∞ X

0 xj , xj , x = {xj } ∈ E, x0 = {x0j } ∈ E β . x0 · x = j=1

We give several examples of β-duals.

Example C.21. Let X be a normed space and assume that the dual space X 0 is equipped with the dual norm. Then ν{X}β = ν β {X 0 }. {x {kxj k} ∈ ν First, let {x0j } ∈ ν β {X 0 } and suppose j } ∈ ν{X}. Then P∞ 0 P∞ 0 β 0

≤ and { xj } ∈ ν . Hence, j=1 xj , xj j=1 xj kxj k < ∞ so 0 β {x0j } ∈ ν{X}β and ν β {X } ⊂ ν{X} . Next, let {x0j } ∈ ν{X}β . If

0 0 β β 0

{xj } ∈ / ν {X }, then { xj } ∈ / ν . Since ν is normal, there exists P∞ t ∈ ν such that j=1 |tj | x 0j = ∞. For each j pick xj ∈ X such j 0

0 and that kxj k = 1 and xj < xj , xj0 + 1/2 . β Then {tj xβj } ∈ βν{X} P∞ 0 0 / ν{X} so ν{X} ⊂ ν {X } and = ∞. Thus, {xj } ∈ j=1 |tj | xj , xj equality follows. Let Xb0 be the dual space of X equipped with the strong topology β(X 0 , X). Proposition C.22. BS(X)β = BV0 (Xb0 ).

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Proof: Let y ∈ BS(X)β . To show that yj → 0 strongly, it suffices to show that hyj , xj i → 0 for every bounded sequence {xj } ⊂ X. If x0 = 0, P∞ then {xj − xj−1 } ∈ BS(X) so j=1 hyj , xj − xj−1 i converges, and we have that limj hyj , xj − xj−1 i = 0 for every bounded sequence {xj }. This implies that lim hyj , xj i = 0 for every bounded sequence {xj } [ define a bounded sequence {zj } by 0, x1 , 0, x3 , 0, ...; then the sequence {hyj , zj+1 − zj i} contains the sequence {hy2j+1, x2j+1 i} as a subsequence so lim hy2j+1, x2j+1 i = 0 and similarly, lim hy2j , x2j i = 0 so lim hyj , xj i = 0 ]. Thus, y ∈ c0 (X). Let {xj } be bounded in X and set zj = xj+1 − xj . Then {zj } ∈ BS(X) P∞ and j=1 hyj , zj i converges. Now (∗)

n X j=1

hyj , zj i =

n X j=1

hyj , xj+1 − xj i =

n X j=1

hyj − yj+1 , xj i − hyn , xn i .

P∞ By what was established above, hyn , xn i → 0 so j=1 hyj − yj+1 , xj i converges absolutely for every bounded sequence {xj } by (∗). This implies P∞ that j=1 (yj − yj+1 ) is absolutely convergent in Xb0 [ if B ⊂ X is bounded, for every j pick xj ∈ B such that pB (yj − yj+1 ) = sup{|hyj − yj+1 , xi| : x ∈ B} < |hyj − yj+1 , xj i| + 1/2j P∞ so j=1 pB (yj − yj+1 ) < ∞ ]. Thus, y ∈ BV0 (Xb0 ). Pn Next, let y ∈ BV0 (Xb0 ) and x ∈ BS(X). Then {sn = j=1 xj } is P∞ bounded so j=1 hyj − yj+1 , sj i converges absolutely. Now (∗∗)

n X j=1

hyj , xj i =

n−1 X j=1

hyj − yj+1 , sj i + hyn , sn i .

Since yn → 0 strongly, hyn , sn i → 0 so (∗∗) implies that converges. That is, y ∈ BS(X)β so BV0 (Xb0 ) ⊂ BS(X)β .

P∞

j=1

hyj , xj i

Proposition C.23. l∞ (X)β = l1 (Xb0 ). Proof: Let y ∈ l∞ (X)β . Suppose that {xj } ⊂ X is bounded and t ∈ P∞ P∞ l . Then {tj xj } ∈ l∞ (X) so j=1 |hyj , tj xj i| = j=1 |tj | |hyj , xj i| < ∞. P ∞ Thus, {hyj , xj i} ∈ l1 or j=1 |hyj , xj i| < ∞. Hence, y ∈ l 1 (Xb0 ) [see the argument in Proposition C.22] and l ∞ (X)β ⊂ l1 (Xb0 ). Let y ∈ l1 (Xb0 ). Suppose that {xj } ∈ l∞ (X) and set B = {xj : j ∈ N}, P∞ P∞ pB (x0 ) = sup{|hx0 , xi| :x ∈ B}. Then j=1 |hyj , xj i| ≤ j=1 pB (yj ) < ∞ so y ∈ l∞ (X)β and l1 (Xb0 ) ⊂ l∞ (X)β . Similarly, we have ∞

Proposition C.24. c0 (X)β = l1 (Xb0 ).

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Let le∞ (X 0 ) be the sequences in l ∞ (Xb0 ) which are equicontinuous (such sequences are always strongly bounded ([Sw2] 19.5)). Proposition C.25. We have the following relationships: (i) le∞ (X 0 ) ⊂ l1 (X)β . (ii) l1 (X)β ⊂ l∞ (Xb0 ). (iii) If X is quasi-barrelled, then l 1 (X)β = l∞ (Xb0 ). Proof: (i): Let y = {yj } ∈ le∞ (X 0 ). Then there exists a continuous semi-norm p on X such that |hyj , xi| ≤ p(x) for all x ∈ X, j ∈ N. Let P∞ P∞ {xj } ∈ l1 (X). Since j=1 |hyj , xj i| ≤ j=1 p(xj ) < ∞, y ∈ l1 (X)β and (i) holds. (ii): Let y ∈ l1 (X)β and let {xj } be bounded. If t ∈ l1 , then {tj xj } ∈ P∞ P∞ 1 l (X) so j=1 |hyj , tj xj i| = j=1 |tj | |hyj , xj i| < ∞. Hence, {hyj , xj i} ∈ l∞ which implies that {yj } is strongly bounded [if B ⊂ X is bounded, then for every j there exists xj ∈ B such that pB (yj ) = sup{|hyj , xi| : x ∈ B} ≤ |hyj , xj i| + 1 so {pB (yj )} is bounded]. Therefore, (ii) holds. (iii) follows from the definition of quasi-barrelled and (i) and (ii) ([Sw2] 19.13, [Wi] 10.1.11). Finally, we compute one example of an operator valued β-dual. Below when X and Y are normed spaces, the dual spaces are always assumed to be equipped with the dual norm. Proposition C.26. Let X and Y be normed spaces with Y complete. Let 1 < p < ∞ and p1 + 1q = 1. Then {Tj } ∈ lp (X)β iff {{Tj0y 0 } : y 0 ∈ Y 0 , ky 0 k ≤ 1} is bounded in l q (X 0 ). Proof: Suppose {{Tj0y 0 } : y 0 ∈ Y 0 , ky 0 k ≤ 1} is bounded in l q (X 0 ) and M is a bound for the norms of the elements in this set. Then

n

n n X

0 0

X 0

X

Tj y kxj k : ky 0 k ≤ 1} ≤ sup

= sup{

hy , T x i T x j j j j

ky 0 k≤1 j=m

j=m

j=m ≤(

n X

j=m

kxj kp )1/p sup {(

P

ky 0 k≤1

n n X X

0 0 q 1/q

Tj y ) ≤ M( kxj kp )1/p

j=m

j=m

which implies that j Tj xj is Cauchy in Y and, therefore, converges. Suppose {Tj } ∈ lp (X)β . First, we claim that {Tj0 y 0 } ∈ lq (X 0 ) for every

P∞

T 0 y 0 q = ∞ y 0 ∈ Y 0 . Suppose that this condition fails to hold. Then j=1 j

P ∞ for some y 0 ∈ Y 0 . Then there exists t ∈ l p such that j=1 Tj0 y 0 |tj | = ∞.

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For every j there exists xj ∈ X, kxj k ≤ 1 with hy 0 , Tj xj i ≥ Tj0 y 0 /2. Then P∞ 0 p / lp (X)β . j=1 |tj | |hy , Tj xj i| = ∞. Since {tj xj } ∈ l (X), {Tj } ∈ Now lq (X 0 ) is complete (Corollary C.7) and the linear map y 0 → {Tj0 y 0 } from Y 0 into lq (X 0 ) has a closed graph. Hence, from the Closed Graph Theorem the map is continuous and the result follows. Other examples of β-duals for vector valued sequence spaces can be found in [FP2], [Fo], [GKR] and [Ros].

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Appendix D The Antosik-Mikusinski Matrix Theorems

In this appendix we will present two versions of the Antosik-Mikusinski Matrix Theorems. These matrix theorems have proven to be very useful in treating applications in functional analysis and measure theory where gliding hump techniques are employed (see [Sw1] for more versions of the matrix theorem and applications). These theorems are used at various points in the text in gliding hump proofs. Let X be a (Hausdorff) TVS. We begin with a simple lemma. Lemma D.1. Let xij ∈ X for i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i and if {Uk } is a sequence of neighborhoods of 0 in X, then there exists an increasing sequence {pi } such that xpi pj ∈ Uj for j˙ > i. Proof: Set p1 = 1. There exists p2 > p1 such that xip1 ∈ U2, xp1 j ∈ U2 for i, j ≥ p2 . Then there exists p3 > p2 such that xip1 , xip2 , xp1 j , xp2 j ∈ U for i, j ≥ p3 . Now just continue the construction. We now establish our version of the Antosik-Mikusinski Matrix Theorem. Theorem D.2. (Antosik-Mikusinski) Let xij ∈ X for i, j ∈ N. Suppose (I) limi xij = xj exists for each j and (II) for each increasing sequence of positive integers {mj } there is a subseP∞ quence {nj } of {mj } such that { j=1 xinj }i is Cauchy. Then limi xij = xj uniformly for j ∈ N. In particular, lim lim xij = lim lim xij = 0 and lim xii = 0. i

j

j

i

i

239

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Proof: If the conclusion fails, there is a closed, symmetric neighborhood U0 of 0 and increasing sequences of positive integers {mk } and {nk } such that xmk nk −xnk ∈ / U0 for all k. Pick a closed, symmetric neighborhood U1 of 0 such that U1 + U1 ⊆ U0 and set i1 = m1 , j1 = n1 . Since xi1 j1 − xj1 = / U1 for (xi1 j1 − xij1 ) + (xij1 − xj1 ), there exists i0 such that xi1 j1 − xij1 ∈ i ≥ i0. Choose k0 such that mk0 > max{i1 , i0 }, nk0 > j1 and set i2 = mk0 , j2 = nk0 . Then xi1 j1 − xi2 j1 ∈ / U1 and xi2 j2 − xj2 ∈ / U0. Proceeding in this manner produces increasing sequences {ik }, {jk } such that xik jk − xjk ∈ / U0 and xik jk − xik+1 jk ∈ / U1 . For convenience, set zk,l = xik jl − xik+1 jl so zk,k ∈ / U1 . Choose a sequence of closed, symmetric neighborhoods of 0, {Un }, such that Un + Un ⊆ Un−1 for n ≥ 1. Note that m X U3 + U 4 + · · · + U m = Uj ⊆ U 2 j=3

for each m ≥ 3. By (I) and (II), limk zkl = 0 for each l and liml zkl = 0 for each k so by Lemma D.1 there is an increasing sequence of positive integers {pk } such that zpk pl , zpl pk ∈ Uk+2 for k > l. By (II), {pk } has a P ∞ subsequence {qk } such that { ∞ k=1 xiqk }i=1 is Cauchy so lim k

Thus, there exists k0 such that m X

z qk 0 ql =

l=1,l6=k0

kX 0 −1

z qk 0 ql +

m X

Ul+2 ⊆

l=k0 +1

P∞

l=1,l6=k0

zqk ql = 0.

l=1 P∞ l=1 zqk0 ql

zqk0 ql ∈ U2. Thus,

z qk 0 qk 0 =

∞ X l=1

∈ U2 . Then for m > k0 ,

m X

l=k0 +1

l=1

+ so zk0 =

∞ X

m+2 X l=3

z qk 0 ql ∈

kX 0 −1

Uk0 +2

l=1

Ul ⊆ U 2

z qk 0 ql − z k0 ∈ U 2 + U 2 ⊆ U 1

This is a contradiction and establishes the result. A matrix [xij ] satisfying conditions (I) and (II) of Theorem D.2 is called a K-matrix . [The appellation ”K” here refers to the Katowice branch of

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the Mathematical Institute of the Polish Academy of Science where the matrix theorems and applications were developed by Antosik, Mikusinski and other members of the institute.] At other points in the text we will also require another version of the matrix theorem which was developed by Stuart ([St1], [St2]) to treat weak sequential completeness of β-duals. Theorem D.3. (Stuart) Let xij ∈ X for i, j ∈ N. Suppose (I) limi xij = xj exists for all j and (II) for each increasing sequence of positive integers {mj } there is a subsequence {nj } and a choice of signs sj ∈ {−1, 1} such that P∞ { j=1 sj xinj }∞ i=1 is Cauchy. Then limi xij = xj uniformly for j ∈ N. In particular,

lim lim xij = lim lim xij = 0 and lim xii = 0. i

j

j

i

i

Proof: If the conclusion fails, there is a closed, symmetric neighborhood of 0,U0 , and increasing sequences of positive integers {mk } and {nk } such / U0 for all k. Pick a closed, symmetric neighborhood of that xmk nk − xnk ∈ 0,U1 , such that U1 + U1 ⊂ U0 and set i1 = m1 , j1 = n1 . Since xi1 j1 − xj1 = (xi1 j1 − xij1 ) + (xij1 − xj1 ), there exists i0 such that xi1 j1 − xij1 ∈ / U1 for i ≥ i0 . Choose k0 such that mk0 > max{i1 , i0 }, nk0 > j1 and set i2 = mk0 , j2 = nk0 . Then / U0 . Proceeding in this manner produces / U1 and xi2 j2 − xj2 ∈ xi 1 j 1 − x i 2 j 1 ∈ / U0 and xik jk − increasing sequences {ik } and {jk } such that xik jk − xjk ∈ / U1 . / U1 . For convenience, set zkl = xik jl − xik+1 jl so zkk ∈ xik+1 jk ∈ Choose a sequence of closed, symmetric neighborhoods of 0, {Un }, such that Un + Un ⊂ Un−1 for n ≥ 1. Note that U3 + U4 + ... + Um =

m X j=3

Uj ⊂ U2 f or each m ≥ 3.

By (I), limk zkl = 0 for each l and by (II), liml zkl = 0 for each k so by Lemma D.1 there is an increasing sequence of positive integers {pk } such that zpk pl , zpl pk ∈ Uk+2 for k > l. By (II) there is a subsequence {qk } of {pk } and a choice of signs sk such that {

∞ X k=1

sk xiqk }∞ i=1

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is Cauchy so ∞ X

lim k

sl zqk ql = 0.

l=1

Thus, there exists k0 such that ∞ X l=1

s l z qk 0 ql ∈ U 2 .

Then for m > k0 , m X

l=1,l6=k0

so

s l z qk 0 ql =

kX 0 −1

s l z qk 0 ql +

l=1

m X

l=k0 +1 ∞ X

z k0 =

l=1,l6=k0

Thus, s k0 z qk 0 qk 0 =

∞ X l=1

since U1 is symmetric

s l z qk 0 ql ∈

kX 0 −1

Uk0 +2 +

l=1

m X

l=k0 +1

Ul ⊂ U 2

s l z qk 0 ql ∈ U 2 .

s l z qk 0 ql − z k0 ∈ U 2 + U 2 ⊂ U 1 z qk 0 qk 0 ∈ U 1

as well. This is a contradiction.

A matrix which satisfies conditions (I) and (II) of Theorem D.3 will be called a signed K-matrix and Theorem D.3 will be referred to as the signed version of the Antosik-Mikusinski Matrix Theorem. We give an example of a matrix which is a signed K-matrix but is not a K-matrix. Example D.4. Let t = {tj } ∈ bs, the space of bounded series, and let X be bs equipped with the topology of coordinatewise convergence, σ(bs, c00 ) [Appendix A]. Define a matrix M = [mij ] with entries from X by mij = ej . Then no row of M has a subseries which converges in X so M is not a P∞ K-matrix. However, given any subsequence {nj } the series j=1 (−1)j enj converges in X so M is a signed K-matrix. Other refinements and comments on the matrix theorems can be found in [Sw1] 2.2. The text [Sw1] contains numerous applications of the matrix theorems to topics in topological vector spaces, measure and integration theory and sequence spaces.

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Appendix E Drewnowski’s Lemma

In this appendix we establish a remarkable result of Drewnowski which asserts that a strongly bounded, finitely additive set function defined on a σ-algebra is in some sense not ”too far” from being countably additive ([Dr]). This result is very useful in treating finitely additive set functions. Let Σ be a σ-algebra of subsets of a set S, X be a TVS whose topology is generated by the quasi-norm |·| and let µ : Σ → X be finitely additive and strongly bounded. (Recall µ is strongly bounded if µ(Ej ) → 0 whenever {Ej } is a pairwise disjoint sequence from Σ.) For E ∈ Σ, set µ0 (E) = sup{|µ(A)| : A ⊂ E, A ∈ Σ};

µ0 is called the submeasure majorant of µ and µ0 is also strongly bounded in the sense that µ0 (Ej ) → 0 whenever {Ej } is a pairwise disjoint sequence from Σ. Lemma E.1. (Drewnowski) If µ : Σ → X is finitely additive and strongly bounded and {Ej } is a pairwise disjoint sequence from Σ, then {Ej } has a subsequence {Enj } such that µ is countably additive on the σ-algebra generated by {Enj }. Proof: Partition N into a pairwise disjoint sequence of infinite sets 0 0 {Kj1 }∞ j=1 . By the strong additivity of µ , µ (∪j∈Ki1 Ej ) → 0 as i → ∞ so 0 there exists i such that µ (∪j∈Ki1 Ej ) < 1/2. Set N1 = Ki1 and n1 = inf N1 . Now partition N1 \ {n1 } into a pairwise disjoint sequence of infinite sub0 2 sets {Kj2 }∞ j=1 . As above there exists i such that µ (∪j∈Ki2 Ej ) < 1/2 . Let 2 N2 = Ki and n2 = inf N2 . Note N2 ⊂ N1 and n2 > n1 . Continuing this construction produces a subsequence nj ↑ ∞ and a sequence of infinite subsets of N, {Nj }, such that Nj+1 ⊂ Nj and µ0 (∪i∈Nj Ei ) < 1/2j . If Σ0 is the σ-algebra generated by {Enj }, then µ is countably additive on Σ0 . 243

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We also have a version of Drewnowski’s Lemma for a sequence of finitely additive set functions. Corollary E.2. Let µi : Σ → X be finitely additive and strongly bounded for each i ∈ N. If {Ej } is a pairwise disjoint sequence from Σ, then there is a subsequence {Enj } such that each µi is countably additive on the σ-algebra generated by {Enj }. 0

Proof: Define a quasi-norm |·| on X N by 0

0

|x| = |{xi }| = N

∞ X i=1

|xi | /((1 + |xi |)2i ).

Define µ : Σ → X by µ(E) = {µi (E)}. Then µ is finitely additive and 0 strongly bounded with respect to |·| so by Lemma 1 there is a subsequence {Enj } such that µ is countably additive on the σ-algebra Σ0 generated by {Enj }. Thus, each µi is countably additive on Σ0 .

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References

[AP1] A. Aizpura and J. Perez-Fernandez, Spaces of s-bounded multiplier convergent series, Acta Math. Hunjar., 87(2000), 135 - 146. [AP2] A. Aizpura and J. Perez-Fernandez, Sequence space associated to a series in Banach space, Indian J. Pure Appl. Math., 33(2002), 1317-1329. [A] P. Antosik, On interchange of limits, Generalized Functions, Convergence Strucctures and Their Applications, Plenum Press, N.Y., 1988, p. 367 - 374. [Ap] T. Apostol, Mathematical Analysis, Addison-Wesley, Reading, 1975. [Ba] S. Banach, Theorie des Operations Lineaires, Warsaw, 1932. [Bar] R. Bartle, A general bilinear vector integral, Studia Math., 15(1956), 337-352. [Bs] B. Basit, On a Theorem of Gelfand and a new proof of the OrliczPettis Theorem, Rend. Inst. Matem. Univ. di Trieste, 18(1986), 159 -162. [Bt] J. Batt, Applications of the Orlicz-Pettis Theorem to operatorvalued measures and compact and weakly compact linear transformations on the space of continuous functions, Revue Roum. Math. Pures Appl., 14(1969), 907 - 945. [Be] G. Bennett, Some inclusion theorems for sequence spaces, Pacific J. Math. 46(1973), 17 - 30. [BK] G. Bennett and N. Kalton, FK spaces containing c0 , Duke Math. J., 39(1972), 561 - 582. [BP] C. Bessaga and A. Pelczynski, On Bases and Unconditional Convergence of series in Banach Space, Studia Math., 17(1958), 151-164. [Bo] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000. [BSS] J. Boos, C. Stuart and C. Swartz, Gliding Hump Properties and

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Matrix Domains, Analysis Math., 30(2004), 243 - 257. [BW] Qingying Bu and Cong Xin Wu, Unconditionally Convergent Series of Operators on Banach Spaces, J. Math. Anal. Appl., 207(1997), 291 - 299. [BM] J. Burzyk and P. Mikusinski, On Normability of Semigroups, Bull. Polon. Acad. Sci., 28(1980), 33-35. [Day] M. Day, Normed Linear Spaces, Springer-Verlag, Berlin, 1962. [DeS] J. DePree and C. Swartz, Introduction to Real Analysis, Wiley, N.Y., 1987. [Die] P. Dierolf, Theorems of Orlicz-Pettis type for locally convex spaces, Man. Math., 20(1977), 73 - 94. [DF] J. Diestel and F. Faires, On Vector Measures, Trans. Amer. Math. Soc., 198(1974), 253 - 271. [DU] J. Diestel and J. Uhl, Vector Measures, Amer. Math. Soc. Surveys #15, Providence, 1977. [Din] N. Dinculeanu, Weak Compactness and Uniform Convergence of Operators in Space of Bochner Integrable Functions, J. Math. Anal. Appl., 1090(1985), 372 - 387. [Dr] L. Drewnowski, Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym Theorems, Bull. Acad. Polon. Sci., 20(1972), 725-731. [DS] N. Dunford and J. Schwartz, Linear Operators I, Interscience, N.Y., 1958. [FL] W. Filter and I. Labudu, Essays on the Orlicz - Pettis Theorem I, Real. Anal. Exch., 16(1990/91), 393 - 403. [FP] M. Florencio and P. Paul, A Note on λ- Multiplier Convergent Series, Casopis Pro. Post. Mat. 113(1988), 421 - 428. [FP2] M. Florencio and P. Paul, Barrelledness conditions on certain vector valued sequence spaces, Arch. Math., 48(1987), 153 - 164. [Fo] J. Fourie, Barrelledness conditions on generalized sequence spaces, South African J. Sci., 84(1988), 346 - 348. [Ga] D.J.H. Garling, The β − and γ− duality of sequence spaces, Proc. Camb. Phil. Soc., 63(1967), 963 - 981. [Gi] D.P. Giesy, A Finite-valued Finitely Additive Unbounded Measure, Amer. Math. Monthly, 77(1970), 508-510. [GDS] H.G. Garnir, M. DeWilde and J. Schmets, Analyse Fontionnelle I, Birkhauser, Basel, 1968. [GR] W. Graves, Proceedings of the Conference on Integration, Topology, and Geometry in Linear Spaces, Amer. Math. Soc., Providence, 1980. [GKR] M. Gupta, P.K. Kamthan and K.L.N. Rao, Duality in Certain

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Generalized Kothe Sequence Spaces, Bull. Inst. Math. Acad. Sinica, 5(1977), 285 - 298. [Ha] H. Hahn, Uber Folgen Linearen Operationen, Monatsch. fur Math. und Phys. 32(1922), 1 - 88. [Hay] R. Haydon, A non-reflexive Grothendieck space that does not contain l∞ , Isreal J. Math, 40(1981), 65-73. [HT] E. Hellinger and O. Toeplitz, Grundlagen fur eine Theorie den unendlichen Matrizen, Math. Ann., 69(1910), 289 - 330. [Ho] J. Howard, The Comparison of an Unconditionally Converging Operator, Studia Math., 33(1969), 295 - 298. [Ka] N. J. Kalton, Spaces of Compact Operators, Math. Ann., 208(1974), 267 - 278. [Ka2] N. Kalton, Subseries Convergence in Topological Groups and Vector Spaces, Isreal J. Math., 10(1971), 402 - 412. [Ka3] N.J. Kalton, The Orlicz - Pettis Theorem, Contemporary Math., Amer. Math. Soc., Providence, 1980. [KG] P.K. Kamthan and M. Gupta, Sequence Spaces and Series, Marcel Dekker, N.Y., 1981. [K1] G. K¨ othe, Topological Vector Spaces I, Springer - Verlag, Berlin, 1969. [K2] G. K¨ othe, Topological Vector Spaces II, Springer-Verlag, Berlin, 1979. [LW] E. Lacey and R.J. Whitley, Conditions under which all the Bounded Linear Maps are Compact, Math. Ann., 158 (1965), 1-5. [LPY] Lee Peng Yee, Sequence Spaces and the Gliding Hump Property, Southeast Asia Bull. Math., Special Issue (1993), 65 - 72. [LPYS] Lee Peng Yee and C. Swartz, Continuity of Superposition Operators on Sequence Spaces, New Zealand J. Math., 24(1995), 41-52. [LB] R. Li and Q. Bu, Locally Convex Spaces Containing no Copy of c0 , J. Math. Anal. Appl., 172(1993), 205-211. [LS] R. Li and C. Swartz, A Nonlinear Schur Theorem, Acta Sci. Math., 58(1993), 497 - 508. [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer-Verlag, Berlin, 1977. [MA] S.D. Madrigal and J.M.B. Arrese, Local Completeness and Series, Simon Stevin, 65(1991), 331 - 335. [Mc] C. W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22(1967), 297 - 303. [MR] C. McArthur and J. Rutherford, Some Applications of an In-

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equality in Locally Convex Spaces, Trans. Amer. Math. Soc., 137(1969), 115 -123. [No] D. Noll, Sequential Completeness and Spaces with the Gliding Humps Property, Manuscripta Math., 66(1990), 237 - 252. [Or] W. Orlicz, Beitr¨ age zur Theorie der Orthogonalent Wichlungen II, Studia Math., 1(1929), 241 - 255. [PBA] J. Perez - Fernandez, F. Benitez - Trujillo and A. Aizpuru, Characterizations of completeness of normed spaces through weakly unconditionally Cauchy series, Czech. Math. J., 50(2000), 889-896. [Pe] B. J. Pettis, On Integration in Vector Spaces, Trans. Amer. Math. Soc., 44(1938), 277 - 304. [Pl] A. Pelczynski, On Strictly singular and Strictly cosingular Operators, Bull. Acad. Polon. Sci., 13 (1965), 31 - 36. [RR] K.P.S. Rao and M. Rao, Theory of Charges, Academic Press, N. Y., 1983. [Ro1] A. Robertson, Unconditional Convergence and the Vitali-HahnSaks Theorem, Bull. Soc. Math, France, Supp;. Mem. 31-32 (1972), 335 341. [Ro2] A. Robertson, On Unconditional Convergence in Topological Vector Spaces, Proc. Royal Soc. Edinburgh, 68(1969), 145-157. [Rol] S. Rolewicz, Metric Linear Spaces, Polish Sci. Publ., Warsaw, 1972. [Ros] R. Rosier, Dual Spaces of Certain Vector Sequence Spaces, Pacific J. Math., 46(1973), 487 - 501. [RS] W. Ruckle and S. Saxon, Generalized Sectional Convergence, J. Math. Anal. Appl., 193(1995), 680 - 705. [SaSe] R. Samaratunga and J. Sember, Summability and Substructures of 2N , Southeast Asia Math. Bull., 66(1990), 237 - 252. [Sm] W. Schachermeyer, On some classical measure-theoretic theorems for non-sigma-complete Boolean algebras, Dissert. Math., Warsaw, 1982. [Sch] H. H. Schaefer, Topological Vector Spaces, MacMillan, N.Y., 1966. ¨ [Sr] J. Schur, Uber lineare Tranformation in der Theorie die unendlichen Reihen, J. Reine Angew. Math., 151(1920), 79 - 111. [Sti] W. J. Stiles, On Subseries Convergence in F-spaces, Israel J. Math., 8(1970), 53 - 56. [St1] C. Stuart, Weak Sequential Completeness in Sequence Spaces, Ph.D. Dissertation, New Mexico State University, 1993. [St2] C. Stuart, Weak Sequential Completeness of β-duals, Rocky Mt. Math. J., 26(1996), 1559 -1568.

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[St3] C. Stuart, Interchanging the limit in a double series, Southeast Asia Bull. Math., 18(1994), 81 - 84. [SS] C. Stuart and C. Swartz, A Projection Property and Weak Sequential Completeness of α-duals, Collect. Math.l, 43(1992), 177 - 185. [Sw1] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ., Singapore, 1996. [Sw2] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N.Y., 1992. [Sw3] C. Swartz, Measure Integration and Function Spaces, World Sci., Pub., Singapore, 1994. [Sw4] C. Swartz, Orlicz-Pettis Theorems for Multiplier Convergent Operator Valued Series, Proy. J. Math., 23(2004), 61-72. [Sw5] C. Swartz, Subseries Convergence in Spaces with Schander Basis, Proc. Amer. Math. Soc.,129(1995), 455-457. [Th] G.E.F. Thomas, L’integration par rapport a une mesure de Radon vectorielle, Ann. Inst. Fourier, 20(1970), 55 - 191. [Thr] B.L. Thorp, Sequential Evaluation Convergence, J. London. Math. Soc., 44(1969), 201-209. [Tw] I. Tweddle, Unconditional Convergence and Vector-valued Measures, J. London Math. Soc., 2(1970), 603 - 610. [Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw - Hill, N.Y., 1978. [Wi2] A. Wilansky, Summability through Functional Analysis, North Holland, Amsterdam, 1984. [Wu] Wu, Junde, The compact sets in the infinite matrix topological algebras, Acta Math. Sinica, to appear. [WL] Wu Junde and Lu Shijie, A Summation Theorem and its Applications, J. Math. Anal. Appl., 257(2001), 29 - 38. [Y] K. Yosida, Functional Analysis, Springer - Verlag, N.Y., 1966.

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Index

A : E → F , 201 δ1 (X, X 0 ), 57 δ2 (X, X 0 ), 62 Dλ (X, X 0 ), 63 FQσ, 219 IQσ, 220 0 τ (X, X P ), 208 M ∞ ( xj ), 140 t(X, X 0 ), 60 tλ (X, X 0 ), 68 s, 214 β-dual, 5 X b , 34 B(S, Σ), 60 Lb (X, Y ), 211 l∞ , 213 bs, 214 k·kbs , 214 bv, 214 k{tj }kbv , 214 M0 , 9, 219 χσ , 216 K(X, Y ), 94 Ac , 37 γ(X, X 0 ), 208 CX (S), 154 cs, 214 c, 213 ca(Σ), 60, 72 Lc (X, Y ), 211 λ(X, X 0 ), 208 c0 -factorable, 224

c0 -invariant, 224 δt , 60 cc , 213 c00 , 213 m0 , 9, 213 ∞-GHP, 15, 85, 224, 234 [m, n], 216 λβ , 5 A : λ → µ, 157 M (λ), 218 bv0 , 214 L→0 (X, Y ), 92, 211 c0 , 213 E βY , 173 µ ˆ (A), 180 LA (X, Y ), 211 LC (X, Y ), 91 Ls (X, Y ), 211 Lpc (X, Y ), 92, 211 χσ t, 216 k·kp , 214 lp , 213, 214 |·|p , 214 s · t, 5 E ββ , 203 AT T , 203 Pn , 231 X s , 34 β(X, X 0 ), 208 Kb (X, Y ), 97 k·k∞ , 213 t · x, 12 251

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ej ⊗ z, 170 AT , 158, 202 E β , 170, 187 l∞ (X), 229 BS(X), 230 BV (X), 230 c(X), 229 CS(X), 230 c0 (X), 229 c00 (X), 229 cc (X), 229 lp (X), 230 m0 (X), 229 BV0 (X), 230 w(E βY , E), 173 σ(X, X 0 ), 208 ω(λβX , λ), 12 0-GHP, 8, 221, 233 AB-space, 215, 231 absolutely convergent, 23, 45 AK-space, 215, 231 Antosik, 146 Antosik Interchange Theorem, 145 Antosik-Mikusinski, 239 Banach Mackey pair, 138 Banach-Mackey space, 34 Banach-Steinhaus property, 173 BK-space, 215 bounded multiplier convergent, 5, 169 Cesaro matrix, 225 conditionally sequentially compact, 14 continuous from above, 182 coordinate functionals, 36 countably additive, 36

finite/co-finite, 37 finitely additive, 36 FK-space, 215 Hahn-Schur Theorem, 193, 196 Hellinger-Toeplitz topology, 6, 51, 170, 188, 209 increasing, 8, 216 infinite gliding hump property, 15, 85, 224, 234 interval, 8, 216 invariant, 218 K-matrix, 240 K-space, 215, 231 LCTVS, 207 linked, 50, 209 locally complete, 27 Mackey topology, 208 matrix domain, 225 monotone, 216, 233 multiplier Cauchy, 5, 169 multiplier convergent, 5, 169 multiplier space, 218 multipliers, 5, 222 Nikodym Convergence Theorem, 200 normal, 10, 216 Orlicz-Pettis Theorem, 49, 53, 85, 90, 152, 187 polar topology, 208 quasi-norm, 207 rearrangement convergent, 22

DF property, 95 Dierolf topology, 57, 62, 63 double series, 145 Drewnowski, 243 exhaustive, 41

Schauder basis, 36 Schauder decomposition, 153 SE, 216, 231 sectional operator, 231 sectional projection, 215

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Index

sections equicontinuous, 216, 231 sections uniformly bounded, 133, 216 semi-variation, 43, 179 SGHP, 9, 219, 233 sign, 9 signed K-matrix, 242 signed strong gliding hump property, 9, 219, 233 signed weak gliding hump property, 12, 216, 233 signed-SGHP, 9, 219, 233 signed-WGHP, 12, 216, 233 signs, 216 solid, 10, 216 Stiles, 151 strong gliding hump property, 9, 219, 233 strong µ-GHP, 138 strong µ gliding hump strong property, 138 strong operator topology, 211 strong topology, 208 strongly additive, 41 strongly bounded, 41, 243 Stuart, 174 SUB, 133, 216 submeasure majorant, 243 subseries convergent, 5 summing operator, 6, 169

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253

TVS, 207 Tweddle topology, 60, 68 unconditional, 36 unconditionally Cauchy, 22 unconditionally convergent, 22 unconditionally converging, 32, 71 uniform operator topology, 211 uniform tails, 63 unordered convergent, 22 variation, 45 weak gliding hump property, 12, 216, 233 weak µ-GHP, 139 weak µ gliding hump property, 139 weak operator topology, 211 weak topology, 208 weakly compact, 71 weakly unconditionally Cauchy, 26 WGHP, 12, 216, 233 wuc, 26 zero gliding hump property, 8, 221, 233

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MULTIPLIER CONVERGENT SERIES Charles Swartz New Mexico State University, USA

World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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TA I P E I

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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MULTIPLIER CONVERGENT SERIES Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-283-387-7 ISBN-10 981-283-387-0

Printed in Singapore.

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To the Memory of My Mother

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Preface

This monograph contains an exposition of the properties and applications of multiplier convergent series with values in a topological vector space. P is a (formal) series If λ is a space of scalar valued sequences and j xjP with values in a topological vector space X, the series j xj is λ multiplier P∞ convergent if the series j=1 tj xj converge in X for every {tj } ∈ λ. For example, if M0 = {χσ : σ ⊂ N}, where χσ is the characteristic function of σ, then M0 multiplier convergence is just subseries convergence. Basic properties of multiplier convergent series are developed in Chapter 2 and applications of multiplier convergent series to topics in topological vector spaces and vector valued measures are given in Chapter 3. A classical result of Orlicz and Pettis states that if a series in a normed linear space is subseries convergent (M0 multiplier convergent) in the weak topology of the space, then the series is actually subseries convergent (M0 multiplier convergent) in the norm topology of the space. Generalizations of this theorem to λ multiplier convergent series with values in a locally convex space are given in Chapters 4, 5 and 6. Another classical theorem of Hahn P and Schur asserts that if j tij is absolutely convergent for every i ∈ N P and if limi j∈σ tij exists for every σ ⊂ N with tj = limi tij , then the series P j tj is absolutely convergent and lim i

∞ X j=1

|tij − tj | = 0.

In Chapter 7 we establish generalizations of the Hahn-Schur Theorem to λ multiplier convergent series with values in a topological vector space. Chapters 8, 9 and 10 contain applications of the Hahn-Schur Theorems to spaces of multiplier convergent series, double series and automatic continuity of matrix mappings between sequence spaces. vii

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Chapter 11 extends the notion of multiplier convergent series to series with operator values and multiplier sequences with values in the domains of the operators. Chapters 12 and 13 extend the Orlicz-Pettis Theorem and Hahn-Schur Theorem to operator valued series and vector valued multipliers. Chapter 13 also contains applications to measures with values in a space of continuous linear operators. Chapter 14 considers automatic continuity results for operator valued matrices acting on vector valued sequence spaces.

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Contents

Preface

vii

1. Introduction

1

2. Basic Properties of Multiplier Convergent Series

5

3. Applications of Multiplier Convergent Series

25

4. The Orlicz-Pettis Theorem

49

5. Orlicz-Pettis Theorems for the Strong Topology

83

6. Orlicz-Pettis Theorems for Linear Operators

89

7. The Hahn-Schur Theorem

101

8. Spaces of Multiplier Convergent Series and Multipliers

133

9. The Antosik Interchange Theorem

145

10. Automatic Continuity of Matrix Mappings

157

11. Operator Valued Series and Vector Valued Multipliers

169

12. Orlicz-Pettis Theorems for Operator Valued Series

187

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13. Hahn-Schur Theorems for Operator Valued Series

191

14. Automatic Continuity for Operator Valued Matrices

201

Appendix A. Topological Vector Spaces

207

Appendix B. Scalar Sequence Spaces

213

Appendix C. Vector Valued Sequence Spaces

229

Appendix D. The Antosik-Mikusinski Matrix Theorems

239

Appendix E. Drewnowski’s Lemma

243

References

245

Index

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Chapter 1

Introduction

One of the most interesting and useful theorems in the early history of functional analysis is a result now known as the Orlicz-Pettis Theorem. The result was originally proven by Orlicz for weakly sequentially complete normed spaces although the result in full generality for normed spaces was known by the Polish mathematicians and appears in Banach’s book ([Or], [Ba]). The first version of the theorem available in English was proven by Pettis and was used to treat topics in vector valued measures and vector valued integrals ([Pe]; see [Ka3] and [FL] for discussions of the history of P the theorem). If X is a topological vector space (TVS), a series j xj in P∞ X is subseries convergent in X if the subseries j=1 xnj converges in X for every subsequence {nj }. The Orlicz-Pettis Theorem for normed spaces P states that if the series j xj is subseries convergent in the weak topology of the space, then the series is actually subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was extended to locally convex spaces by McArthur ([Mc]). If σ is any subset of N and χσ is the P characteristic function of σ, then a series j xj in a TVS X is subseries P∞ P convergent iff the series j=1 χσ (j)xj = j∈σ xj converges in X for every σ ⊂ N. Thus, if m0 = span{χσ : σ ⊂ N}, the sequence space of real P valued sequences with finite range, a series j xj in a TVS X is subseries P∞ convergent iff the series j=1 tj xj converges for every t = {tj } ∈ m0 . To obtain a generalization of the notion of subseries convergence, we may replace the space m0 by a general vector space λ of real valued sequences. If λ is a vector space of real valued sequences and {xj } is a sequence in the P TVS X, the (formal) series j xj is said to be λ multiplier convergent if P∞ the series j=1 tj xj converges in X for every t = {tj } ∈ λ; the elements t ∈ λ are called multipliers. This suggests that generalizations of the Orlicz1

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Multiplier Convergent Series

Pettis Theorem might be obtained by replacing subseries convergent series by λ multiplier convergent series for certain sequence spaces λ. We show that such generalizations are possible in Chapters 4, 5, and 6. Another classical result which involves subseries convergent series is a result which is often referred to as the Hahn-Schur Theorem. One verP sion of the Hahn-Schur Theorem states that if j xij is subseries conP∞ vergent for every i ∈ N, limi j=1 xinj exists for every subsequence {nj } P and if xj = limi xij , then the series j xj is subseries convergent and P P limi j∈σ xij = j∈σ xj uniformly for σ ⊂ N ([Ha], [Sc], [Sw1]; this version of the theorem actually holds for series with values in an Abelian P topological group). A series j xj in a TVS is said to be bounded mulP tiplier convergent if the series j xj is l∞ multiplier convergent ([Day]). There is a version of the Hahn-Schur Theorem for bounded multiplier conP vergent series which states that if j xij is bounded multiplier converP∞ gent for every i ∈ N, limi j=1 tj xij exists for every t = {tj } ∈ l∞ and P if xj = limi xij , then the series j xj is bounded multiplier convergent P∞ P∞ and limi j=1 tj xij = j=1 tj xj uniformly for t ∈ l∞ , ktk∞ ≤ 1 ([Sw2]). Again this suggest that one might obtain generalizations of both versions of the Hahn-Schur Theorem by replacing subseries and bounded multiplier convergent series by λ multiplier convergent series for certain sequence spaces λ. We show in Chapter 7 that versions of the Hahn-Schur Theorem are obtainable for λ multiplier convergent series if the sequence space λ satisfies sufficient conditions. There are further applications of λ multiplier convergent series to topics in Banach space theory, sequence spaces and matrix mappings. For example, a result of Bessaga and Pelczynski states that a Banach space X contains no subspace isomorphic to c0 iff every c0 multiplier convergent series in X is subseries convergent (or bounded multiplier convergent) ([BP]). A generalization of this result to sequentially complete locally convex topological vector spaces (LCTVS) is given in Chapter 3.15. A characterization of dual spaces not containing c0 is given in terms of subseries convergent series in 3.20, a characterization of locally complete LCTVS in terms of c0 multiplier convergent series is given in 3.10 and a characterization of Banach-Mackey spaces in terms of l 1 multiplier convergent series is given in 3.23. In Chapter 3, we also give applications of multiplier convergent series to vector valued measures. We give a characterization of bounded vector measures in terms of c0 multiplier convergent series in 3.33 and a characterization of strongly bounded (strongly additive) vector measures in terms of subseries convergence in 3.43.

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Introduction

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3

Further applications to various topics in sequence spaces and matrix mappings between sequence spaces are given in later chapters. Multiplier convergent series are interesting in their own right and we develop their basic properties in Chapter 2. In the last four chapters we consider operator valued series and vector valued spaces of multipliers. Let X, Y be TVS, L (X, Y ) the space of all continuous linear operators from X into Y and E be a vector space of X P valued sequences. A series j Tj in L(X, Y, ) is E multiplier convergent P∞ if the series j=1 Tj xj converges in Y for every sequence {xj } ∈ E. The basic properties of operator valued series with vector valued multipliers sometimes closely parallels the properties of series with scalar multipliers but sometimes require additional assumptions. We present these properties in Chapter 11. Versions of the Orlicz-Pettis Theorem and the Hahn-Schur Theorem for operator valued series and vector valued multipliers are presented in Chapters 12 and 13. The basic notations, definitions and terminology are presented in Appendices A, B and C. Appendices D and E contain material not easily accessible and which is used in the text.

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Chapter 2

Basic Properties of Multiplier Convergent Series

In this chapter we will develop the basic properties of multiplier convergent series. In what follows λ will denote a scalar sequence space which contains the subspace c00 of sequences which are eventually 0 and Λ ⊂ λ will denote a subset while X will denote a Hausdorff topological vector space (TVS). P Definition 2.1. A (formal) series j xj in X is Λ multiplier convergent P∞ in X if the series j=1 tj xj converges in X for every t = {tj } ∈ Λ . The P∞ series is Λ multiplier Cauchy in X if the series j=1 tj xj satisfies a Cauchy condition for every t = {tj } ∈ Λ. The elements t = {tj } ∈ Λ are called multipliers. P subseries A series j xj which is m0 multiplier convergent is said to be P convergent; thus, a series is subseries convergent iff the subseries j xnj is P convergent for every subsequence {nj }. A series j xj which is l∞ multiplier convergent is said to be bounded multiplier convergent. We now establish the basic properties of Λ multiplier convergent series. We begin by considering boundedness properties of Λ multiplier convergent series. P convergent series. The summing operator Let j xj be a λ multiplier P S (with respect to λ and j xj ) is the linear map S : λ → X defined by P∞ St = S({tj }) = j=1 tj xj for t = {tj } ∈ λ. Recall that the β-dual of λ, λβ , is defined to be ∞ X λβ = {sj } : sj tj = s · t converges for every t ∈ λ j=1

β

and λ and λ form a dual pair under the bilinear pairing s · t (Appendix A). 5

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Multiplier Convergent Series

6

If Y, Y 0 are a dual pair, a Hellinger-Toeplitz topology defined for dual pairs is a topology w(Y, Y 0 ) with the property that whenever a linear map T : Y → Z is σ(Y, Y 0 ) − σ(Z, Z 0 ) continuous, then T is w(Y, Y 0 ) − w(Z, Z 0 ) continuous (Appendix A.1). Theorem 2.2. Let X be a Hausdorff locally convex TVS (LCTVS) and P The summing operator j xj a λ multiplier convergent series in X. S : λ → X is σ(λ, λβ ) − σ(X, X 0 ) continuous and, therefore, w(λ, λβ ) − w(X, X 0 ) continuous with respect to any Hellinger-Toeplitz topology w. Proof: Let x0 ∈ X 0 , t ∈ λ. Then hx0 , Sti =

∞ X j=1

tj hx0 , xj i = {hx0 , xj i} · t

since {hx0 , xj i} ∈ λβ by the convergence of the series. This implies that S is σ(λ, λβ ) − σ(X, X 0 ) continuous. The last statement follows from the definition of Hellinger-Toeplitz topologies. Theorem 2.2 gives a boundedness result for multiplier convergent series Corollary 2.3. If B is σ(λ, λβ ) bounded, then SB = { is bounded in X.

P∞

j=1 tj xj

: t ∈ B}

For topological sequence spaces we also have a boundedness result for the sums of multiplier convergent series. Corollary 2.4. Let λ be a K-space. If λβ ⊂ λ0 and B ⊂ λ is bounded, then P SB = { ∞ j=1 tj xj : t ∈ B} is bounded in X.

Proof: Since B is bounded in X and λβ ⊂ λ0 , B is σ(λ, λβ ) bounded so the result follows from Corollary 2.3. For a general condition which guarantees that λβ ⊂ λ0 , we have Proposition 2.5. (i) If λ is a barrelled K-space, then λβ ⊂ λ0 . (ii) If λ is an AK-space, then λ0 ⊂ λβ . (iii) If λ is a barrelled AK-space, then λ0 = λβ .

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Proof: (i): Let s ∈ λβ . For each n define fn : λ → R by fn (t) = Pn j=1 sj tj . Since λ is a K-space, each fn is continuous. Now fn (t) → P∞ j=1 sj tj = s · t so the linear functional t → s · t is continuous since λ is barrelled. Therefore, s ∈ λ0 .

P∞ j (ii): Let f ∈ λ0 . Set sj = f, e j . If t ∈ λ, then t = j=1 tj e P P∞ ∞ j (convergence in λ) so hf, ti = j=1 tj f, e = j=1 tj sj = s · t so f = s ∈ λβ . (iii) follows from (i) and (ii). Proposition 2.5 is applicable, in particular, if λ is a Banach or Frechet space. Concerning the strong boundedness of partial sums of multiplier convergent series, we have Corollary 2.6. Suppose that λ is a barrelled AB space (Appendix B.3) and Pn P j=1 tj xj : j xj is λ multiplier convergent. If B ⊂ λ is bounded, then { n ∈ N, t ∈ B} is β(X, X 0 ) bounded. Pn Proof: Let Pn : λ → λ be the sectional operator Pn (t) = j=1 tj ej . By the AB assumption {Pn : n} is pointwise bounded on λ and, therefore, {Pn : n} is equicontinuous by the barrelledness assumption. Since λ is barrelled, 0 0 λ has the strong topology β(λ, λ ) so {Pn B : n} is β(λ, λ ) bounded. By Proposition 2.5, λβ ⊂ λ0 so {Pn B : n} is β(λ, λβ ) bounded. The result now follows from Theorem 2.2 since the strong topology is a Hellinger-Toeplitz topology. In particular, Corollary 2.6 is applicable to bounded multiplier convergent series. P Corollary 2.7. Let j xj be bounded multiplier convergent ( l ∞ multiplier P∞ convergent). Then { j=1 tj xj : k{tj }k∞ ≤ 1} is β(X, X 0 ) bounded. Since m0 is barrelled (Theorem 7.59 or [Sw1] 4.7.9), we also have P Corollary 2.8. Let j xj be subseries convergent (m0 multiplier converP gent). Then { j∈σ xj : σ ⊂ N} is β(X, X 0 ) bounded. The condition that λβ ⊂ λ0 in Corollaries 2.6, 2.7 and 2.8 is important.

Example 2.9. Let λ = c00 with the sup-norm. For any {xj } ⊂ X the P series j xj is c00 multiplier convergent. Take any unbounded sequence

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{xj } in X. Then {ek : k ∈ N} is bounded in c00 , but { 0

1

k ∈ N} is not bounded in X. Note that c00 = l ⊂

cβ00

Pk

j=1

ekj xj = xk :

= s.

We have another result for boundedness in topological sequence spaces. This result requires a gliding hump assumption. An interval in N is a set of the form I = {k ∈ N : m ≤ k ≤ n}, where m ≤ n. A sequence of intervals {Ij } is increasing if max Ik < min Ik+1 for all k. If x = {xk } is any sequence (scalar or vector) and σ ⊂ N, χσ x will denote the coordinatewise product of χσ and x. Definition 2.10. Let λ be a K-space. Then λ has the zero gliding hump property (0-GHP) if whenever {Ij } is an increasing sequence of intervals and {tj } ⊂ λ converges to 0 in λ, there is a subsequence {nj } such that the P∞ coordinatewise sum of the series j=1 χInj tnj belongs to λ. For examples of spaces with 0-GHP, see Appendix B. P Theorem 2.11. Let λ be a K-space with 0-GHP. If j xj is λ multiplier convergent in X, then the summing operator S : λ → X is sequentially continuous and, therefore, bounded.

Proof: Suppose the conclusion fails. Then there exist a closed neighborP∞ /U hood of 0, U , in X and a null sequence {ti } in λ such that j=1 tij xj ∈ P n 1 m1 for every i. Set m1 = 1. There exists n1 > m1 such that j=1 t j xj ∈ / U. Pick a closed symmetric neighborhood of 0, V , such that V + V ⊂ U . Since P 1 m2 t xj ∈ V . limi tij = 0 for every j, there exists m2 > m1 such that nj=1 P 2 j m P n 2 m2 / t 2 xj ∈ / U . Hence, nj=n There exists n2 > n1 such that j=1 tj xj ∈ 1 +1 j V . Continuing this construction produces increasing sequences {mj }, {nj } such that (∗)

X

j∈Ik

k tm / V, where Ik = [nk + 1, nk+1 ]. j xj ∈

P∞ By 0-GHP, there is a subsequence {pk } such that t = k=1 χIpk tpk ∈ P∞ P P∞ mp k xj should converge. But, this λ. Hence, j=1 tj xj = k=1 j∈Ipk tj contradicts (∗). Corollary 2.12. Let the assumptions be as in Theorem 2.11. If B ⊂ λ is bounded, the SB ⊂ X is bounded. Remark 2.13. Note that λ = c00 does not have 0-GHP so this assumption in Theorem 2.11 cannot be dropped. See Example 2.9.

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We next consider uniform convergence for multiplier convergent series when the multipliers range over certain bounded subsets of the multiplier space. These results also require gliding hump assumptions. In the following definition a sign is a variable which assumes the values {±1}. Definition 2.14. Let λ be a K-space and Λ ⊂ λ. The set Λ has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {tk } in Λ and every increasing sequence of intervals {Ik }, there is a subsequence {nk } and a sequence of signs {sk } such that the coordinatewise P∞ sum k=1 sk χInk tnk belongs to Λ. The set has the strong gliding hump property (SGHP) if the signs in the definition above can all be chosen to be equal to 1. See Appendix B for examples. For example, l ∞ has SGHP and bs has signed-SGHP but not SGHP. Λ = {χσ : σ ⊂ N} = M0 has SGHP whereas m0 = spanM0 does not. We first establish a lemma. P Lemma 2.15. Let j xj be Λ multiplier convergent where Λ ⊂ λ. If the P series ∞ j=1 tj xj do not converge uniformly for t ∈ B ⊂ Λ, then there exist a symmetric neighborhood of 0,V , in X , tk ∈ B and an increasing sequence P / V. of intervals {Ik } such that j∈Ik tkj xj ∈ P∞ Proof: If the series j=1 tj xj do not converge uniformly for t ∈ B, there exists a symmetric neighborhood, U , of 0 such that for every k there P∞ / U . For k = 1, let m1 tk x ∈ exist tk ∈ B, mk ≥ k such that j=m P∞k j j1 1 / U . Pick a symmetric and t ∈ B satisfy this condition so j=m1 tj xj ∈ neighborhood of 0, V , such that V + V ⊂ U . There exists n1 > m1 such P∞ that j=n1 +1 t1j xj ∈ V . Then n1 X

j=m1

t1j xj

=

∞ X

j=m1

t1j xj

−

∞ X

j=n1 +1

t1j xj ∈ / V.

Put I1 = [m1 , n1 ] and continue the construction. Theorem 2.16. Let λ be a K-space and Λ ⊂ λ have signed-SGHP. If the P P series j xj is Λ multiplier convergent, then the series ∞ j=1 tj xj converge uniformly for t belonging to bounded subsets of Λ. P∞ Proof: Suppose that B ⊂ Λ is bounded but the series j=1 tj xj do not converge uniformly for t ∈ B. Let the notation be as in Lemma 2.15. Let nk and sk be as in the definition of signed-SGHP above and

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let t = P

P∞

In k t k=1 sk χP

j∈Ink tj xj

= sk Cauchy condition.

nk

j∈Ink

P∞ ∈ Λ. Then tj xj does not converge since j=1 P∞ tnj k xj ∈ / V , i.e., j=1 tj xj does not satisfy the

Theorem 2.16 implies two well known results for bounded multiplier and subseries convergent series which we now state. P Corollary 2.17. Let j xj be l∞ multiplier convergent. Then the series P∞ j=1 tj xj converge uniformly for k{tj }k∞ ≤ 1. Proof: l∞ has SGHP.

P Corollary 2.18. Let Then the series j xj be subseries convergent. P∞ t x converge uniformly for t ∈ M = {χ : σ ⊂ N}. j j 0 σ j=1 Proof: M0 has SGHP and is a bounded subset of m0 .

A sequence space λ is normal (solid ) if t ∈ λ and |sj | ≤ |tj | for all j implies that s = {sj } ∈ λ. For example, c0 and lp , 0 < p ≤ ∞, are normal whereas c and m0 are not normal. From Corollary 2.17, we have Corollary 2.19. Let λ be a normal K-space with signed-SGHP and with P the property that {s ∈ λ : |s| ≤ |t|} is bounded for every t ∈ λ. If j xj P∞ is λ multiplier convergent and t ∈ λ, then the series s x converge j=1 j j uniformly for |sj | ≤ |tj |. Without some assumptions on the multiplier space, the conclusion of Theorem 2.16 may fail even when the multiplier space satisfies WGHP or 0-GHP (see Appendix B). P Example 2.20. The series j ej is lp multiplier convergent in (l p , k·kp ) P∞ for any 1 ≤ p < ∞, but the series j=1 tj ej do not converge uniformly P k j k p for k{tj }kp ≤ 1 [ take tk = ek so ∞ j=1 tj e = e ]. Note that l has both WGHP and 0-GHP (Appendix B). The SGHP assumption in Theorem 2.16 is only a sufficient condition for the uniform convergence conclusion. P Example 2.21. Let x = {xj } ∈ l2 . Then the series j xj ej is l2 multiplier convergent in l 1 . If B ⊂ l2 is bounded, M = sup{ktk2 : t ∈ B} and t ∈ B, then

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2

∞

∞ ∞ ∞ ∞ X X X X

X

2 2 2 j 2

t x e = ( |t x |) ≤ |t | |x | ≤ M |xj | → 0 j j j j j j

j=n

j=n j=n j=n j=n 1

so the conclusion of Theorem 2.16 holds but l 2 does not have SGHP. Another uniform convergence result holds for multiplier spaces satisfying 0-GHP. P Theorem 2.22. Let λ be a K-space with 0-GHP and let j xj be λ mulP ∞ i tiplier convergent. If ti → 0 in λ, then the series j=1 tj xj converge uniformly for i ∈ N. Proof: Suppose the conclusion fails to hold. Then there exists a closed neighborhood, U, of 0 such that for every k there exist pk , mk > k such P pk that ∞ / U . For k = 1, let p1 , m1 > 1 satisfy this condition so j=mk tj xj ∈ ∞ X

j=m1

tpj 1 xj ∈ / U.

Pick a symmetric neighborhood of 0,V, such that V + V ⊂ U . There exists P∞ n1 > m1 such that j=n1 +1 tpj 1 xj ∈ V . Then n1 X

tpj 1 xj =

∞ X

j=m1

j=m1

tpj 1 xj −

Pn

i j=m tj xj

∞ X

j=n1 +1

tpj 1 xj ∈ / V.

∈ V for 1 ≤ i ≤ p1 , n ≥ m ≥ N1 . Let There exists N1 such that p2 , m2 > N1 , n2 satisfy the conditions above for N1 . Note that we must have p2 > p1 . Continuing this construction produces increasing sequences {pj }, {mj }, {nj } such that (#)

nk X

j=mk

tpj k xj ∈ / V.

Set Ik = [mk , nk ]. By 0-GHP, since {tpk } → 0, there is a subsequence {qk } P∞ P∞ of {pk } such that t = k=1 χIqk tqk ∈ λ. Since j=1 tj xj converges, we P should have that j∈Iq tqj k xj → 0 contradicting the condition (#). k

Without the 0-GHP assumption the conclusion of Theorem 2.22 may fail. P∞ Example 2.23. Let λ = c00 = X. Then j=1 jej is c00 multiplier converP∞ gent in c00 . Now ti = ei /i → 0 but the series j=1 tij jej do not converge uniformly.

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We can obtain another uniform convergence result from the continuity of the summing operator. Theorem 2.24. Let λ be a complete, metrizable AK-space and let X be a P Mackey space (i.e., X has the Mackey topology). If j xj is λ multiplier P∞ convergent in X and K ⊂ λ is compact, then the series j=1 tj xj converge uniformly for t ∈ K. Proof: Since λ is metrizable, λ carries the Mackey toplology ([Sw2] 18.8) so the summing operator S : λ → X is continuous (Theorem 2.2). Since K P∞ is compact, limn j=n tj ej = 0 uniformly for t ∈ K ([Sw2] 10.15). Thus, P∞ P∞ limn j=n tj Sej = limn j=n tj xj = 0 uniformly for t ∈ K.

We next consider uniform convergence for families of multiplier convergent series. The β-dual of Λ ⊂ λ with respect to X is defined to be X xj is Λ multiplier convergent}. ΛβX = {{xj } : j

P∞ If t ∈ λ and x ∈ λβX , we write t · x = j=1 tj xj [see Appendix B]. We define the topology ω(λβX , λ) on λβX to be the weakest topology on λβX such that the mappings x = {xj } →

∞ X j=1

t j xj = t · x

from λβX into X are continuous for every t ∈ λ. Thus, if X is the scalar field, then λβX = λβ and ω(λβX , λ) = σ(λβ , λ). We now give a definition for another gliding hump property which will be used. Definition 2.25. Let Λ ⊂ λ. The space Λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ Λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } and a sequence of signs P∞ {sj } such that the coordinatewise sum of the series j=1 sj χInj t ∈ Λ. The space Λ has the weak gliding hump property (WGHP) if the signs above can all be chosen equal to 1 for every t ∈ λ. Examples of spaces with signed-WGHP and WGHP are given in Appendix B. For example, any monotone space such as c00 , c0 , lp (0 < p ≤ ∞), and m0 has WGHP while the space bs has signed-WGHP but not WGHP.

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Theorem 2.26. Let Λ ⊂ λ have signed-WGHP. If {xk } ⊂ ΛβX is such that limk t · xk exists for every t ∈ Λ and limk xkj exists for every j, then P∞ for every t ∈ Λ the series j=1 tj xkj converge uniformly for k ∈ N. Proof: If the conclusion fails, then

(∗) there exists a neighborhood of 0, U , in X such that f or every n P nn there exist kn , nn > mn > n such that j=m tj xkj n ∈ / U. n

P 1 tj xkj 1 ∈ / U. By (∗) for n = 1, there exist k1 , n1 > m1 such that nj=m 1 P n 0 k 0 There exists m > n1 such that t x ∈ U for n > m > m , 1 ≤ j j j=m Pn2 k2 0 k ≤ k1 . By (∗) there exist k2 , n2 > m2 > m such that j=m2 tj xj ∈ / U. Hence, k2 > k1 . Continuing this construction produces increasing sequences {ki }, {mi }, {ni } with mi < ni < mi+1 and (#)

x k i · χ Ii t ∈ / U, where Ii = [mi , ni ].

Define the matrix M by M = [mij ] = [xki · χIj t]. We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Second, given any increasing sequence of integers, there is a subsequence {pk } and a sequence of signs {sk } such P∞ that z = {zj } = j=1 sj χIpj t ∈ Λ. Then ∞ X j=1

sj mipj =

∞ X j=1

s j x k i · χ Ip j t = x k i · z

and lim xki · z exists. Hence, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (#). From Theorem 2.26, we can obtain an important weak sequential completeness result due to Stuart ([St1], [St2], [Sw1]). First, we establish a lemma. Lemma 2.27. Let Λ ⊂ λ. If {xk } ⊂ ΛβX is such that limk t · xk exists for every t ∈ Λ, limk xkj = xj exists for each j and for each t ∈ Λ the series P∞ k βX such that j=1 tj xj converge uniformly for k ∈ N, there exists x ∈ Λ k t · x → t · x for every t ∈ Λ.

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Proof: Set x = {xj }. We claim that x ∈ ΛβX and t · xk → t · x for every P∞ t ∈ Λ. Put u = lim t · xk . It suffices to show that u = j=1 tj xj . Let U be a balanced neighborhood of 0 in X and pick a balanced neighborhood P∞ V such that V + V + V ⊂ U . There exists p such that j=n tj xkj ∈ V for P∞ n ≥ p, k ∈ N. Fix n ≥ p. Pick k = kn such that j=1 tj xkj − u ∈ V and Pn k j=1 tj (xj − xj ) ∈ V . Then

n X j=1

t j xj − u = (

∞ X j=1

tj xkj − u) +

n X j=1

tj (xj − xkj ) −

∞ X

j=n+1

tj xkj ∈ V + V + V ⊂ U

and the result follows. Stuart’s result now follows from Lemma 2.27 and Theorem 2.26. Corollary 2.28. (Stuart) Let λ have signed-WGHP and let X be sequentially complete. Then (λβX , ω(λβX , λ)) is sequentially complete. Proof: If {xk } is ω(λβX , λ) Cauchy and X is sequentially complete, then limk t · xk exists for every t ∈ λ so Theorem 2.26 and Lemma 2.27 apply. Since any monotone space has WGHP, Corollary 2.28 applies to monotone spaces, in particular to c0 , lp (0 < p ≤ ∞) and m0 . Corollary 2.28 also applies to the space of bounded series bs which has signed-WGHP but not WGHP as originally noted by Stuart (Appendix B). A subset F of λβX is said to be conditionally ω(λβX , λ) sequentially compact if every sequence {xk } ⊂ F has a subsequence which is such that lim t · xk exists for every t ∈ λ ([Din]). From Theorem 2.26 we have Corollary 2.29. Let λ have signed-WGHP. If F ⊂ λβX is conditionally P∞ ω(λβX , λ) sequentially compact and t ∈ λ, then the series j=1 tj xj converge uniformly for x ∈ F . Without the gliding hump assumptions, the conclusions in Theorem 2.26 and Corollary 2.28 may fail. Example 2.30. Let λ = c so λβ = l1 . Then {ek } is ω(l1 , c) = σ(l1 , c) P∞ Cauchy, but if e is the constant sequence {1}, the series j=1 ekj ej do not converge uniformly and the sequence {ek } is not σ(l1 , c) convergent. We next consider another uniform convergence result with another gliding hump assumption.

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Definition 2.31. The space λ has the infinite gliding hump property (∞GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate P∞ sum j=1 apj χIpj t ∈ λ. The term ”infinite gliding hump” is used to suggest that the ”humps”, χIpj t, are multiplied by a sequence of scalars which tend to ∞; there are other gliding hump properties where the humps are multiplied by elements of classical sequence spaces (Appendix B). Examples of spaces with ∞-GHP are given in Appendix B. For example, λ = lp (0 < p < ∞) and λ = cs have ∞-GHP. The spaces l ∞ , m0 , bs and bv do not have ∞-GHP. Theorem 2.32. Assume that λ has ∞-GHP. If B ⊂ λβX is pointwise P∞ bounded on λ , then for every t ∈ λ the series j=1 tj xj converge uniformly for x ∈ B. Proof: If the conclusion fails, there exist > 0, a continuous semi-norm p on X, {xk } ⊂ B and subsequences {mk }, {nk } with m1 < n1 < m2 < ... and nk X (∗) p( tl xkl ) > . l=mk

Put Ik = [mk , nk ]. By ∞-GHP there exist {pk }, apk > 0, apk → ∞ such that any subsequence of {pk } has a further subsequence {qk } such that ∞ X

k=1

Define a matrix

aqk χIqk t ∈ λ.

M = [mij ] by mij =

X

apj tl xpl i /api .

l∈Ipj

We claim that M is a K matrix (Appendix D.2). First, the columns of M converge to 0 since B is pointwise bounded on λ and 1/api → 0. Next, given any subsequence of {pj } there is a further subsequence {qj } such that P∞ u = k=1 aqk χIqk t ∈ λ . Then ∞ X j=1

miqj = (1/api )

∞ X l=1

ul xpl i = (1/api )xpi · u → 0.

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Hence, M is a K matrix so the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). From Lemma 2.27 and Theorem 2.32 we obtain another weak sequential completeness result. Corollary 2.33. Assume that λ has ∞-GHP and X is sequentially complete. Then (λβX , ω(λβX , λ)) is sequentially complete. Remark 2.34. The signed-WGHP and ∞-GHP are independent so Corollaries 2.28 and 2.33 cover different spaces. For example, the space bs has signed-WGHP but not ∞-GHP while the space bv0 has ∞-GHP but not signed-WGHP. We next consider uniform convergence results when the elements range over both subsets of λβX and λ. These results have stronger conclusions but require stronger assumptions. First, we consider an improvement of Theorem 2.26. Theorem 2.35. Assume that Λ ⊂ λ has signed-SGHP and limk xkj = xj exists for each j. If {xk } ⊂ λβX is such that lim t · xk exists for every t ∈ Λ P∞ and B ⊂ Λ is bounded, then the series j=1 tj xkj converge uniformly for k ∈ N, t ∈ B. Proof: If the conclusion fails, (∗) there is a neighborhood U of 0 such that for every n there exist P nn tn xkj n ∈ / U. kn , tn ∈ B, nn > mn > n such that j=m n j 1 By (∗) for n = 1 there exist k1 , t ∈ B, n1 > m1 > 1 such that P n1 / U. By Theorem 2.16 there exists m0 > n1 such that t 1 xk 1 ∈ 1 j j Pj=m q k for 1 ≤ k ≤ k1 , t ∈ B, q ≥ p ≥ m0 . By (∗) there j=p tj xj ∈ U Pn2 2 k2 exist k2 , t2 ∈ B, n2 > m2 > m0 such that ∈ / U . Hence, j=m2 tj xj k2 > k1 . Continuing this construction produces increasing sequences {ki }, {mi }, {ni }, mi+1 > ni > mi , ti ∈ B such that (∗∗)

ni X

j=mi

/ U. tij xkj i ∈

Set Ii = [mi , ni ] and define a matrix M = [mij ] = [xki · χIj tj ].

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We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Next, for any increasing sequence of positive integers, there is a subsequence {pj } and a sequence of signs {sj } such that P∞ z = j=1 sj χIpj tpj ∈ Λ. Then ∞ X j=1

ki

sj mipj =

∞ X j=1

xki · sj χIpj tpj = xki · z

and lim x ·z exists. Thus, M is a signed K-matrix so by the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗∗). Since M0 ⊂ m0 has SGHP and l∞ has SGHP, we have the following corollaries. P x be subseries convergent for every i ∈ N Corollary 2.36. Let P j ij and suppose that limi j∈σ xij exists for every σ ⊂ N. Then the series P converge uniformly for i ∈ N, σ ⊂ N. j∈σ xij P Corollary 2.37. Let j xij be bounded multiplier convergent for every P ∞ i ∈ N and suppose limi j=1 tj xij exists for every {tj } ∈ l∞ . Then the P∞ series j=1 tj xij converge uniformly for i∈ N, k{tj }k∞ ≤ 1. Corollary 2.38. Assume that λ has signed-SGHP. If F ⊂βX is conditionally ω(λβX ,λ) sequentially compact and B ⊂ λ is bounded, then the series P∞ j=1 tj xj converge uniformly for x ∈ F, t ∈ B.

Example 2.30 shows that the gliding hump property in Theorem 2.35 is important. We have another uniform convergence result generalizing Theorem 2.22.

Theorem 2.39. Assume that λ has 0-GHP. If {xk } ⊂ λβX is such that P∞ lim t · xk exists for every t ∈ λ and tk → 0 in λ, then the series j=1 tlj xkj converge uniformly for k, l ∈ N. Proof: If the conclusion fails, (∗) there is a neighborhood U of 0 such that for every n there exist kn , ln , nn > mn > n such that nn X / U. tljn xkj n ∈ j=mn

P n1 / U. tl1 xkj 1 ∈ By (∗) for n = 1, there exist k1 , l1 , m1 < n1 such that j=m 1 j P q 0 i k By Theorem 2.22 there exists m > n1 such that t x ∈ U for j=p j j

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i ∈ N, 1 ≤ k ≤ k1 , q > p > m0 . By (∗) there exist k2 , l2 , n2 > m2 > m0 P n2 l 2 k2 such that / U . Hence, k2 > k1 and l2 > l1 . Continuing j=m2 tj xj ∈ this construction produces increasing sequences {ki }, {li }, {mi }, {ni } with mi+1 > ni > mi and

(∗∗)

ni X

j=mi

tlji xkj i ∈ / U.

Set Ii = [mi , ni ]. Define a matrix M = [mij ] = [xki · χIj tlj ]. We claim that M is a K-matrix (Appendix D.2). First the columns of M converge by hypothesis. Next, given any increasing sequence of positive P pj ∈ λ. Then integers there is a subsequence {pj } such that z = ∞ j=1 χIpj t P∞ ki ki · z and limi x · z exists. By the Antosik-Mikusinski j=1 mipj = x Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗∗). The 0-GHP hypothesis in Theorem 2.39 is important. Pk j Example 2.40. Let λ = c00 so λβ = s. Let xk = j=1 e ∈ s and P l tl = j=1 ej /l so {xk } is σ(s, c00 ) Cauchy and tl → 0 in (c00 , k·k∞ ). Then P∞ l k j=N tj xj = (k − l)/l if k ≥ l ≥ N so the series do not converge uniformly for k, l ∈ N although the series do converge uniformly for fixed k or l. This shows that the 0-GHP cannot be dropped in Theorem 2.39. Corollary 2.41. Assume that λ has 0-GHP. If F ⊂ λβX is conditionally P∞ ω(λβX ,λ) sequentially compact and tl → 0 in λ, then the series j=1 tlj xj converge uniformly for l ∈ N, x ∈ F . We next consider compactness in the range of the summing operator associated with a multiplier convergent series. We first establish a basic lemma. P Lemma 2.42. Let Λ ⊂ λ. If j xj is Λ multiplier convergent and the P∞ series j=1 tj xj converge uniformly for t ∈ Λ, then the summing operator P∞ S : Λ → X, St = j=1 tj xj , is continuous with respect to the topology p of coordinatewise convergence on Λ and the topology of X. Proof: Let tδ = {tδj } be a net in Λ which converges to t ∈ Λ with respect to p. Let U be a neighborhood of 0 in X and pick a symmetric

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neighborhood V such that V + V + V ⊂ U . There exists n such that P∞ t x ∈ V for every t ∈ Λ. There exists δ such that α ≥ δ implies Pj=n jα j (t j 0 for every P P j j. Equip m0 with the norm ktks = ∞ e is j=1 sj |tj |. Then the series subseries convergent in (m0 , k·ks ) but is not bounded multiplier convergent P since, for example, the series j ej /j does not converge to an element of m0 . From Theorem 2.54 and Corollary 2.46, we have P Corollary 2.56. Let X be a sequentially complete LCTVS. If j xj is P∞ subseries convergent, then { j=1 tj xj : ktk∞ ≤ 1} is compact in X.

We now define two additional notions of convergence for series in TVS. P Definition 2.57. The series j xjPis unconditionally convergent (rear∞ rangement convergent) if the series j=1 xπ(j) converges for every permuP tation π : N → N. The series j xj is unconditionally Cauchy if the series P∞ j=1 xπ(j) is Cauchy for every permutation. P Definition 2.58. The series j xj is unordered convergent if the net P { j∈σ xj : σ ∈ F} converges, where F is the family of all finite subP sets of N ordered by inclusion. We write limF j∈σ xj for the limit of this net when the net converges. We have the following relationships. Theorem 2.59. Let X be a TVS and {xj } ⊂ X. Consider the following conditions: P (i) the series j xj is unconditionally convergent, P (ii) the series j xj is unordered convergent, P (iii) the series j xj is subseries Cauchy,

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P (iv) the series j xj is unconditionally Cauchy. Then (i)⇒(ii)⇒(iii)⇒(iv). P∞ P Proof: Assume (i). Let x = j=1 xj . Assume the net { j∈σ xj : σ ∈ F} does not converge to x. Then there exists a symmetric neighborhood, U , of 0 in X such that for every σ ∈ F there exists σ 0 ∈ F, P σ 0 ⊃ σ with x − j∈σ0 xj ∈ / U . Pick a symmetric neighborhood of 0, V , Pn such that V + V ⊂ U . There exists N such that x − j=1 xj ∈ V for n ≥ N . Let d1 = {1, ..., N } and let d01 be as above. Set d2 = {1, ..., max d01 } and let d02 be as above. Continue in this way to obtain a sequence d1 , d01 , d2 , d02 , ... . Define a permutation π of N by enumerating the elements P of d1 , d01 \d2 , d2 \d01 , d02 \d2 , ... . The series ∞ j=1 xπ(j) is not convergent since X X X xj ) ∈ / V. xj = ( xj − x) + (x − j∈d0n \dn

j∈d0n

j∈dn

Hence, (i) implies (ii). P Assume (ii). Note that since limF j∈σ xj is unique, every rearrangeP P ment of j xj converges to the same limit, namely, limF j∈σ xj . Let U be a symmetric neighborhood of 0 in X. There exists σ0 ∈ F such that P P P xnj be a subseries of j xj . j∈σ xj − x ∈ U for every σ ⊃ σ0 . Let Pk j P Pick N > max σ0 . If k > j ≥ N , then i=j xi ∈ U so xnj is Cauchy and (iii) holds. Assume (iii). If (iv) fails, there exist a symmetric neighborhood of 0, U , in X and a permutation π of N and an increasing sequence {mn } such Pmn+1 xπ(i) ∈ / U . Choose a subsequence {mnj } of {mn } such that that i=m n +1 min{π(i) : mnj + 1 ≤ i ≤ mnj+1 +1 } > max{π(i) : mnj ≤ i ≤ mnj +1 }.

Arrange the integers π(i), mnj + 1 ≤ i ≤ mnj+1 , j ∈ N into an increasing P sequence {ij }. Then ∞ j=1 xij does not satisfy the Cauchy condition and (iii) fails. Hence, (iii) implies (iv). Corollary 2.60. Let X be a sequentially complete TVS. Then (i) and (ii) of Theorem 2.59 are equivalent to: P (iii)’ the series j xj is subseries convergent.

Finally, for LCTVS there is the notion of absolute convergence. P Definition 2.61. Let X be an LCTVS. The series j xj is absolutely P∞ convergent if j=1 p(xj ) < ∞ for every continuous semi-norm p on X.

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The notion of absolute convergence is a very strong condition. For example, in a Banach space every convergent series is absolutely convergent iff the space is finite dimensional (Dvoretsky-Rogers Theorem ([Sw2] 30.1)); in a Frechet space every convergent series is absolutely convergent iff the space is nuclear ([Sch] 10.7.2)). However, we do have P Proposition 2.62. If the series j xj is absolutely convergent in the LCTVS X, then the series is subseries Cauchy; if X is sequentially complete, then the series is subseries convergent. The sequential completeness statement in the last part of Theorem 2.62 is important. P Example 2.63. Let X = c00 . Set xj = ej /j 2 . Then the series j xj is absolutely convergent in X but not convergent.

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Chapter 3

Applications of Multiplier Convergent Series

In this chapter we will give several applications of multiplier convergent series to various topics in locally convex spaces and vector valued measures. As before, throughout this chapter λ will denote a sequence space containing c00 , the space of sequences which are eventually 0 and X will denote a (Hausdorff) LCTVS. We begin by establishing a generalization of a result of G. Bennett ([Be]). Proposition 3.1. If λβ for every x0 ∈ X 0 .

P

j

xj is λ multiplier Cauchy in X, then {hx0 , xj i} ∈

Proof: Let t ∈ λ, x0 ∈ X 0 . Then

converges so {hx0 , xj i} ∈ λβ .

E D P P∞ ∞ 0 x0 , j=1 tj xj = j=1 tj hx , xj i

We consider the converse of Proposition 3.1 under additional assumptions. Theorem 3.2. Let (λ, τ ) be a metrizable AK-space such that λ0 = λβ . P Then j xj is λ multiplier Cauchy in (X, τ (X, X 0 )) iff {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Proof: Suppose that {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Define a linear P 0 0 0 map T : c00 → X by T t = ∞ j=1 tj xj . If x ∈ X , t ∈ c00 , then hx , T ti = P∞ 0 0 j=1 tj hx , xj i = t · {hx , xj i} which implies by hypothesis that T is σ(c00 , λβ )−σ(X, X 0 ) continuous and ,therefore, τ (c00 , λβ )−τ (X, X 0 ) continuous. Since λ0 = (c00 , τ |c00 )0 and τ |c00 = τ (λ, λ0 ) |c00 (any metrizable space carries the Mackey topology ([Sw2] 18.8)), τ |c00 = τ (c00 , λ0 ) = τ (c00 , λβ ). P∞ Now, if s ∈ λ, s = j=1 sj ej , where the convergence is in τ = τ (λ, λβ ) 25

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Pn by the AK assumption, so { j=1 sj ej }n is τ (c00 , λβ ) Cauchy. Therefore, Pn Pn Pn { j=1 T (sj ej )}n = { j=1 sj T ej }n = { j=1 sj xj }n is τ (X, X 0 ) Cauchy. The converse is given in Proposition 3.1. Corollary 3.3. If λ satisfies the assumptions of Theorem 3.2 and P (X, τ (X, X 0 )) is sequentially complete, then j xj is λ multiplier convergent in (X, τ (X, X 0 )) iff {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Corollary 3.4. Let λ be as in Theorem 3.2 and let X be an FK-space. Then X contains λ iff c00 ⊂ X and {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Remark 3.5. Bennett’s result corresponds to the case where λ = l p , 1 ≤ p < ∞ or λ = c0 in Corollaries 3.3 and 3.4. Note that Corollaries 3.3 and 3.4 also apply to the spaces cs and bv0 . For conditions which guarantee that λ0 = λβ , see Proposition 2.5. Without some additional assumptions on the multiplier space λ, the converse of Proposition 3.1 may fail. Example 3.6. Let λ = l ∞ so λβ = l1 . Let X = c0 and consider the series P j P j e is not e in c0 . If s ∈ X 0 = l1 , then { s, ej } = {sj } ∈ l1 = λβ , but ∞ l multiplier convergent in c0 . We next consider results which involve series which are c0 multiplier Cauchy and c0 multiplier convergent. These series are often described in a different way which we now consider. P Definition 3.7. A series j xj in X is said to be weakly unconditionally P∞ Cauchy (wuc) if j=1 |hx0 , xj i| < ∞ for every x0 ∈ X 0 . P P Note that a series j xj is wuc iff the series j xj is subseries Cauchy in the weak topology σ(X, X 0 ). A series which is subseries convergent in the weak topology σ(X, X 0 ) is wuc, but a wuc series may not be subseries P convergent in the weak topology (consider the series j ej in c0 ). We give several characterizations of wuc series. Proposition 3.8. Let {xj } ⊂ X. The following are equivalent: P (i) The series j xj is wuc. (ii) {hx0 , xj i} ∈ l1 for every x0 ∈ X 0 . P (iii) The series j xj is c0 multiplier Cauchy. P (iv) { j∈σ xj : σ f inite} is bounded in X.

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(v) For every continuous semi-norm p on X, there exists M > 0 such that P p( j∈σ tj xj ) ≤ M ktk∞ for every t ∈ l ∞ and σ finite. P∞ (vi) The map T : c00 → X, T t = j=1 tj xj , is linear and continuous. P (vii) The series j xj is c0 multiplier Cauchy in σ(X, X 0 ).

Proof: Clearly (i) and (ii) are equivalent, and (ii) and (iii) are equivalent by Bennett’s result in Theorem 3.2. Assume that (i) holds. If x0 ∈ X 0 and σ is finite, then * X + X ∞ 0 x, x |hx0 , xj i| < ∞ ≤ j j=1 j∈σ P so { j∈σ xj : σ finite} is σ(X, X 0 ) bounded and, therefore, bounded in X so (iv) holds. Assume that (iv) holds. Let p be a contimuous semi-norm on X. Set P M = 2 sup{p( j∈σ xj ) : σ finite}. By the McArthur/Rutherford inequality P (Lemma 2.53), p( j∈σ tj xj ) ≤ M ktk∞ for every t ∈ l ∞ so (v) holds. That (v) implies (vi) is immediate. Suppose that (vi) holds. Then the adjoint operator T 0 : X 0 → c000 = l1 P 0 so T 0 x0 = {hx0 , xj i} ∈ l1 . Therefore, ∞ j hx , xj i converges for every j=1 s P∞ P 0 s ∈ c0 and j=1 sj xj is σ(X, X ) Cauchy or j xj is c0 multiplier Cauchy in σ(X, X 0 ). Thus, (vii) holds. P∞ Assume that (vii) holds. Then j=1 sj hx0 , xj i converges for every x0 ∈ P ∞ X 0 and for every s ∈ c0 . Hence, j=1 |hx0 , xj i| < ∞ for every x0 ∈ X 0 and (i) holds.

Note that it follows from Proposition 3.8 that a continuous linear operator between LCTVS carries wuc series into wuc series (condition (iv)). P Corollary 3.9. Let j xj be c0 multiplier convergent in X. Then P (i) j xj is wuc, (ii) for every continuous semi-norm p on X there exists M > 0 such that P p( ∞ t∈c , j=1 tj xj ) ≤ M ktk∞ for every P∞ 0 (iii) the linear map T : c0 → X, T t = j=1 tj xj , is continuous.

Proof: (i) follows from Proposition 3.8 (iii); (ii) follows from Proposition 3.8 (v); (iii) follows directly from (ii). We can now use the notions of wuc series and c0 multiplier convergent series to give a characterization of a locally complete LCTVS due to Madrigal and Arrese ([MA]). Recall that an LCTVS X is locally complete if for every

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closed, bounded, absolutely convex set B ⊂ X, the space XB = spanB equipped with the Minkowski functional pB of B in XB is complete ([K2]). Theorem 3.10. The LCTVS X is locally complete iff every wuc series in X is c0 multiplier convergent. P Proof: Suppose that X is locally complete and let j xj be a wuc series P in X. Then S = { j∈σ xj : σ finite} is bounded in X by Proposition 3.8. Let B be the closed, absolutely convex hull of S so (XB , pB ) is complete. P Since S is bounded in (XB , pB ), j xj is wuc in (XB , pB ) by Proposition 3.8. By the completeness of (XB , pB ) and condition (iii) of Proposition P 3.8, j xj is c0 multiplier convergent in (XB , pB ). Since the inclusion of P (XB , pB ) into X is continuous, j xj is c0 multiplier convergent in X. Let B be a closed, bounded, absolutely convex subset of X and suppose that {xj } is Cauchy in (XB , pB ). Pick an increasing sequence {nj } such that pB xnj+1 − xnj < 1/j2j P∞ for every j and set yj = xnj+1 − xnj . Then j=1 jyj is pB absolutely P∞ P∞ j convergent ( p (jy ) ≤ 1/2 < ∞ ) so by Proposition 3.8 j j=1 B j=1 P∞ P∞ By wuc in (XB , pB ) and, therefore, j=1 jyj is wuc in X. j=1 jyj isP P hypothesis ∞ jy is c multiplier convergent in X so the series j 0 j=1 j yj Pk is convergent to, say, y ∈ X. Thus, j=1 yj = xnk+1 − xn1 → y or xnj+1 → y + xn1 = z in X. Now, {xnj } is Cauchy in (XB , pB ), {xnj } converges in X to z and the topology pB is linked to the relative topology of XB from X so {xnj } converges to z in XB (Appendix A.4). Thus, XB is complete with respect to pB . Theorem 3.10 has an interesting corollary due to Madrigal and Arrese ([MA]). Corollary 3.11. Let X be a locally complete LCTVS. The following are equivalent: (i) every wuc series in X is subseries convergent, (ii) every wuc series in X is l ∞ multiplier convergent, (iii) every continuous linear operator T : c0 → X has a compact extension T : l∞ → X. P Proof: Suppose that (i) holds. Let j xj be wuc and let t ∈ l∞ . By P Theorem 3.10, j xj is c0 multiplier convergent. By Proposition 3.8 and

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P P Corollary 3.9, the series j tj xj is wuc. Hence, j tj xj converges by (i) and (ii) holds. Suppose that (ii) holds. Let T : c0 → X be linear and continuous. P P j P Since e is wuc in c0 , j T ej is wuc in X. By (ii), j T ej is l∞ multiP∞ plier convergent. By Corollary 2.46, { j=1 tj T ej : ktk∞ ≤ 1} is compact. P j Therefore, by Theorem 2.2, T t = j tj T e , defines a compact operator ∞ from l into X which extends T . Hence, (iii) holds. P Suppose that (iii) holds. Let j xj be wuc in X. By Theorem 3.10, P∞ P j xj is c0 multiplier convergent so T t = j=1 tj xj defines a continuous linear operator from c0 into X by Corollary 3.9. By (iii) T is compact so P P S = { j∈σ xj : σ finite} is relatively compact. By Theorem 2.48, j xj is subseries convergent. Bessaga and Pelczynski have shown that a Banach space X contains no subspace isomorphic to c0 iff every wuc series in X is subseries convergent ([BP]). We now extend this characterization to LCTVS. For this we require several preliminary lemmas. Lemma 3.12. Let xij ∈ R, εij > 0 for every i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i, then there exists an increasing sequence {mj } such that xmi mj ≤ εij for i 6= j.

Proof: Set m1 = 1. There exists m2 > m1 such that |xm1 j | < ε12 and |xim1 | < ε21 for all i, j ≥ m2 . There exists m3 > m2 such that |xm1 j | < ε13 , |xm2 j | < ε23 , |xim1 | < ε31 , |xim2 | < ε32 for all i, j ≥ m3 . Now just continue.

Lemma 3.13. Let X be a semi-normed space and xij ∈ X for i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i, then given > 0 there exists a subsequence {mj } such that ∞ X X

xmi mj < . i=1 j6=i

P∞ P∞ Proof: Pick ij > 0 such that i=1 j=1 ij < . Let {mj } be the subsequence from Lemma 3.12 applied to the double sequence kxij k. Then

xmi mj ≤ ij for i 6= j so the result follows.

Lemma 3.14. Let X be a semi-normed space that contains a c0 multiplier P convergent series j xj with kxj k ≥ δ > 0 for every j. Then there exists a subsequence {mj } such that for any subsequence {nj } of {mj }, T {tj } = P∞ T t = j=1 tj xnj defines a topological isomorphism of c0 into X.

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Proof: By replacing X by the linear subspace spanned by {xj }, we may

0 0 0 assume that X is separable. For each j pick x ∈ X , x j j ≤ 1, such that

0 xj , xj = kxj k. By the Banach-Alaoglu Theorem, {x0j } has a subsequence which is weak* convergent to an element x0 ∈ X 0 ; to avoid cumbersome notation later, assume that {x0j } is weak* convergent to x0 . Then 0 x − x0 , xj ≥ δ − |hx0 , xj i| > δ/2 j 0 for large hx 0 j since , xj i → 0; again to avoid cumbersome notation assume 0 that xj − x , xj ≥ δ/2 for all j. The matrix

M = [hx0i − x0 , xj i]

satisfies the assumption of Lemma 3.13 so let {mj } be the subsequence from Lemma 3.13 with = δ/4. Now define a continuous linear operator T : c0 → X by T t = P∞ 0 0 0 j=1 tj xmj (Corollary 3.9). If zi = xmi − x , then by the conclusion of Lemma 3.13, we have 2 kT {tj }k ≥ |hzi0 , T {tj }i| ≥ |ti hzi0 , xmi i| − ≥ |ti | δ/2 − k{tj }k∞ δ/4.

X

tj zi0 , xmj j6=i

Taking the supremum over all i in the inequality above gives kT {tj }k ≥ (δ/8) k{tj }k∞ so T has a bounded inverse. The same computation applies to any subsequence {nj } of {mj } so the result follows. We now give a characterization of sequentially complete LCTVS which have the property that any wuc series is subseries convergent. In the statement below, if X is a semi-normed space, B(X) denotes the closed unit ball of X. Theorem 3.15. Let X be a sequentially complete LCTVS. The following are equivalent: (i) X contains no subspace (topologically) isomorphic to c0 . P (ii) If j xj is c0 multiplier convergent in X, then xj → 0. P P (iii) If j xj is c0 multiplier convergent in X, then j xj is subseries convergent in X.

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P P (iv) If j xj is c0 multiplier convergent in X, then j xj is bounded multiplier convergent in X. P P∞ (v) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges ∞ uniformly for {tj } ∈ B(l ). P P∞ (vi) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges uniformly for {tj } ∈ B(c0 ). P P∞ (vii) If j xj is c0 multiplier convergent in X, then j=1 tj xj converges 1 uniformly for {tj } ∈ B(l ). (viii) Every continuous linear operator T : c0 → X is compact and has a compact extension to l ∞ . Proof: (i) implies (ii): Suppose there exists a c0 multiplier convergent P series j xj with xj 9 0. Then we may assume there exists a continuous semi-norm p on X and δ > 0 such that p(xj ) ≥ δ for all j. By Lemma P∞ 3.14 there is a subsequence {mi } such that H{tj } = j=1 tj xmj defines a topological isomorphism from c0 onto (Hc0 , p). Let I be the continuous inP∞ clusion operator from X onto (X, p). By Corollary 3.9, T {tj } = j=1 tj xmj defines a continuous linear operator from c0 into X, and T −1 = H −1 I is continuous so T defines a linear homeomorphism from c0 into X. (ii) implies (iii): Suppose there exists a c0 multiplier convergent series P P X such that j xj diverges. Since X is sequentially complete, j xj in P n {sn } = { j=1 xj } is not Cauchy. Hence, there exist a neighborhood of 0, V , in X and an increasing sequence {nj } such that yj = snj+1 − snj ∈ /V P∞ P x is c multiplier convergent, the series for all j. Since 0 j j j=1 tj yj converges for every {tj } ∈ c0 . By (ii), yj → 0. This contradiction shows that (ii) implies (iii). That (iii) implies (iv) is given in Theorem 2.54. That (iv) implies (v) is given in Theorem 2.54. That (v) implies (vi) and (vi) implies (vii) is clear. P (vii) implies (ii): Suppose there is a c0 multiplier convergent series j xj P∞ 1 in X such that the series j=1 tj xj converges uniformly for {tj } ∈ B(l ) but xj 9 0. There exists a neighborhood of 0,V , and a subsequence {xnj } such that xnj ∈ / V for every j. Let tk = {tkj } = enk ∈ B(l1 ). Then P∞ k P / V so the series ∞ j=1 tj xj fail to converge uniformly for j=1 tj xj = xnk ∈ {tj } ∈ B(l1 ). (viii) implies (i) since no continuous, linear, 1-1 map from c0 into X can have a continuous inverse by the compactness of the map. Finally, (iv) implies (viii): Let T : c0 → X be linear and continuous and P set T ej = xj . Then j xj is c0 multiplier convergent and, hence, bounded

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multiplier convergent by (iv). By Corollary 2.46, {T {tj } : k{tj }k∞ ≤ 1} = {

∞ X j=1

tj xj : k{tj }k∞ ≤ 1}

is compact so (viii) holds. Remark 3.16. The equivalence of (i) and (iii) for the case when X is a Banach space is a well known result of Bessaga and Pelczynski ([BP]). Bessaga and Pelczynski derive their result from results on basic sequences in B-spaces; Diestel and Uhl give a proof based on Rosenthal’s Lemma ([DU] I.4.5). The equivalence of (i) and (viii) was noted by Li. The conditions (v), (vi) and (vii) are contained in [LB]. Without the sequential completeness assumption, the conclusions in Theorem 3.15 may fail. P j Example 3.17. The series e is wuc in c00 with the sup-norm but is not subseries convergent. However, c00 being of countable algebraic dimension does not contain a subspace isomorphic to c0 . We next derive a result of Pelczynski on unconditionally converging operators. A continuous linear operator T from a Banach space X into a Banach space Y is said to be unconditionally converging if T carries wuc series into subseries convergent series ([Pl]). A weakly compact operator is unconditionally converging [we give a proof of this fact in Chapter 4 after we establish the Orlicz-Pettis Theorem; recall an operator is weakly compact if it carries bounded sets into relatively weakly compact sets]. The identity on l1 gives a example of an unconditionally converging operator which is not weakly compact [recall that a sequence in l 1 is weakly convergent iff the sequence is norm convergent; this result will be established in Chapter 7 when Hahn-Schur Theorems are derived; see also, [Sw2] 16.14]. Theorem 3.18. Let X, Y be Banach spaces and T : X → Y a continuous linear operator which is not unconditionally converging. Then there exist topological isomorphisms I1 : c0 → X and I2 : c0 → Y such that T I1 = I2 [i.e., T has a bounded inverse on a subspace isomorphic to c0 ]. P Proof: By hypothesis there exists a wuc series j xj in X such that P P j T xj is not subseries convergent. Since j T xj contains a subseries P which is not convergent, we may as well assume that the series j T xj diverges. Thus, there exist δ > 0 and a subsequence {nj } such that kzj k ≥

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Pnj+1 δ, where zj = T uj and uj = i=n xi . By Proposition 3.8, the series j +1 P P T u are both wuc. Since kxk ≥ kT xk / kxk for x ∈ X, kuj k ≥ u and j j j j P P δ/ kT k. Applying Lemma 3.14 to the series j uj and j T uj , there is a P∞ P subsequence {mj } such that I1 {tj } = j=1 tj umj and I2 = ∞ j=1 tj T umj define isomorphisms from c0 into X and Y , respectively. Obviously, T I1 = I2 . Remark 3.19. The converse of Theorem 3.18 holds and gives an interesting characterization of unconditionally converging operators (see [Ho]). We next consider wuc series in the strong dual of an LCTVS. Theorem 3.20. Let X be a barrelled LCTVS. The following are equivalent: (i) (X 0 , β(X 0 , X)) contains no subspace isomorphic to c0 , P (ii) every wuc series j x0j in X 0 is β(X 0 , X) subseries 0 convergent, P P < ∞ for every (iii) every series j x0j in X 0 which satisfies ∞ j=1 xj , x 0 x ∈ X is β(X , X) subseries convergent, (iv) every continuous linear operator T : X → l 1 is compact [an operator T is compact if T carries bounded sets into relatively compact sets]. Proof: Conditions (i) and (ii) are equivalent by Theorem 3.15 since β(X 0 , X) is sequentially complete by the barrelledness of X ([Wi] 6.1.16 and 9.3.8). 0 P P x ,x < ∞ Assume that (ii) holds. Let j x0j be such that ∞ j j=1 P for every x ∈ X. Then { j∈σ x0j : σ finite} is weak* bounded and, thereP fore, β(X 0 , X) bounded since X is barrelled. Therefore, j x0j is wuc in P (X 0 , β(X 0 , X)) by Proposition 3.8. Hence, j x0j is β(X 0 , X) subseries convergent by (ii) and (iii) holds. Assume that (iii) holds. Let T : X → l 1 be linear and continuous. Set x0j = T 0 ej . Now T 0 is β(l∞ , l1 ) − β(X 0 , X) continuous so {x0j } is β(X 0 , X) bounded. For x ∈ X, T x ∈ l1 we have ∞ ∞ ∞ X j 0 j X 0 X e , T x < ∞. T e ,x = x ,x = j j=1 0 j xj

j=1

j=1

is β(X , X) subseries convergent and, therefore, l ∞ mulBy (iii), tiplier convergent since β(X 0 , X) is sequentially complete as noted above (Theorem 2.54). Therefore, if B ⊂ X is bounded, then ∞ ∞ X X 0 j e , T x = 0. lim sup xj , x = lim sup P

0

n x∈B j=n

n x∈B j=n

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Hence, T B is relatively compact in l 1 ([Sw2] 10.15) and (iv) holds. P 0 0 0 Assume that (iv) holds. Let j xj be wuc in (X , β(X , X)). Define T :

X → l1 by T x = { x0j , x }. T is obviously linear and is σ(X, X 0 ) − σ(l1 , l∞ ) continuous since if t ∈ l ∞ , x ∈ X, *∞ + ∞ X X

0 0 t · Tx = t j xj , x = t j xj , x j=1

j=1

[ the series j tj x0j is σ(X 0 , X) Cauchy and ,therefore, σ(X 0 , X) convergent since X is barrelled ([Wi] 9.3.8)]. Thus, T is β(X, X 0 )−β(l1 , l∞ ) continuous. By (iv), T is compact. If B ⊂ X is bounded, T B is relatively compact in l1 so ∞ ∞ X X j 0 xj , x = 0 lim sup e , T x = lim sup P

n x∈B j=n

n x∈B j=n

P ([Sw2] 10.15) and j x0j is β(X 0 , X) convergent. The same argument can P be applied to every subseries of j x0j so (ii) holds.

Remark 3.21. If X is barrelled, then X 0 is weak* sequentially complete P so condition (iii) is equivalent to the statement that every series j x0j in X 0 which is σ(X 0 , X) subseries convergent is β(X 0 , X) subseries convergent. This is the statement of an Orlicz-Pettis type Theorem which we will consider in Chapter 4. Without the barrelledness assumption, the conclusion of Theorem 3.20 may fail. P j Example 3.22. Let X = c00 with the sup-norm. The series j e in 1 0 l = X satisfies the condition (iii) in Theorem 3.20 but is not strongly subseries convergent in l 1 and l1 contains no subspace isomorphic to c0 . We next give a characterization of Banach-Mackey spaces in terms of multiplier convergent series. Recall that an LCTVS X is a Banach-Mackey space if every σ(X, X 0 ) bounded subset of X is β(X, X 0 ) bounded; i.e., if B ⊂ X is pointwise bounded on X 0 , then B is uniformly bounded on σ(X 0 , X) bounded subsets of X 0 ([Wi] 10.4.3). The Banach-Mackey Theorem states that any sequentially complete LCTVS is a Banach-Mackey space ([Wi] 10.4.8). Let X be an LCTVS. Let X b (X s ) be the space of all bounded (sequentially continuous) linear functionals on X. Since X 0 ⊂ X s ⊂ X b ,

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(X, X s ) and (X, X b ) both form dual pairs. We now give a characterization of Banach-Mackey spaces in terms of l 1 multiplier convergent series and the spaces X s and X b . Theorem 3.23. Let X be an LCTVS. The following are equivalent: (i) (ii) (iii) (iv)

X is a Banach-Mackey space. P∞ If {x0j } is σ(X 0 , X) bounded and {tj } ∈ l1 , then j=1 tj x0j ∈ X s . P∞ 0 b If {x0j } is σ(X 0 , X) bounded and {tj } ∈ l1 , then j=1

t j xj ∈ X . 0 0 0 0 If {xj } is σ(X , X) Cauchy and hx , xi = lim xj , x for x ∈ X, then x0 ∈ X b .

Proof: Suppose that (i) holds. Let xj → 0 in X. Then {xj } is bounded in X and, therefore, β(X, X 0 ) bounded by (i). Hence, M = sup{|hx0i , xj i| : i, j ∈ N} < ∞

and

X ∞ X ∞ 0 ≤M t x , x |tj | j i j j=n j=n

0 P for {tj } ∈ l1 . Therefore, the series ∞ j=1 tj xj , xi converge uniformly for i ∈ N. Hence, ∞ ∞ X

X

lim tj lim x0j , xi = 0 tj x0j , xi = i

j=1

j=1

i

P∞

0 j=1 tj xj

∈ X s and (ii) holds. That (ii) implies (iii) is immediate. Assume that (iii) holds. We show that (i) holds. Let A ⊂ X be σ(X, X 0 ) bounded and B ⊂ X 0 be σ(X 0 , X) bounded. We show that sup{|hx0 , xi| : x0 ∈ B, x ∈ A} < ∞. If this fails to hold, there exist {x0j } ⊂ B and {xj } ⊂ A such that so

(#)

Consider the matrix

|hx0i , xi i| > i2 for every i.

M = [mij ] = [(1/j) x0j , (1/i)xi ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since {xi } is σ(X, X 0 ) bounded. Given any subsequence {mj } P∞ pick a further subsequence {nj } such that j=1 1/nj < ∞. By (iii) + *∞ ∞ E D X X 0 (1/nj ) x0nj , (1/i)xi → 0. (1/nj )xnj , (1/i)xi = j=1

j=1

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Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (#). We next show that (i) implies (iv). Let A ⊂ X be bounded.

Then A is σ(X, X 0 ) bounded by (i). Since {x0j } is β(X 0 , X) bounded, { x0j , x : x ∈ A, j ∈ N} is bounded. Therefore, {hx0 , xi : x ∈ A} is bounded. Therefore, x0 ∈ X b and (iv) holds. Suppose that (iv) holds. We show that (iii) holds and this will com- P plete the proof. X and {tj } ∈ l1 , then limn nj=1 tj x0j , x

0 If x ∈ P P∞ ∞ = j=1 tj xj , x . By (iv), j=1 tj x0j ∈ X b and (iii) holds.

Theorem 3.23 is contained in [LS], Theorem 7, where other characterizations of Banach-Mackey spaces are given. We make an interesting observation concerning Banach spaces with an unconditional Schauder basis. Let X be a Banach space. A sequence {bj } ⊂ X is a Schauder basis for X if every x ∈ X has a unique series P∞ representation x = j=1 tj bj ; the linear functionals fj : X → R defined by hfj , xi = tj are called the coordinate functionals associated with the basis {bj }. It is known that the coordinate functionals are equicontinuous P∞ ([Sw2] 10.10). If the series x = j=1 hfj , xi bj is unconditionally convergent (subseries convergent) for every x, the basis {bj } is said to be unconditional. Theorem 3.24. Let {bj } be an unconditional basis for the Banach space P∞ X. If x ∈ X, the series j=1 tj bj converge uniformly for |tj | ≤ |hfj , xi|. P∞ Proof: Since X is complete, the series j=1 hfj , xi bj is also bounded multiplier convergent (Theorem 2.54) so the result follows from Theorem 2.54. In the last part of this chapter we present several applications of convergent series to topics in vector valued measures. Let A (Σ) be an algebra (σ-algebra) of subsets of a set S and let X be a TVS. A set function µ : A →X is finitely additive (countably additive) if µ(∅) = 0 and P µ(A∪B) = µ(A)+µ(B) when A, B ∈ A with A∩B = ∅ (µ(A) = ∞ j=1 µ(Aj ) A ∈ A). Note that if when {Aj } ⊂ A, is pairwise disjoint and A = ∪∞ j P∞ j=1 µ is countably additive, then the series µ(A ) is unconditionally j j=1 convergent since the union ∪∞ j=1 Aj is independent of the ordering of the {Aj }. Finitely additive set functions, even scalar valued functions, defined on algebras or σ-algebras are not necessarily bounded as the following examples

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show. First, we give a simple example of an unbounded, finitely additive set function defined on an algebra. Example 3.25. Let A be the algebra of finite/co-finite subsets of N; i.e., A ∈ A iff either A or the complement of A, Ac , is finite. Define µ : A → R by µ(A) equals the number of elements in A when A is finite and µ(A) equals minus the number of elements in Ac when Ac is finite. Then µ is finitely additive but not bounded. To present an example of a finitely additive set function defined on a σ-algebra which is unbounded is more complicated. We present an example due to Giesy ([Gi]). Lemma 3.26. Let A, B be algebras of subsets of a set S with A ⊂ B and let α : A → R be finitely additive. If B ∈ BA and b ∈ R, there exists β : B → R finitely additive such that β is an extension of α with β(B) = b. Proof: Let S(A) (S(B)) be the vector space of all A (B) simple functions. Then α Rinduces a linear functional α0 : S(A) → R via integration, i.e., α0 (f ) = f dα. The linear functional α0 has a linear extension, β 0 , to S(B) such that β 0 (χB ) = b. Then β(E) = β 0 (χE ) defines the desired finitely additive extension of α. We now give an example of a real valued, finitely additive set function defined on the σ-algebra of Lebesgue measurable subsets of R which is not bounded. Example 3.27. Let {Ej }∞ j=0 be a pairwise disjoint sequence of bounded intervals whose union is R. Let Ak be the algebra generated by {R, E0 , E1 , ..., Ek } so A0 ⊂ A1 ⊂ ... ⊂ M, where M is the σ-algebra of Lebesgue measurable subsets of R. Set α0 = 0 on A0 ; let α1 be a finitely additive extension of α0 to A1 such that α1 (E1 ) = 1 (Lemma 3.26). Inductively, there is a sequence {αk } of finitely additive set functions such that αk : Ak → R, αk+1 extends αk and αk (Ek ) = k. Now A = ∪∞ k=0 Ak is an α is finitely additive on A. By Lemma 3.26 there is algebra and α = ∪∞ k k=0 a finitely additive extension of α, µ, to M and we have that µ(Ek ) = k for every k so µ is not bounded. We now give several conditions which characterize bounded, finitely additive set functions with values in LCTVS.

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Theorem 3.28. Let X be an LCTVS and let µ : A → R be finitely additive. The following are equivalent: (i) µ is bounded, (ii) for every pairwise disjoint sequence {Aj } from A, {µ(Aj )} is bounded, P (iii) for every pairwise disjoint sequence {Aj } from A, the series j µ(Aj ) is c0 multiplier Cauchy. Proof: Clearly, (i) implies (ii). Suppose that (ii) holds and µ is not bounded. If E ∈ A, set AE = {A ∩ E: A ∈ A}. Suppose that µ(AE ) is not absorbed by the absolutely convex neighborhood of 0, U , in X. Pick an absolutely convex neighborhood of 0, V , such that V + V ⊂ U . We claim that for every k there exist nk > k and a partition (Ak , Bk ) of E with Ak , Bk ∈ A and µ(Ak ) ∈ / nk V, µ(Bk ) ∈ / nk V . For, there exists nk > k such that µ(E) ∈ nk V . But, µ(AE ) * nk (V + V ) since V + V ⊂ U . Therefore, there exists Ak ∈ AE such that µ(Ak ) ∈ / nk (V + V ). Note that µ(Ak ) ∈ / nk V . Put Bk = EAk . Then µ(Bk ) ∈ / nk V since otherwise µ(Ak ) = µ(E) − µ(Bk ) ∈ nk (V + V ). Since µ(A) is assumed to be unbounded, there exists an absolutely convex neighborhood of 0, U , in X such that µ(A) is not absorbed by U . Pick V as above. By the observation above there exist n1 > 1 and a partition (A1 , B1 ) of S such that µ(A1 ) ∈ / n1 V and µ(B1 ) ∈ / n1 V . Either µ(AA1 ) or µ(AB1 ) is not absorbed by U since otherwise there exists m such that µ(AA1 ) ⊂ mU and µ(AB1 ) ⊂ mU and µ(AS ) = µ(A) ⊂m(U +U ) ⊂m(2U ) since U is convex. Pick whichever of A1 or B1 satisfies this condition, label it F1 and set E1 = SF1 . Now treat F1 as above to obtain a partition (E2 , F2 ) of F1 and n2 > n1 such that µ(E2 ) ∈ / n2 V, µ(F2 ) ∈ / n2 V and µ(AF2 ) is not absorbed by U . Continuing this construction produces a pairwise disjoint sequence {Ek } such that {µ(Ek )} is not absorbed by U . Thus, (ii) fails to hold so (ii) implies (i). Suppose that (iii) holds, {Aj } ⊂ A is pairwise disjoint and t ∈ c0 . Then P j tj µ(Aj ) is Cauchy so lim tj µ(Aj ) = 0. Since t ∈ c0 is arbitrary, {µ(Aj )} is bounded. Thus, (ii) holds. Suppose that (i) holds and let {Aj } ⊂ A be pairwise disjoint. Then X µ(Aj ) : σ finite} = {µ(∪j∈σ Aj ) : σ finite j∈σ

is bounded. Therefore, 3.8.

P

j

µ(Aj ) is c0 multiplier Cauchy by Proposition

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If the LCTVS X is sequentially complete, the condition (iii) in Theorem 3.28 can be strengthened. To establish this we employ a very intersting and simple result of Li Ronglu ([LS] Corollary 10) which will be used several times later. Lemma 3.29. Let {Ej } be a sequence of sets. Let G be an Abelian (HausP∞ dorff ) topological group and fj : Ej → G. If the series fj (tj ) conj=1P ∞ verges for every sequence {tj } with tj ∈ Ej , then the series j=1 fj (tj ) converge uniformly for all sequences {tj } with tj ∈ Ej . Proof: If the conclusion fails to hold, there exist a neighborhood, U , of 0 in G and sequences {tij }j , tij ∈ Ej , and an increasing sequence {ni } P∞ such that j=ni fj (tij ) ∈ / U . Pick a symmetric neighborhood of 0,V , such P∞ P∞ 1 that V + V ⊂ U . Since limk j=k fj (t1j ) = 0 and / U, j=n1 fj( tj ) ∈ Pm1 1 there exists m1 > n1 such that j=n1 f (tj ) ∈ / V . Put N1 = 1 and pick P∞ i2 ni2 = N2 > m1 such that / U . As before pick m2 > j=N2 fj (tj ) ∈ Pm2 i2 / V . Continuing this construction produces N2 such that j=N2 fj (tj ) ∈ increasing sequences {Nk }, {mk } and {ik } such that Nk < mk < Nk+1 P mk and j=N fj (tijk ) ∈ / V . Pick an arbitrary sequence {uj } with uj ∈ Ej for k every j. Define a sequence {sj } with sj ∈ Ej by sj = tijk if Nk ≤ j ≤ mk P∞ and sj = uj otherwise. If the series j=1 fj (sj ) converges, there exists Pmk Pn N such that j=m fj (sj ) ∈ V for n > m ≥ N . But, j=Nk fj (sj ) = P∞ P mk ik / V for large k so the series j=1 fj (sj ) does not satisfy j=Nk fj (tj ) ∈ the Cauchy condition and, therefore, does not converge. This contradicts the hypothesis. To illustate the utility of Lemma 3.29, we derive a couple of previous results for series which were established by other means. First, we consider a version of Corollary 2.18. P Corollary 3.30. Let X be a TVS and j xj a series in X which is subP series convergent. Then the series j∈σ xj converge uniformly for σ ⊂ N. Proof: Let Ej = {0, 1} for every j and define fj : Ej → X by fj (0) = 0 and fj (1) = xj . Then the conclusion follows directly from Lemma 3.29. Next, we consider an improvement of Corollary 2.19. P Corollary 3.31. Let X be a TVS and let λ be a normal space and let j xj P∞ be λ multipier convergent. If t ∈ λ, then the series j=1 sj xj converge uniformly for |sj | ≤ |tj |.

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Proof: Let Ej = {t ∈ R : |t| ≤ |tj |} and define fj : Ej → X by fj (t) = txj . Then the conclusion follows directly from Lemma 3.29. Note that the space λ in Corollary 3.31 is not assume to be a K-space with signed-SGHP as in Corollary 2.19. From Corollary 3.31 we can obtain immediately Corollary 2.17. P Corollary 3.32. Let X be a TVS and let j xj be l∞ multiplier converP∞ gent. Then the series j=1 tj xj converge uniformly for k{tj }k∞ ≤ 1. Proof: Let t be the constant sequence with 1 in each coordinate. Then the result follows immediately from Corollary 3.31.

Notice that Corollaries 3.31 and 3.32 were proven in reverse order in Chapter 2. We now show that Lemma 3.29 can be used to obtain an improvement in Theorem 3.28 when the space X is a sequentially complete LCTVS. Corollary 3.33. Let X be a sequentially complete LCTVS and µ : A →X finitely additive. The following are equivalent: (i) µ is bounded, (ii) for every pairwise disjoint sequence {Aj } from A, {µ(Aj )} is bounded, P (iii)’ for every pairwise disjoint sequence {Aj } from A, the series j µ(Aj ) is c0 multiplier convergent, (iv) for every pairwise disjoint sequence {Aj } from A and t ∈ c0 , the series P∞ j=1 sj µ(Bj ) converge uniformly for Bj ⊂ Aj , Bj ∈ A and |sj | ≤ |tj |. Proof: (i), (ii) and (iii)’ are equivalent by Theorem 3.28. Obviously, (iv) implies (iii)’. Assume (iii)’. We apply Lemma 3.29. Set Ej = {(B, s) : B ∈ A, B ⊂ Aj , |s| ≤ |tj |} and define fj : Ej → X result.

by fj (B, s) = sµ(B). Lemma 3.29 now gives the

We also have the following boundedness result for countably additive set functions defined on σ-algebras. Corollary 3.34. Let X be an LCTVS and Σ a σ-algebra. If µ : Σ → X is countably additive, then µ is bounded.

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Proof: µ satisfies condition (ii) of Theorem 3.28. Remark 3.35. The locally convex assumption in Corollary 3.34 is important. Turpin has given an example of a countably additive set function defined on a σ-algebra with values in a (non-locally convex) TVS which is unbounded ([Rol] 3.6.4). We next consider an important property for vector valued set functions which was introduced by Rickart which lies between finite additivity and countable additivity. Definition 3.36. Let X be a TVS and µ : A →X be finitely additive. Then µ is strongly bounded (strongly additive, exhaustive) if µ(Aj ) → 0 for every pairwise disjoint sequence {Aj } from A. A countably additive set function defined on a σ-algebra is obviously strongly bounded. We show below that bounded, finitely additive scalar valued set functions are strongly bounded and give an example of a bounded, finitely additive set function defined on a σ-algebra with values in a Banach space which is not strongly bounded. Lemma there exists M ≥ 0 such that P 3.37. Let {tj } ⊂ R and assume thatP ∞ j∈σ tj ≤ M for every finite σ ⊂ N. Then j=1 |tj | ≤ 2M .

Proof: Let σ be finite. Set σ+ = {j ∈ σ : tj ≥ 0} and σ− = {j ∈ σ : tj < 0}. Then X X tj ≤ M |tj | = j∈σ+

j∈σ+

and X

j∈σ−

so

P

j∈σ

|tj | = −

X

j∈σ−

|tj | ≤ 2M . Since σ is arbitrary,

Corollary 3.38. Let µ : A → R iff µ is strongly bounded.

tj ≤ M

P∞

j=1

|tj | < 2M .

be finitely additive. Then µ is bounded

Proof: If µ is strongly bounded, then µ is bounded since condition (ii) of Theorem 3.28 is satisfied.

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Suppose that µ is bounded with sup{|µ(A)| : A ∈ A } = M < ∞. Let {Aj } ⊂ A be pairwise disjoint. If σ ⊂ N is finite, then X µ(Aj ) = |µ(∪j∈σ Aj )| ≤ M j∈σ P so by Lemma 3.37, ∞ j=1 |µ(Aj )| ≤ 2M . In particular, µ(Aj ) → 0 so µ is strongly bounded. Remark 3.39. The proof of Corollary 3.38 shows that if µ : A → R is bounded and finitely additive and {Aj } is pairwise disjoint, then the P absolutely convergent. Thus, if µ fails to be countably series ∞ j=1 µ(Aj ) is P additive, the series ∞ j=1 µ(Aj ) converges but may fail to converge to the ”proper value”, namely, µ(∪∞ j=1 Aj ). For vector valued set functions we have the following boundedness result. Corollary 3.40. Let X be an LCTVS. If µ : A →X is strongly bounded, then µ is bounded. Proof: For each x0 ∈ X 0 , x0 ◦ µ = x0 µ : A → R is strongly bounded so {hx , µ(A)i : A ∈ A} is bounded by Corollary 3.38. Thus, {µ(A) : A ∈ A} is weakly bounded in X and, therefore, bounded in X. 0

The example indicated in Remark 3.35 shows that the local convex assumption in Corollary 3.40 is important. The converse of Corollary 3.38 is false, in general. Example 3.41. Let M be the σ-algebra of Lebesgue measurable subsets of [0, 1]. Define µ : M → L∞ [0, 1] by µ(E) = χE . Then µ is bounded, finitely additive but not strongly bounded [take any pairwise disjoint sequence from M with positive Lebesgue measure]. We have a series characterization of strongly additive set functions. Theorem 3.42. Let X be a TVS and µ : A →X be finitely additive. The following are equivalent: (i) µ is strongly bounded, P (ii) for every pairwise disjoint sequence {Aj } ⊂ A , the series j µ(Aj ) is Cauchy.

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Proof: That (ii) implies (i) is clear. Assume that (ii) fails to hold. Then there exist a neighborhood U of 0 in P X and an increasing sequence of intervals {Ij } such that j∈Ik µ(Aj ) ∈ /U for all k. If Bk = ∪j∈Ik Aj , then {Bk } is pairwise disjoint and µ(Bk ) 9 0 so (i) fails. If X is sequentially complete, using Li’s Lemma 3.29 we can strengthen condition (ii). Corollary 3.43. Let X be a sequentially complete TVS and µ : A →X be finitely additive. The following are equivalent: (i) µ is strongly bounded, P (ii)’ for every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) converges, P (iii) for every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Bj ) converge uniformly for Bj ⊂ Aj , Bj ∈ A. Proof: That (i) and (ii)’ are equivalent follows from Theorem 3.42. Obviously (iii) implies (ii)’. Assume that (ii)’ holds. We establish (iii) by using Lemma 3.29. Set Ej = {B ∈ A : B ⊂ Aj } and define fj : Ej → X by fj (B) = µ(B). Lemma 3.29 now gives the result. We consider the semi-variation of set functions µ : A →X with values in a normed space X. The semi-variation is useful in discussing topics in vector measures and vector integration ([DS], [DU] I.1). Definition 3.44. For A ∈ A the semi-variation of µ on A is defined to be

n

X

: {Aj }nj=1 a partition of A and |tj | ≤ 1 . kµk (A) = sup t µ(A ) j j

j=1

We have the following properties of the semi-variation. In the proposition below, the variation of a real valued set function ν is denoted by |ν| ([Sw3] 2.2.1.7). Proposition 3.45. Let µ : A →X.

(i) kµk (A) = sup{|x0 µ| (A) : kx0 k ≤ 1}, (ii) sup{kµ(B)k : B ⊂ A, B ∈ A} ≤ kµk (A) ≤ 2 sup{kµ(B)k : B ⊂ A, B ∈ A}.

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Proof: (i): Let {A1 , ..., An } be a partition of A and let |tj | ≤ 1 for j = 1, ..., n. Then D P E

Pn

n

j=1 tj µ(Aj ) = sup{ x0 , j=1 tj µ(Aj ) : kx0 k ≤ 1} Pn ≤ sup{ j=1 |hx0 , tj µ(Aj )i| : kx0 k ≤ 1} Pn ≤ sup{ j=1 |hx0 , µ(Aj )i| : kx0 k ≤ 1} Pn ≤ sup{ j=1 |x0 µ| (Aj ) : kx0 k ≤ 1} = sup{|x0 µ| (A) : kx0 k ≤ 1}. Therefore, kµk (A) ≤ sup{|x0 µ| (A) : kx0 k ≤ 1}. For the reverse inequality, let x0 ∈ X 0 ,kx0 k ≤ 1 and {A1 , ..., An } be a partition of A. Then Pn P |hx0 , µ(Aj )i| = nj=1 (signx0 µ(Aj ))x0 µ(Aj ) j=1 D P E = x0 , nj=1 (signx0 µ(Aj ))µ(Aj )

P

n ≤ j=1 (signx0 µ(Aj ))µ(Aj ) ≤ kµk (A). Therefore, |x0 µ| (A) ≤ kµk (A) and kµk (A) ≥ sup{|x0 µ(A)| : kx0 k ≤ 1}. Thus, (i) holds. For (ii), recall that for scalar set functions ν we have that sup{|ν(B)| : B ⊂ A, B ∈ A} ≤ |ν| (A) ≤ 2 sup{|ν(B)| : B ⊂ A, B ∈ A} ([Sw3] 2.2.7). Let kx0 k ≤ 1. Then sup{kµ(B)k : B ⊂ A, B ∈ A} = sup{|x0 µ(B)| : B ⊂ A, B ∈ A, kx0 k ≤ 1} ≤ sup{|x0 µ| (B) : B ⊂ A, B ∈ A, kx0 k ≤ 1} = sup{|x0 µ| (A) : kx0 k ≤ 1} ≤ 2 sup{|x0 µ(B)| : B ⊂ A, B ∈ A, kx0 k ≤ 1} = 2 sup{kµ(B)k : B ⊂ A, B ∈ A}. Thus. (ii) follows from (i).

Thus, from Proposition 3.45 µ has finite semi-variation iff µ is bounded. Conditions for µ to be bounded are given in Theorem 3.28 and Corollary 3.33. We next show that the stronger conclusion of Corollary 3.43 can be used to establish a strong boundedness property for strongly bounded set functions. Proposition 3.46. Let X be a Banach space and µ : A →X be strongly bounded. If {Aj } ⊂ A is pairwise disjoint, then for every ε > 0 there exists an N such that kµk (∪nj=m Aj ) < ε for n > m ≥ N . In particular, the semi-variation is strongly bounded in the sense that kµk (Aj ) → 0.

P

Proof: By Corollary 3.43 there exists N such that ∞ j=m µ(Bj ) < ε

P

n for Bj ⊂ Aj , Bj ∈ A, m ≥ N . Therefore, j=m µ(Bj ) ≤ 2ε for Bj ⊂

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Aj , Bj ∈ A, n > m ≥ N . Suppose n > m ≥ N

Pand B ∈ A with

B ⊂

n

n n ∪j=m Aj . Then B = ∪j=m B ∩ Aj so kµ(B)k = j=m µ(B ∩ Aj ) ≤ 2ε. By Proposition 3.45, kµk (∪nj=m Aj ) ≤ 4ε. Using Theorem 3.15 we can derive a result of Diestel connecting bounded, finitely additive set functions and strongly bounded set functions ([DU] I.4.2). For this we first present an example. Example 3.47. Let A be the algebra of finite/co-finite subsets of N; A = {A : either A or Ac is finite}. Define µ : A →c0 by µ(A) = χA if A is finite and µ(A) = −χAc if Ac is finite. Then µ is bounded and finitely additive but not strongly bounded since µ({j}) = ej 9 0. Theorem 3.48. Let X be a sequentially complete LCTVS. Then X contains no subspace isomorphic to c0 iff every bounded, finitely additive X valued set function defined on an algebra of sets is strongly bounded. Proof: Example 3.47 shows that if X contains a subspace isomorphic to c0 , then there is a bounded, finitely additive X valued set function defined on an algebra which is not strongly bounded. Suppose that X contains a subspace isomorphic to c0 and µ : A →X is a bounded, finitely additive set function defined on an algebra A. Let {Aj } P be a pairwise disjoint sequence from A. By Corollary 3.33 j µ(Aj ) is c0 P multiplier convergent and by Theorem 3.15 the series j µ(Aj ) is subseries convergent. Hence, µ(Aj ) → 0 and µ is strongly bounded. Finally, in this section we consider the class of vector valued measures of bounded variation. For simplicities sake, we consider only the case of set functions with values in a normed space. P Recall that a series j xj in a normed space is absolutely convergent P∞ iff j=1 kxj k < ∞. An absolutely convergent series is obviously subseries Cauchy so if X is a Banach space an absolutely convergent series is subseries convergent. The converse holds in a finite dimensional space but P not in infinite dimensional spaces [consider j (1/j)ej in c0 or recall the Dvoretsky-Rogers Theorem ([Day], [Sw2] 30.1.1)]. Definition 3.49. Let X be a normed space and µ: A →X be finitely additive. If E ∈ A, the variation of µ on E is defined to be n X |µ| (E) = sup kµ(Aj )k : {Aj }nj=1 is a partition of E with Aj ∈ A . j=1

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If |µ| (S) < ∞, µ is said to have bounded variation. It is routine to show that the variation |µ| : A → [0, ∞] is finitely additive. Since kµ(A)k ≤ |µ| (A) for A ∈ A , if µ has bounded variation, then µ is bounded. If µ : Σ → R is countably additive, then µ has bounded variation ([Sw3] 2.2.1). This statement is false for vector valued set functions as the following example shows. Example 3.50. Let P be power set of N. Let X be a normed space and P P j∈σ xj . j xj subseries convergent in X. Define µ : P →X by µ(σ) = Then it is easily seen that µ is countably additive and bounded [indeed {µ(σ) : σ ⊂ N} is relatively compact by Theorem 2.47]. However, µ has P∞ P∞ P bounded variation iff j=1 kµ({j})k = j=1 kxj k < ∞, i.e., iff j xj is absolutely convergent. Thus, if X is infinite dimensional, by the DvoretskyRogers Theorem ([Day], [Sw2] 30.1.1), there is a countably additive X valued set function defined on a σ-algebra which is of infinite variation. We have a characterization of set functions having bounded variation in terms of absolutely converging series. Theorem 3.51. Let µ : A →X equivalent:

be finitely additive. The following are

(i) µ has bounded variation, P (ii) for every pairwise disjoint sequence {Aj } ⊂ A , j µ(Aj ) is absolutely convergent, P∞ (iii) for every pairwise disjoint sequence {Aj } ⊂ A, the series j=1 kµ(Bj )k converge uniformly for Bj ⊂ Aj with Bj ∈ A. Proof: That (i) implies (ii) is clear. Suppose that (ii) holds. We establish (iii) by using Li’s Lemma 3.29. Set Ej = {B ∈ A : B ⊂ Aj } and define fj : Ej → R by fj (B) = kµ(B)k. Then Lemma 3.29 gives (iii) immediately. Clearly (iii) implies (ii). Suppose that (ii) holds but (i) fails. Note that µ is bounded by Theorem 3.28. Set M = sup{kµ(A)k : A ∈ A }. There exists a partition {A11 , ..., A1n , A1n+1 } of S such that n+1 X j=1

µ(A1j ) > M + 1,

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where some {A1j }, say, A1n+1 satisfies |µ| (A1n+1 ) = ∞ since |µ| is finitely additive. Then n X

µ(A1j ) ≥ 1 + M − µ(A1n+1 ) ≥ 1. j=1

Now treat A1n+1 as S above to obtain a partition of A1n+1 , Pm 2 2 2 2

≥ 2 and |µ| (A2m+1 ) = {A1 , ..., Am , Am+1 } with j=1 µ(Aj ) ∞. Continuing this construction produces a pairwise disjoint sequence {A11 , ..., A1n , A21 , ..., A2m , ...} which violates condition (ii). The equivalence of (i) and (ii) was established by Thorpe ([Thr]).

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Chapter 4

The Orlicz-Pettis Theorem

As noted earlier the classical version of the Orlicz-Pettis Theorem for normed spaces asserts that a series in a normed space which is subseries convergent in the weak topology of the space is subseries convergent in the norm topology of the space ([Or], [Pe]). The theorem was originally established by Orlicz for weakly sequentially complete spaces but was evidently known in full generality by the Polish mathematicians as it appears as a statement in Banach’s book ([Ba]). The first version available in English was established by Pettis in [Pe] where it was used to treat topics in vector valued integration — the Pettis integral. The theorem was extended to locally convex spaces by McArthur ([Mc]). For historical discussions of the theorem see [Ka3], [DU], or [FL]. Since a series is subseries convergent iff the series is m0 multiplier convergent, it is natural to ask what sequence spaces λ have the property that series which are λ multiplier convergent in the weak topology are λ multiplier convergent in some stronger topology such as the Mackey topology. We will refer to such results as Orlicz-Pettis Theorems. The locally convex topologies which we utilize will all be polar topologies which are described briefly in Appendix A. We record the polar topologies which we will encounter. Let X, X 0 be a pair of spaces in duality with the duality pairing h, i. The weak topology σ(X, X 0 ) (strong topology β(X, X 0 )) is the polar topology generated by the finite subsets (σ(X 0 , X) bounded subsets) of X 0 . The Mackey topology is the polar topology τ (X, X 0 ) generated by the absolutely convex, σ(X 0 , X) compact subsets of X 0 . We will also use two other polar topologies. The polar topology λ(X, X 0 )(γ(X, X 0 )) is the polar topology on X generated by the family of all σ(X 0 , X) compact subsets of X 0 [conditionally σ(X 0 , X) sequentially 49

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compact subsets of X 0 ; a subset B ⊂ X 0 is conditionally σ(X 0 , X) sequentially compact if every sequence {x0j } ⊂ B has a subsequence {x0nj } such that limhx0nj , xi exists for every x ∈ X ([Din])]. Obviously, λ(X, X 0 ) is stronger than the Mackey topology τ (X, X 0 ) and can be strictly stronger ([K1] 21.4). The topologies λ(X, X 0 ) and γ(X, X 0 ) are not comparable. We recall some basic results from Appendix A.3-6. Definition 4.1. Let X be a vector space and σ and τ two vector topologies on X. We say that τ is linked to σ if τ has a neighborhood base at 0 consisting of σ closed sets. [The terminology is that of Wilansky ([Wi] 6.1.9).] For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are linked to the weak topology σ(X, X 0 ). Lemma 4.2. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. (i) If {xj } ⊂ X is τ Cauchy and if σ − lim xj = x, then τ − lim xj = x. (ii) If (X, σ) is sequentially complete and σ ⊂ τ, then (X, τ ) is sequentially complete. Remark 4.3. It is important that the topologies σ and τ are linked in Lemma 4.2. For example, consider the space c with its weak topology P σ(c, l1 ) and the topology of pointwise convergence p. The series j ej is p convergent, the partial sums of the series are σ(c, l 1 ) Cauchy, but the series is not σ(c, l1 ) convergent. Lemma 4.4. Let X be a vector space and σ and τ two vector topologies P on X such that τ is linked to σ. If every series j xj which is σ subseries convergent satisfies τ − lim xj = 0, then every series in X which is σ subseries convergent is τ subseries convergent. The proofs of the lemmas can be found in Appendix A.3-6. Throughout this chapter λ will denote a scalar sequence space which contains c00 , the space of sequences which are eventually 0. If ∞ X λβ = {sj } : sj tj converges for every {tj } ∈ λ j=1 P∞ is the β-dual of λ, we write s · t = j=1 sj tj for {sj } ∈ λβ and {tj } ∈ λ. Note that λ and λβ are in duality with respect to the bilinear pairing s · t.

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Recall that if λ has a vector topology τ , then (λ, τ ) is an AK-space if Pn P∞ t = τ − lim j=1 tj ej = j=1 tj ej for each t ∈ λ [Appendix B.2]. We show that the conclusion of any Orlicz-Pettis Theorem for a Hellinger-Toeplitz topology is characterized by the AK-property. Recall that a locally convex topology w(X, X 0 ) defined for dual pairs X, X 0 is said to be a HellingerToeplitz topology if whenever a linear map T : X → Y is σ(X, X 0 )−σ(Y, Y 0 ) continuous, then T is also w(X, X 0 )−w(Y, Y 0 ) continuous ([Wi] 11.1.5 or see Appendix A.1; note that Hellinger-Toeplitz topologies must be defined for dual pairs). For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are Hellinger-Toeplitz topologies [Appendix A.2]. Theorem 4.5. Let w be a Hellinger-Toeplitz topology for dual pairs. The following are equivalent: (i) For every dual pair X, X 0 a series which is λ multiplier convergent for the weak topology σ(X, X 0 ) is λ multiplier convergent with respect to w(X, X 0 ). (ii) (λ, w(λ, λβ )) is an AK-space. P j Proof: Assume (i). Then j e is λ multiplier convergent with reP spect to σ(λ, λβ ) so by (i), j ej is λ multiplier convergent with respect P∞ to w(λ, λβ ). But, this means that if t ∈ λ, then t = j=1 tj ej , where the series is w(λ, λβ ) convergent so (ii) holds. P Assume (ii). Let to j xj be λ multiplier convergent with respect P σ(X, X 0 ). Consider the summing operator S : λ → X , St = ∞ t x j j j=1 [σ(X, X 0 ) limit ]. By Theorem 2.2, S is σ(λ, λβ ) − σ(X, X 0 ) continuous and, therefore, w(λ, λβ ) − w(X, X 0 ) continuous. If t = {tj } ∈ λ, then t = Pn Pn P∞ w(λ, λβ ) − lim j=1 tj ej so T t = w(X, X 0 ) − lim j=1 tj xj = j=1 tj xj . Hence, (i) holds. Condition (i) is, of course, just the conclusion of the Orlicz-Pettis Theorem for the Hellinger Toeplitz topology w(X, X 0 ). Thus, in order to establish an Orlicz-Pettis Theorem for a Hellinger-Toeplitz topology, it suffices to check the AK-property for the topology w(λ, λβ ). We now give several examples where this is the case. P Corollary 4.6. Let λ be a barrelled AK-space. If j xj is λPmultiplier convergent with respect to the weak topology σ(X, X 0 ), then j xj is λ multiplier with respect to the strong topology β(X, X 0 ).

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Proof: A barrelled space always carries the strong topology so the result follows from Theorem 4.5 since the strong topology is a Hellinger-Toeplitz topology. Remark 4.7. Corollary 4.6 applies to any Banach [Frechet ] AK-space. In particular, Corollary 4.6 applies to the spaces λ = c0 , lp (1 ≤ p < ∞), cs or bv0 [Appendix B]. In general, Orlicz-Pettis Theorems do not hold for the strong topology even in the case of subseries convergent series as the following example shows. P Example 4.8. The series j ej is subseries convergent in l ∞ with respect to the weak topology σ(l ∞ , l1 ) = σ(l∞ , (l∞ )β ) but is not subseries convergent with respect to the strong topology β(l ∞ , l1 ) = k·k∞ . We consider Orlicz-Pettis Theorems for the strong topology in Chapter 5. These results require more stringent assumptions on the multiplier space λ. We next establish an AK theorem for a general class of sequence spaces. Recall that a sequence space λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, then there exist a sequence of signs {sj } and a subsequence {nj } P such that the coordinate sum ∞ j=1 sj χInj t ∈ λ ; if the signs can all be chosen to be equal to 1, then λ is said to have the weak gliding hump property (WGHP). For examples, see Appendix B. Theorem 4.9. Assume that λ has signed-WGHP. Then (i) (λ, γ(λ, λβ )) is an AK-space. (ii) (λ, λ(λ, λβ )) is an AK-space. Proof: (i): Since γ(λ, λβ ) is linked to σ(λ, λβ ), it suffices to show that for P∞ every t ∈ λ the series j=1 tj ej is γ(λ, λβ ) Cauchy (Lemma 4.2). Suppose that there exist > 0, K ⊂ λβ which is conditionally σ(λβ , λ) sequentially compact and increasing intervals {Ij } such that X sup u · tj ej > . u∈K j∈Ik

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For each k pick uk ∈ K such that k X j tj e > . (∗) u · j∈Ik

There exists an increasing sequence {nk } such that {unk } is σ(λβ , λ) Cauchy. Define the matrix X tl el ]. M = [mij ] = [uni · l∈Inj

We show that M is a signed K-matrix (Appendix D.3). First the columns of M converge since {uni } is σ(λβ , λ) Cauchy. Next, if {pj } is an increasing sequence, there is a subsequence {qj } of {pj } and a sequence of signs {sj } P∞ P such that v = j=1 sj l∈Inq tl el ∈ λ. Then j

∞ X j=1

P∞

sj miqj = uni · v

so lim j=1 sj miqj exists since {uni } is σ(λβ , λ) Cauchy. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (∗) and establishes (i). (ii): Consider λ with the Mackey topology τ (λ, λβ ) so the dual of (λ, τ (λ, λβ )) is λβ . We claim that (λ, τ (λ, λβ )) is τ (λ, λβ ) separable. This follows since (λ, σ(λ, λβ )) is an AK-space so the σ(λ, λβ ) closure of S = span{ek : k ∈ N} is σ(λ, λβ ) dense in λ. But, S has the same closure in σ(λ, λβ ) and τ (λ, λβ ) so S is τ (λ, λβ ) dense in λ and (λ, τ (λ, λβ )) is τ (λ, λβ ) separable. This implies that σ(λβ , λ) compact sets are sequentially compact ([Wi] 9.5.3). Now, the proof of part (i) may be repeated using a σ(λβ , λ) compact (sequentially compact ) set K ⊂ λβ . From Theorems 4.5 and 4.9, we obtain an Orlicz-Pettis Theorem for λ multiplier convergent series. Corollary 4.10. Assume that λ has signed-WGHP and let X be an P LCTVS. If j xj is λ multiplier convergent with respect to the weak topolP with respect to the ogy σ(X, X 0 ), then j xj is λ multiplier convergent P topologies γ(X, X 0 ) and λ(X, X 0 ). In particular, if j xj is λ multiplier P convergent with respect to σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the Mackey topology τ (X, X 0 ).

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Corollary 4.10 contains an Orlicz-Pettis Theorem with respect to the topology γ(X, X 0 ) for a multiplier space λ which has the signed-WGHP. We observe here that any multiplier space for which the Orlicz-Pettis Theorem holds with respect to the topology γ(X, X 0 ) has the property that the topology σ(λβ , λ) is sequentially complete [thus, Corollary 4.10 implies Stuart’s sequential completeness result in 2.28]. For this observation, let {y k } be σ(λβ , λ) Cauchy. Set yj = limk y k · ej = limk yjk and y = {yj }. P We claim that y ∈ λβ and y k → y in σ(λβ , λ). Then j ej is λ multiplier convergent with respect to σ(λ, λβ ) and is, therefore, λ multiplier convergent with respect to γ(λ, λβ ) by hypothesis. Let t ∈ λ and let > 0. There Pn exists N such that j=m yjk tj < for n > m ≥ N and for all k ∈ N P by the γ(λ, λβ ) convergence. Then nj=m yj tj ≤ for n > m ≥ N so P y t converges and y ∈ λβ . Pick M such that k ≥ M implies that Pj j j N −1 k j=1 (yj tj − yj tj ) < . If k ≥ M , then X X ∞ NX ∞ X ∞ k X ∞ k −1 k + + ≤ < 3 y t y t − y t y t (y t − y t ) j j j j j j j j j j j j j=1 j=1 j=1 j=N j=N so y k → y in σ(λβ , λ). Appendix B gives a list of sequence spaces with signed-WGHP to which Corollary 4.10 applies. In particular, the space m0 has signed-WGHP (being monotone and having WGHP) so Corollary 4.10 applies to subseries convergent series. We give a formal statement of the subseries result. P is subseries convergent Corollary 4.11. Let X be an LCTVS. If j xj P with respect to the weak topology σ(X, X 0 ), then j xj is subseries con0 vergent with respect to the topologies γ(X, X ) and λ(X, X 0 ). In particuP lar, then j xj is subseries convergent with respect to the Mackey topology τ (X, X 0 ).

The usual statement of the Orlicz-Pettis Theorem for subseries convergent series and the Mackey topology was established by McArthur ([Mc]). The statement for the topology λ(X, X 0 ) was established Bennett and Kalton in [BK]. The version for γ(X, X 0 ) is given by Dierolf in [Die]. We give an example covered by Corollary 4.10 but not by Corollary 4.11. P Example 4.12. Consider the series j (1/j)ej in cs, the space of convergent series (Appendix B). This series is obviously not subseries convergent

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in cs with respect to the norm topology. However, if λ = bs, the space P of bounded series (Appendix B), and if t ∈ bs, then the series j (tj /j)ej converges in cs since {tj /j} ∈ cs because {1/j} ∈ bv0 = (bs)β , the space of P null sequences with bounded variation (Appendix B). Thus, j (1/j)ej is bs multiplier convergent in cs but not subseries convergent in cs. Without some assumption on the multiplier space λ, the conclusion of Corollary 4.10 may fail. Example 4.13. Let cc = c0 ⊕span{1, 1, 1, ...}, the space of sequences which are eventually constant (Appendix B). If X is a TVS, then a series P P multiplier convergent in X iff the series j xj converges j xj in X is cc P in X. The series j (ej+1 − ej ) is σ(c0 , l1 ) convergent in c0 (to −e1 ) and, therefore, cc multiplier convergent with respect to σ(c0 , l1 ) but is not cc multiplier convergent with respect to the norm or Mackey topology of c0 . The space l∞ is monotone and, therefore, has WGHP so Corollary 4.10 applies to l∞ or bounded multiplier convergent series. We give a formal statement of this version of the Orlicz-Pettis Theorem. P ∞ multiCorollary 4.14. Let X be an LCTVS. If the series j xj is l 0 plier convergent with respect to the weak topology σ(X, X ), then the series P ∞ multiplier convergent with respect to the topologies γ(X, X 0 ) j xj is l P 0 and λ(X, X ). In particular,if the series j xj is l∞ multiplier convergent with respect to the weak topology, then the series is l ∞ multiplier convergent with respect to the Mackey topology τ (X, X 0 ). We can use Corollary 4.10 to give a generalization of an interesting and useful Orlicz-Pettis Theorem due to Kalton ([Ka3]). Theorem 4.15. Let λ have signed-WGHP. Let X, X 0 be a pair of vector spaces in duality and suppose that τ is a polar topology from this dualP ity which is separable. If j xj is λ multiplier convergent with respect to P σ(X, X 0 ), then j xj is λ multiplier convergent with respect to τ .

Proof: Let D = {dk : k ∈ N} be τ dense in X and let A be a family of σ(X 0 , X) bounded sets which generate the polar topology τ, τ = τA (Appendix A). If A ∈ A, we show that A is conditionally σ(X 0 , X) sequentially compact and the result will follow from Corollary 4.10. Let {yk } ⊂ A. Since {yk } is pointwise bounded on D, a diagonal procedure implies that there exists a subsequence{ynk } of {yk } which converges pointwise on D ([DeS] 26.10). We claim that {ynk } is σ(X 0 , X) Cauchy. Let x ∈ X. There

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exists a net {dα } in D such that {dα } converges to x with respect to τ . If β > 0, there exists β such that y, x − d < /3

for y ∈ A. There exists n such that j, k ≥ n implies that ynj − ynk , dβ < /3. If j, k ≥ n, then

ynj − yn , x ≤ yn , x − dβ + ynj − yn , dβ + ynj , dβ − x < . k k k Hence, lim hynk , xi exists so {ynk } is σ(X 0 , X) Cauchy as claimed. Theorem 4.15 has an application to the strong topology. Corollary 4.16. Let λ have signed-WGHP. If (X, β(X, X 0 )) is separable P P and j xj is λ multiplier convergent with respect to σ(X, X 0 ), then j xj is λ multiplier convergent with respect to β(X, X 0 ). Example 4.8 shows that the separability assumption in Theorem 4.15 and Corollary 4.16 is important. The subseries version of Theorem 4.15 is due to Kalton ([Ka3]). Remark 4.17. If X is a barreled AK-space, Corollary 4.16 applies. In particular, Corollary 4.16 applies to c0 , lp (1 ≤ p < ∞), cs or bv0 with respect to their normed topologies. In Corollary 4.11 we showed that a series which is subseries convergent with respect to the weak topology σ(X, X 0 ) is actually subseries convergent with respect to two stronger polar topologies. Dierolf has shown that there is a strongest polar topology which has the same subseries convergent series as the weak topology ([Die]). He has established a similar result for bounded multiplier convergent series. We now establish both of these results of Dierolf and then give a generalization of his results to λ multiplier convergent series. As established in Theorem 4.5 the conclusion of the Orlicz-Pettis Theorem with respect to a Hellinger-Toeplitz topology is associated with the AK property for the Hellinger-Toeplitz topology on the multiplier space. A series is subseries convergent iff the series is m0 multiplier convergent, but m0 is not an AK-space with respect to its ”natural” topology, the k·k∞ topology. We now define the Dierolf topology and show that mo is an AKspace with respect to this topology. We also show that the Dierolf topology is a Hellinger-Toeplitz topology so Theorem 4.5 is applicable. In treating the Dierolf topology we use some basic properties of l 1 which we now state for convenience. Proposition 4.18. For the space l 1 , we have the following properties:

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(i) The topologies k·k1 , σ(l1 , l∞ ) and σ(l1 , m0 ) have the same convergent sequences, Cauchy sequences, the same bounded sets and the same compact sets. P∞ (ii) A subset K of l 1 is k·k1 relatively compact iff limn j=n |tj | = 0 uniformly for {tj } ∈ K. For Proposition 4.18 see [K1] 22.4. For (ii) see [Sw2] 10.1.15. Part (i) will be established later in Chapter 7. Let X, X 0 be vector spaces in duality. Let M = M(X, X 0 ) be the family of all subsets M ⊂ X 0 such that M is σ(X 0 , X) bounded and for every linear, continuous map T : (X 0 , σ(X 0 , X)) → (l1 , σ(l1 , m0 )),

T M is relatively compact in (l 1 , k·k1 ).

Definition 4.19. The Dierolf topology, δ1 (X, X 0 ), on X is the polar topology, τM , of uniform convergence on the elements of M (Appendix A). From Theorem A.2 of Appendix A, we easily have Theorem 4.20. δ1 (X, X 0 ) is a Hellinger-Toeplitz topology. We now show that m0 is an AK-space under the Dierolf topology δ1 (X, X 0 ). Actually, we have a stronger result. Pn Theorem 4.21. For t ∈ m0 , δ1 (m0 , l1 ) − limn j=1 tj ej = t, uniformly for ktk∞ ≤ 1. In particular, (m0 , δ1 (m0 , l1 )) is an AK-space. Proof: Let M ∈ M (relative to the duality between m0 and l1 ). Then M P∞ is relatively compact in (l 1 , k·k1 ) so by Proposition 4.18, limn j=n |sj | = 0 uniformly for s ∈ M . Thus, for s ∈ M and t ∈ m0 with ktk∞ ≤ 1, we have X X ∞ ∞ ≤ |sj | . s t j j j=n j=n P j Therefore, limn s · ∞ j=n tj e = 0 uniformly for s ∈ M, ktk∞ ≤ 1, so P n 1 j δ1 (m0 , l ) − limn j=1 tj e = t, uniformly for ktk∞ ≤ 1.

From Theorems 4.5 and 4.21, we obtain an Orlicz-Pettis Theorem for the Dierolf topology δ1 (X, X 0 ). P Theorem 4.22. Let X be an LCTVS. If the series j xj is subseries P convergent with respect to σ(X, X 0 ), then j xj is subseries convergent with respect to δ1 (X, X 0 ).

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Actually, from Theorems 4.20, 4.21 and 4.22, we have an improvement of Corollary 4.11. P is subseries Theorem 4.23. Let X be an LCTVS. If the series j xjP n convergent with respect to σ(X, X 0 ), then δ1 (X, X 0 ) − limn j=1 tj xj = P∞ j=1 tj xj uniformly for t ∈ m0 , ktk∞ ≤ 1.

If X is an LCTVS, it is clear that δ1 (X, X 0 ) is stronger than λ(X, X 0 ), the topology of uniform convergence on σ(X 0 , X) compact sets, since any σ(X 0 , X) compact set belongs to the family M [relative to the duality between X and X 0 ] by Proposition 4.18. Thus, Theorem 4.22 gives an improvement to the Orlicz-Pettis Theorem given in Corollary 4.11. Remark 4.24. The topology δ1 (X, X 0 ) is stronger than the topology γ(X, X 0 ), the topology of uniform convergence on the conditionally σ(X 0 , X) sequentially compact sets. For if A ⊂ X 0 is conditionally σ(X 0 , X) sequentially compact and T : (X 0 , σ(X 0 , X))) → (l1 , σ(l1 , m0 ))

is linear and continuous, then T A is conditionally σ(l 1 , m0 ) sequentially compact. By Proposition 4.18, T A is relatively k·k1 compact since k·k1 and σ(l1 , m0 ) have the same Cauchy sequences. Thus, Theorem 4.22 gives an improvement of Corollary 4.11 for the topology γ(X, X 0 ). We now show that δ1 (X, X 0 ) is the strongest polar topology with the same subseries convergent series as σ(X, X 0 ). For this we require a slight refinement of the statement in Proposition 4.18 (i). Proposition 4.25. Let K ⊂ l 1 . The following are equivalent: (i) (ii) (iii) (iv)

K is relatively k·k1 compact, P∞ limn j=n |tj | = 0 uniformly for t ∈ K, P∞ for each s ∈ l ∞ , limn j=n sj tj = 0 uniformly for t ∈ K, P∞ for each s ∈ m0 , limn j=n sj tj = 0 uniformly for t ∈ K.

Proof: (i) and (ii) are equivalent by Proposition 4.18 and clearly (ii) implies (iii) implies (iv). Suppose that (iv) holds but (ii) fails. Then there exists > 0 such that P∞ k for every k there exist mk > k and tk ∈ K such that i=m 1 k ti > 5. In P ∞ 1 particular, there exist m1 > 1, t ∈ K such that i=m1 ti > 5. There Pn1 1 P∞ t exists n1 > m1 such that i=n1 +1 t1i < . Therefore, i=m i < 4. 1 Put I1 = [m1 , n1 ], I1+ = {i ∈ I1 : t1i ≥ 0} and I1− = {i ∈ I1 : t1i < 0}. Either

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P i∈I1− t1i > 2; pick one of these which satisfies this P inequality and label it J1 so i∈J1 t1i > 2. Continuing this construction produces an increasing sequence of finite subsets Jk of N, tk ∈ K, mk < nk < mk+1 < ... with

P i∈I1+ t1i > 2 or

∞ X X k k ti < . ti > 2, Jk ⊂ [mk , nk ], i=nk +1

i∈Jk

Put s =

P∞

i=1

χJi [ coordinate sum ] so s ∈ m0 . ∞ ∞ X X X tki − si tki ≥ i=mk

i=nk +1

i∈Jk

so (iv) fails to hold.

Then

k t i >

Theorem 4.26. δ1 (X, X 0 ) is the strongest polar topology with the same subseries convergent series as σ(X, X 0 ). Proof: From Theorem 4.22, δ1 (X, X 0 ) and σ(X, X 0 ) have the same subseries convergent series. Suppose α is a polar topology with the same subseries convergent series as σ(X, X 0 ) and let α be the topology of uniform convergence on the family A of σ(X 0 , X) bounded subsets of X 0 (Appendix A). Let A ∈ A. We show that A ∈ M [with respect to the duality between X and X 0 ]. Let T : (X 0 , σ(X 0 , X))) → (l1 , σ(l1 , m0 )) be linear and continuous. Then T 0 : (m0 , σ(m0 , l1 )) → (X, σ(X, X 0 )) P is linear and continuous. Now j ej is m0 multiplier convergent with reP spect to σ(m0 , l1 ) so j T 0 ej is m0 multiplier convergent with respect to σ(X, X 0 ) and, therefore, with respect to α. Thus, if s ∈ m0 , lim n

0

∞ X

j=n

0

0 j

sj x , T e

= lim n

∞ X

j=n

s j T x0 , e j = 0

uniformly for x ∈ A. From Proposition 4.25, T A is relatively k·k1 compact so A ∈ M. Thus, A ⊂ M and α is weaker than δ1 (X, X 0 ). As we have seen the Dierolf topology is the strongest polar topology which has the same subseries convergent series as the weak topology. Tweddle has shown that there is a strongest locally convex topology which has

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the same subseries convergent series as the weak topology ([Tw]). We now give a description of the Tweddle topology. Let X be an LCTVS. Let O be P the set of all X valued series j xj which are σ(X, X 0 ) subseries converP∞ P gent. If j xj ∈ O, we write j=1 xj for the σ(X, X 0 ) sum of this series. Let X # be the space of all linear functionals x0 on X such that + * ∞ ∞ X X X 0 0 xj ∈ O. for all xj hx , xj i = x , j=1

j=1

j

The Mackey topology, τ (X, X # ), is the Tweddle topology on X and is denoted by t(X, X 0 ). We have the following important property of the Tweddle topology. Theorem 4.27. The Tweddle topology t(X, X 0 ) is the strongest locally convex topology on X which has the same subseries convergent series as the weak topology σ(X, X 0 ). Proof: Suppose that ν is a locally convex topology on X which has the 0 0 same subseries convergent series as σ(X, X 0 ). Let E . Then for D H P= (X, ν) P P ∞ ∞ 0 # 0 0 0 0 j xj ∈ O and x ∈ H , we have j=1 hx , xj i = x , j=1 xj so x ∈ X

and H 0 ⊂ X # . Therefore, τ (X, H 0 ) is weaker than τ (X, X # ) = t(X, X 0 ). But, ν ⊂ τ (X, H 0 ) so ν is weaker than t(X, X 0 ). We give an example where the Tweddle topology is strictly stronger than the Mackey topology by computing the space X # and comparing it to X 0 . The example uses the Nikodym Boundedness Theorem for countably additive set functions which we prove later in Theorem 4.60 (see also [Sw3] 2.8.8 for a statement).

Example 4.28. Let Σ be a σ-algebra of subsets of a set S. Let B(S, Σ) be the space of all bounded, real valued Σ-measurable functions defined on S. Let ca(Σ) be the space of all real valued, countably additive set functions defined on Σ and let Γ = span{δt : t ∈ S}, where δt is the Dirac measure concentrated at t. The weak topology σ(B(S, Σ), Γ) is just the topology of pointwise convergence, p, on B(S, Σ) so (B(S, Σ), p)0 = Γ. We show that B(S, Σ)# (with respect to p) is ca(Σ) so t(B(S, Σ), Γ) = τ (B(S, Σ), ca(Σ)) is strictly stronger than τ (B(S, Σ), Γ). First, suppose f ∈ B(S, Σ)# . Then f induces a set function on Σ, still denoted by f , defined by f (E) = hf, χE i for E ∈ Σ. We claim that P f ∈ ca(Σ). Let {Ej } ⊂ Σ be pairwise disjoint. Then j χEj is subseries

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P

χEj converges to χ∪∞ j=1 Ej

convergent with respect to p and the series with respect to p. Therefore, D

f, χ∪∞ j=1 Ej

E

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= f (∪∞ j=1 Ej ) =

∞ X

j=1

j

∞ X f (Ej ), f, χEj = j=1

#

and f is countably additive. Thus, B(S, Σ) ⊂ ca(Σ). P Next, let ν ∈ ca(Σ) and let j gj be subseries convergent with respect P to p. We claim that { j∈σ gj : σ finite} is bounded with respect to the sup-norm, k·k∞ , on B(S, Σ). If this is not the case, for every k there exist finite σk and tk ∈ S such that X gj (tk ) > k. (∗) j∈σk P Since j gj is subseries convergent in B(S, Σ) with respect to p, the P ∞ ∞ series with respect to the j {gj (tk )}k=1 is subseries convergent in l ∞ topology of coordinatewise convergence in l . Define µk : 2N → R by P µk (σ) = j∈σ gj (tk ). Note that µk ∈ ca(2N ) and if σ ⊂ N, X sup |µk (σ)| = sup ( gj )(tk ) < ∞. k j∈σ By the Nikodym Boundedness Theorem (Theorem 4.60 and or [Sw3] 2.8.8), X gj )(tk ) < ∞. sup sup |µk (σ)| = sup sup ( σ σ k k j∈σ But, this contradicts (∗). P If {nk } is a subsequence, then { kj=1 gnj } is uniformly bounded on S by the claim established above so by the Bounded Convergence Theorem, Z X k Z ∞ X gnj dν = gnj dν. lim k

j=1

S

S j=1

Therefore, ν ∈ B(S, Σ)# and B(S, Σ)# = ca(Σ). There is an analogous Dierolf topology for bounded multiplier convergent series which we will now describe. The proofs of the various properties of the Dierolf topology for bounded multiplier convergent series are almost identical to the proofs for the Dierolf topology for the subseries case so we will give the appropriate statements but omit the proofs.

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Let X, X 0 be in duality. Let N = N (X, X 0 ) be the family of all σ(X 0 , X) bounded subsets N ⊂ X 0 such that for every continuous linear operator T : (X 0 , σ(X 0 , X)) → (l1 , σ(l1 , l∞ )) T N is relatively compact in (l 1 , k·k1 ) [again note Proposition 4.25]. Definition 4.29. The Dierolf topology for bounded multiplier convergent series, δ2 (X, X 0 ), on X is the polar topology, τN , of uniform convergence on the elements of N = N (X, X 0 ) [Appendix A]. We now state the analogues of Theorems 4.20, 4.21, 4.22 and 4.23 for the Dierolf topology δ2 (X, X 0 ). Theorem 4.30. The topology δ2 (X, X 0 ) is a Hellinger-Toeplitz topology. Theorem 4.31. For t ∈ l ∞ , δ2 (l∞ , l1 ) − lim n

n X

tj e j = t

j=1

uniformly for ktk∞ ≤ 1. In particular, (l ∞ , δ2 (l∞ , l1 )) is an AK-space. P Theorem 4.32. Let X be an LCTVS. If the series j xj is bounded mulP tiplier convergent in the weak topology σ(X, X 0 ), then the series j xj is bounded multiplier convergent in the Dierolf topology δ2 (X, X 0 ). P Theorem 4.33. Let X be an LCTVS. If the series j xj is bounded multiplier convergent in the weak topology σ(X, X 0 ), then δ2 (X, X 0 ) − Pn limn j=1 tj ej = t uniformly for ktk∞ ≤ 1. The analogues of the statements in Remark 4.24 also hold for the Dierolf topology δ2 (X, X 0 ).

Remark 4.34. δ2 (X, X 0 ) is stronger than λ(X, X 0 ), the topology of uniform convergence on the σ(X 0 , X) compact subsets of X 0 . Thus, Theorem 4.32 gives an improvement to the statement in Corollary 4.14 for the topology λ(X, X 0 ). Remark 4.35. δ2 (X, X 0 ) is stronger than γ(X, X 0 ), the topology of uniform convergence on the σ(X 0 , X) bounded sets which are conditionally σ(X 0 , X) sequentially compact. Thus, Theorem 4.32 gives an improvement to the statement in Corollary 4.14 for the topology γ(X, X 0 ).

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Finally, the analogue of Theorem 4.26 holds for the Dierolf topology δ2 (X, X 0 ). Theorem 4.36. The Dierolf topology δ2 (X, X 0 ) is the strongest polar topology on an LCTVS X with the same bounded multiplier convergent series as the weak topology σ(X, X 0 ). The proof of this theorem proceeds as the proof of Theorem 4.26 except that the statement in Proposition 4.25 (iii) is used in place of the statement in Proposition 4.25 (iv). Using the Dierolf topologies for m0 and l∞ multiplier convergent series as models, we show that an analogous topology can be defined for arbitrary sequence spaces of multipliers, and when the multiplier space λ satisfies the signed-WGHP, we compare the general Dierolf topology to the topologies λ(X, X 0 ) and γ(X, X 0 ). Definition 4.37. Let λ be a sequence space containing c00 . A subset P K ⊂ λβ has uniform tails if for every s ∈ λ, limn ∞ j=n sj tj = 0 uniformly for t ∈ K. From Proposition 4.25, if λ = m0 or λ = l∞ , a subset K ⊂ l1 = λβ has uniform tails iff K is relatively k·k1 compact. Let X be an LCTVS. We say that a σ(X 0 , X) bounded subset D ⊂ X 0 belongs to Dλ iff for every continuous linear operator T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ))

the subset T D ⊂ λβ has uniform tails.

Definition 4.38. The general Dierolf topology on X, denoted by Dλ (X, X 0 ), is the polar topology τDλ of uniform convergence on the elements of Dλ . From Proposition 4.25 it follows that δ1 (X, X 0 ) = Dm0 (X, X 0 ) and δ2 (X, X 0 ) = Dl∞ (X, X 0 ) so it is reasonable to refer to Dλ (X, X 0 ) as a Dierolf topology with respect to λ. In order to establish the basic property of the general Dierolf topology we prove the following result. Proposition 4.39. There is a 1-1 correspondence between σ(X, X 0 ) λ mulP tiplier convergent series j xj and continuous linear operators T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ)).

The correspondence is given by xj = T 0 ej .

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P Proof: Suppose that j xj is λ multiplier convergent with respect to P∞ σ(X, X 0 ). Define the summing operator S : λ → X by St = j=1 tj xj [σ(X, X 0 ) sum]. Then S is σ(λ, λβ ) − σ(X, X 0 ) continuous by Theorem 2.2. Therefore, T = S 0 : X 0 → λβ is σ(X 0 , X) − σ(λβ , λ) continuous and T 0 x0 = {hx0 , xj i}. If T : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ))

is linear and continuous, then S = T 0 : λ → X is σ(λ, λβ ) − σ(X, X 0 ) P∞ P continuous. Now j ej is σ(λ, λβ ) λ multiplier convergent so j=1 T 0 ej = P∞ P ∞ j 0 j=1 Se = j=1 xj is σ(X, X ) λ multiplier convergent and the correspondence follows. Remark 4.40. Thus, it follows from Proposition 4.39, to check that a subset D ⊂ X 0 belongs to Dλ , it suffices to show that if t ∈ λ, then P∞ P limn j=n tj hx0 , xj i = 0 uniformly for x0 ∈ D whenever j xj is λ multiplier convergent with respect to σ(X, X 0 ). Theorem 4.41. The general Dierolf topology Dλ (X, X 0 ) is the strongest polar topology on X with the same λ multiplier convergent series as σ(X, X 0 ). P Proof: Let j xj be λ multiplier convergent with respect to σ(X, X 0 ). P Let S : λ → X be the summing operator with respect to j xj , St = P∞ 0 β 0 j=1 tj xj [σ(X, X ) sum]. By Theorem 2.2, S is σ(λ, λ ) − σ(X, X ) con0 0 0 β β tinuous and S = T : (X , σ(X , X)) → (λ , σ(λ , λ)) is linear and continuous. Let D ∈ Dλ . Then T D has uniform tails. Therefore, for every s ∈ λ, P∞ limn j=n sj hx0 , xj i = 0 uniformly for x0 ∈ D since T x0 = {hx0 , xj i}. That P is, the series j sj xj converges in Dλ (X, X 0 ). Suppose that α is a polar topology with the same λ multiplier convergent series as σ(X, X 0 ). Let α be the polar topology of uniform convergence on P the family A of σ(X 0 , X) bounded sets (Appendix A). Let A ∈ A. If j xj P is λ multiplier convergent with respect to σ(X, X 0 ), then j xj is λ multiP∞ plier convergent with respect to α so if t ∈ λ, then limn j=n tj hx0 , xj i = 0 uniformly for x0 ∈ A. Then A ∈ Dλ by Remark 4.40 and α is weaker than Dλ (X, X 0 ). As in Theorem 4.20 we can show that Dλ (X, X 0 ) is a Hellinger-Toeplitz topology. Theorem 4.42. Dλ (X, X 0 ) is a Hellinger-Toeplitz topology.

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Proof: From Theorem A.2 of Appendix A, we need to show that if D ∈ Dλ with respect to the dual pair Y, Y 0 and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is linear and continuous, where X, X 0 is another dual pair, then T 0 D ∈ Dλ with respect to the dual pair X, X 0 . Let U : (X 0 , σ(X 0 , X)) → (λβ , σ(λβ , λ)) be linear and continuous. Then U T 0 : (Y 0 , σ(Y 0 , Y )) → (λβ , σ(λβ , λ)) is linear and continuous so U T 0 D has uniform tails in λβ so T 0 D ∈ Dλ with respect to the dual pair X, X 0 . From Theorems 4.5, 4.41 and 4.42, we have Theorem 4.43. (λ, Dλ (λ, λβ )) is an AK-space. We next compare the general Dierolf topology to the topologies λ(X, X 0 ) and γ(X, X 0 ). This will require an additional assumption on the multiplier space λ. We first establish a lemma. P Lemma 4.44. Suppose M ⊂ X 0 is such that there exist a series j xj 0 which is λ multiplier convergent with respect to σ(X, X ) and a t ∈ λ such P∞ that the series j=1 tj hx0 , xj i do not converge uniformly for x0 ∈ M . Then there exist > 0, {x0k } ⊂ M and an increasing sequence of intervals {Ik } with X 0 t hx , x i j j > k j∈Ik for every k.

Proof: If the series do not converge uniformly for x0 ∈ M , then there 0 0 exists > 0 such that for every k there exist mk > k, x = x (k) ∈ M such P∞ 0 that j=mk tj hx , xj i > 2. In particular, there exist m1 , x01 ∈ M with P P ∞ j=m1 tj hx01 , xj i > 2. Since the series j tj xj is σ(X, X 0 ) convergent, P 0 there exists n1 > m1 such that ∞ j=n1 tj hx1 , xj i < . Thus, if I1 = P [m1 , n1 ], then j∈I1 tj hx01 , xj i > . Now just continue the construction.

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We first compare the general Dierolf topology with the topology γ(X, X 0 ) of uniform convergence on σ(X 0 , X) bounded sets which are conditionally σ(X 0 , X) sequentially compact. Theorem 4.45. Let λ have signed-WGHP. Then γ(X, X 0 ) ⊂ Dλ (X, X 0 ). Proof: Suppose that K ⊂ X 0 is σ(X 0 , X) bounded and conditionally σ(X 0 , X) sequentially compact. If K does not belong to Dλ , then by Remark 4.40 and Lemma 4.44 there exist a multiplier convergent series P 0 0 j xj with respect to σ(X, X ), t ∈ λ, > 0, {xk } ⊂ K and an increasing sequence of intervals {Ik } such that X 0 (∗) tj hxk , xj i > j∈Ik

for every k. By the conditional σ(X 0 , X) sequential compactness of K, we may assume that lim hx0k , xi exists for every x ∈ X. Define a matrix X tl hx0i , xl i]. M = [mij ] = [ l∈Ij

We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by the compactness condition. Next, if {pj } is an increasing sequence there is a further subsequence {qj } of {pj } and a sequence of signs P∞ {sj } such that u = {uj } = j=1 sj χIqj t ∈ λ . Then + * ∞ ∞ X X sj miqj = x0i , u j xj j=1

j=1

P∞

so limi j=1 sj miqj exists. Thus, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗). We next consider the topology λ(X, X 0 ) of uniform convergence on σ(X 0 , X) compact subsets of X 0 . Theorem 4.46. Let λ have signed-WGHP. Then λ(X, X 0 ) ⊂ Dλ (X, X 0 ). Proof: Let K ⊂ X 0 be σ(X 0 , X) compact. Assume that K does not belong to Dλ and let the notation be as in the proof of Theorem 4.45. Let X0 = span{xk : k ∈ N}. The set {x0k : k ∈ N} is relatively σ(X00 , X0 ) compact and, therefore, relatively σ(X00 , X0 ) sequentially compact since X0 is separable ([Wi] 9.5.3). Therefore, we may assume that lim hx0k , xi

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exist for every x ∈ X. The proof can now be completed as in the proof of Theorem 4.45. Without the assumption on the multiplier space λ, the inclusions in Theorems 4.45 and 4.46 may fail to hold. Example 4.47. Let λ = cc , the space of sequences which are eventually constant (Appendix B). Then a series in a TVS is λ multiplier convergent iff the series is convergent. If one has a series which is σ(X, X 0 ) convergent but not τ (X, X 0 ) convergent, then the series is λ multiplier convergent with respect to σ(X, X 0 ) but not λ multiplier convergent with respect to τ (X, X 0 ). P [For example, take j (ej+1 − ej ) in c0 .] Thus, τ (X, X 0 ) is not contained in Dλ (X, X 0 ) by Theorem 4.41. Since both topologies λ(X, X 0 ) and γ(X, X 0 ) contain τ (X, X 0 ), this shows that the containments in Theorems 4.45 and 4.46 do not hold. Remark 4.48. Theorems 4.45 and 4.46 contain the results in Corollaries 4.11 and 4.14 as special cases. We now compare the general Dierolf topology with the strong topology when the multiplier space has the ∞-GHP. Recall that λ has ∞-GHP if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate sum of the P∞ series j=1 apj χIpj t ∈ λ (Appendix B; examples are given in Appendix B). Theorem 4.49. Let λ have ∞-GHP. Then β(X, X 0 ) ⊂ Dλ (X, X 0 ).

Proof: Let B ⊂ X 0 be σ(X 0 , X) bounded. Assume that B does not belong to Dλ . Let the notation be as in Theorem 4.45 so X 0 (∗) tj hxk , xj i > j∈Ik

for every k with x0k ∈ B. By the ∞-GHP there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such P that s = {sj } = ∞ k=1 aqk χIqk t ∈ λ. Define a matrix M = [mij ] = [

X

l∈Ij

tl apj hx0i /api , xl i].

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We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since x0i /api → 0 in σ(X 0 , X). Next, given a subsequence of {pj } let {qj } be a subsequence as above. Then + * ∞ ∞ ∞ X X X X 0 0 sl xl → 0, sl hxi /api , xl i = xi /api , miqj = j=1

j=1 l∈Iqj

P∞

l=1

0

where l=1 sl xl is the σ(X, X ) sum of the series. Thus, M is a K-matrix so the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). There is also an analogue of the Tweddle topology for λ multiplier convergent series when the multiplier space has signed-WGHP. If X is an LCTVS, let X # (= Xλ# ) be the space of all linear functionals x0 on X satisfying * + ∞ ∞ X X 0 0 tj hx , xj i = x , t j xj j=1

j=1

0

P for every t ∈ λ and every σ(X, X ) λ multiplier convergent series j xj , P∞ where j=1 tj xj is the σ(X, X 0 ) sum of the series. We define the Tweddle topology on X to be Dλ (X, X # ) and denote the topology by tλ (X, X 0 ). Thus, tm0 (X, X 0 ) = t(X, X 0 ). We show that tλ (X, X 0 ) is the strongest locally convex topology on X with the same λ multiplier convergent series as σ(X, X 0 ). Theorem 4.50. Let λ have signed-WGHP. Then tλ (X, X 0 ) is the strongest locally convex topology on X with the same λ multiplier convergent series as σ(X, X 0 ). P Proof: If j xj is λ multiplier convergent with respect to σ(X, X 0 ), then P with respect to σ(X, X # ) by the definition j xj is λ multiplier convergent P # of X . By Theorem 4.41, j xj is λ multiplier convergent with respect to tλ (X, X 0 ) = Dλ (X, X #). Suppose α is a locally convex topology with the same λ multiplier convergent series as σ(X, X 0 ). Put H 0 = (X, α)0 . Then H 0 ⊂ X # . By Theorem 4.41, α ⊂ τ (X, H 0 ) ⊂ Dλ (X, H 0 ) ⊂ Dλ (X, X # ) = tλ (X, X 0 ). We now present several applications of the Orlicz-Pettis theorems to various topics in functional analysis and measure theory. As a first application we present the original application by Pettis to vector valued measures.

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Theorem 4.51. Let Σ be a σ-algebra of subsets of a set S and let X be an LCTVS. If µ : Σ → X is such that x0 ◦ µ = x0 µ : Σ → R is countably additive for every x0 ∈ X 0 , then µ is countably additive with respect to the original topology of X. That is, if µ is countably additive with respect to the weak topology, then µ is countably additive with respect to the original topology. Proof: Let {Aj } ⊂ Σ be pairwise disjoint. If {Anj } isa subsequence, P∞ then A = ∪∞ Σ and hx0 , µ(A)i = j=1 x0 , µ(Anj ) for every x0 ∈ j=1 Anj ∈ P X 0 . Thus, the series j µ(Aj ) is subseries convergent with respect to the weak topology σ(X, X 0 ). By the Orlicz-Pettis Theorem in Corollary 4.11, the series is subseries convergent with respect to the original topology of X. That is, µ is countably additive with respect to the original topology of X. For vector valued set functions defined on algebras, we have Theorem 4.52. Let A be an algebra of subsets of a set S and let X be a weakly sequentially complete LCTVS. If µ : A → X is such that x0 µ is countably additive for every x0 ∈ X 0 , then µ is countably additive with respect to the original topology of X. ∈ A. Then Proof: Let {Aj } ⊂ A be pairwise disjoint with A = ∪∞ j=1 AjP P∞ 0 0 0 0 hx , µ(A)i = j=1 hx , µ(Aj )i for every x ∈ X and the series j µ(Aj ) is unconditionally convergent with respect to σ(X, X 0 ) since the union ∪∞ j=1 Aj is independent of the ordering of the {Aj } [note that we cannot assert that not belong to A for the series is subseries convergent since ∪∞ j=1 Anj mayP arbitrary subsequences]. By Corollary 2.60 the series j µ(Aj ) is subseries convergent with respect to σ(X, X 0 ). By the Orlicz-Pettis Theorem in P Corollary 4.11, the series j µ(Aj ) is subseries convergent with respect to the original topology of X. Thus, µ is countably additive with respect to the original topology of X. As an application of Theorem 4.52 and the Orlicz-Pettis Theorem, we have Theorem 4.53. Let X be a weakly sequentially complete LCTVS. Then X contains no subspace isomorphic to c0 . P Proof: Let j xj be c0 multiplier convergent in X. By Proposition 3.8, P∞ 0 ∞ 0 0 j=1 tj hx , xj i converges for every t ∈ l , x ∈ X . Hence, the partial

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P sums of the series j tj xj are σ(X, X 0 ) Cauchy and, therefore, σ(X, X 0 ) P ∞ convergent for every t ∈ l ∞ . That is, the series multiplier j xj is l convergent in the weak topology. By the Orlicz-Pettis Theorem in Corollary P 4.14 the series j xj is l∞ multiplier convergent in the original topology of X. Thus, condition (iv) of Theorem 3.15 holds. The proofs that (iv) implies (viii) implies (i) in Theorem 3.15 do not employ the sequential completeness of X so the result follows from these proofs. Corollary 4.54. If X is a semi-reflexive space, then X contains no subspace isomorphic to c0 . Proof: (X, σ(X, X 0 )) is boundedly complete and, therefore, sequentially complete ([Wi] 10.2.4) so the result follows from Theorem 4.53. Corollary 4.55. If X is a barrelled LCTVS, then (X 0 , σ(X 0 , X)) contains no subspace isomorphic to c0 . Proof: (X 0 , σ(X 0 , X)) is sequentially complete ([Wi] 9.3.8). But, (X 0 , σ(X 0 , X))0 = X so (X 0 , σ(X 0 , X)) is weakly sequentially complete. As another application of the Orlicz-Pettis Theorem, we derive a result similar in spirit to Theorem 3.2 but with different hypothesis ([KG] 3.10.5). Theorem 4.56. Let X be weakly sequentially complete and let λ be monoP tone. Then a series j xj in X is λ multiplier convergent in X iff 0 β {hx , xj i} ∈ λ for every x0 ∈ X 0 . P Proof: If j xj is λ multiplier convergent, then {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 by Proposition 3.1. Conversely, suppose that {hx0 , xj i} ∈ λβ for every x0 ∈ X 0 . Let t = {tj } ∈ λ. Since λ is monotone, the partial sums of any subseries P of j tj xj is σ(X, X 0 ) Cauchy and, therefore, σ(X, X 0 ) convergent by hyP pothesis. That is, the series j tj xj is σ(X, X 0 ) subseries convergent and, therefore, convergent in X by the Orlicz-Pettis Theorem in Corollary 4.11. Note that the assumption that λ is monotone in Theorem 4.56 is an algebraic condition whereas the assumptions in Theorem 3.2 are topological. Theorem 4.56, however, has the weak sequential completeness assumption. We can also use the Orlicz-Pettis Theorem to derive a result of Pelczynski on weakly compact and unconditionally converging operators (recall

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Theorem 3.18). A continuous linear operator T between normed spaces X and Y is weakly compact if T carries bounded sets into relatively weakly compact sets and T is unconditionally converging if T carries wuc series into subseries convergent series. Theorem 4.57. ([Pl]) If T : X → Y is a weakly compact operator from the normed space X into the normed space Y , then T is unconditionally converging. P P Proof: Let j xj be wuc. Then { j∈σ xj : σ f inite} is bounded by P Proposition 3.8. Thus, { j∈σ T xj : σ f inite} is relatively σ(Y, Y 0 ) comP pact. By Theorem 2.48, the series j T xj is σ(Y, Y 0 ) subseries convergent. P By the Orlicz-Pettis Theorem in Corollary 4.11, the series j T xj is subseries convergent in Y . The identity operator on l 1 shows that the converse of Theorem 4.57 is false, in general; however, there are spaces for which the converse does hold (see [Pl]). As another application of the Orlicz-Pettis Theorem, we derive a version of the Nikodym Boundedness Theorem for countably additive set functions. Dunford and Schwartz refer to the Nikodym Boundedness Theorem as a ”striking improvement of the principle of uniform boundedness” ([DS] p. 309). The theorem states that a family of countably additive signed measures defined on a σ-algebra which is pointwise bounded on the σ-algebra is uniformly bounded on the entire σ-algebra. For the proof of the theorem, we first establish a result which is central to most of the proofs of the theorem (see, however, Dunford and Schwartz where there is a proof based on the Baire Category Theorem ([DS] IV.9.8)). Let A be an algebra of subsets of a set S, and let ba(A) be the space of all real valued, bounded, finitely additive set functions defined on A. Lemma 4.58. Let M ⊂ ba(A) be pointwise bounded on A. Then sup{|µ(A)| : µ ∈ M, A ∈ A} < ∞ iff sup{|µi (Ai )| : i ∈ N} < ∞ for every pairwise disjoint sequence {Ai } from A and every sequence {µi } ⊂ M. Proof: Suppose sup{|µ(A)| : µ ∈ M, A ∈ A} = ∞ . Note that for each r > 0 there exist a partition (E, F ) of S with E, F ∈ A and µ ∈ M such that min{|µ(E)| , |µ(F )|} > r. [ This follows since |µ(E)| > r + sup{|ν(S)| : ν ∈ M } implies |µ(S \ E)| ≥ |µ(E)| − |µ(S)| > r. ] Hence, there exist µ1 ∈ M and a partition (E1 , F1 ) of S such that min{{|µ1 (E1 )| , |µ1 (F1 )|} > 1.

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Now either sup{|µ(E1 ∩ A)| : µ ∈ M, A ∈ A} = ∞ or sup{|µ(F1 ∩ A)| : µ ∈ M, A ∈ A} = ∞. Pick whichever of E1 or F1 satisfies this condition and label it B1 and set A1 = S \ B1 . Now treat B1 as S above to obtain a partition (A2 , B2 ) of B1 and µ2 ∈ M such that |µ2 (A2 )| > 2 and sup{|µ(B2 ∩ A)| : µ ∈ M, A ∈ A} = ∞. Continuing this construction produces a sequence {µi } ⊂ M and a pairwise disjoint sequence {Ai } from A such that |µi (Ai )| > i. This establishes the sufficiency; the necessity is clear. Let Σ be a σ-algebra of subsets of a set S. Let S(Σ) be the vector space of all real valued Σ-simple functions and let ca(Σ) be the space of all countably additive signed measures µ : Σ → R.R Then S(Σ), ca(Σ) form a dual pair via the integration pairing hµ, f i = f dµ, f ∈ S(Σ), µ ∈ca(Σ). We now give our proof of the Nikodym Boundedness Theorem. Theorem 4.59. Let M ⊂ ca(Σ) be such that sup{|µ(E)| : µ ∈ M } < ∞ for every E ∈ Σ. Then sup{|µ(E)| : µ ∈ M, E ∈ Σ} < ∞. Proof: By Lemma 4.58 it suffices to show that sup{|µi (Ai )| : i ∈ N} < ∞ for every {µi } ⊂ M and pairwise disjoint sequence {Ai } from Σ or that P (1/i)µi (Ai ) → 0. The series i χAi is σ(S(Σ),ca(Σ)) subseries convergent by the countable additivity of the members of ca(Σ). By the version of the P Orlicz-Pettis Theorem for the topology λ(S(Σ),ca(Σ)) the series i χ Ai P∞ converges in λ(S(Σ), ca(Σ)). In particular, limn j=n (1/k)µk (Aj ) = 0 uniformly for k ∈ N since {(1/k)µk } is σ(ca(Σ), S(Σ)) convergent to 0 by the pointwise boundedness assumption. In particular, (1/i)µi (Ai ) → 0 as desired. It should be noted that a version of the theorem for countably additive set functions defined on σ-algebras with values in an LCTVS follows immediately from Theorem 4.59. Corollary 4.60. Let X be an LCTVS and let M be an family of countably additive set functions defined on Σ with values in X. If {µ(E) : µ ∈ M } is bounded in X for every E ∈ Σ, then {µ(E) : µ ∈ M, E ∈ Σ} is bounded.

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Proof: Let x0 ∈ X 0 . Then the subset {x0 µ : µ ∈ M } ⊂ ca(Σ) is pointwise bounded on Σ so the set {x0 µ : µ ∈ M } is uniformly bounded on Σ by Theorem 4.59. That is, the set {x0 µ : µ ∈ M, E ∈ Σ} is σ(X, X 0 ) bounded and, therefore, bounded in X. A few remarks pertaining to the Nikodym Boundedness Theorem are in order. First, the local convex assumption in Corollary 4.60 is important. Turpin has given an example of a countably additive set function defined on a σ-algebra with values in a (non-locally convex) TVS which is unbounded ([Rol]). Theorem 4.59 actually holds for bounded, finitely additive set functions defined on σ-algebras; we will give a proof of this version of the Nikodym Boundedness Theorem in Chapter 7 based on the Hahn-Schur Theorem. As the following example shows, the conclusion of Theorem 4.59 is false for set functions defined on algebras. Example 4.61. Let A be the algebra of finite/co-finite subsets of N. Let δn be the Dirac measure concentrated at n; δn (E) = 1 if n ∈ E and δn (E) = 0 if n ∈ / E. Define µn (E) = n(δn+1 (E) − δn (E)) if E is finite and µn (E) = −n(δn+1 (E) − δn (E)) if E c = N \ E is finite. Then {µn } is pointwise bounded on A but not uniformly bounded on A [µn ({n}) = n]. Despite Example 4.61, there are algebras for which the conclusion of Theorem 4.59 holds. The treatise by Schachermeyer contains examples, references and other discussions concerning the Nikodym Boundedness Theorem ([Sm]); see also Diestel and Uhl ([DU]). Finally, we present several versions and applications of the Orlicz-Pettis Theorem in an abstract setting. Let E, F be vector spaces such that there is a bilinear mapping from · : E × F → X, (x, y) → x · y, x ∈ E, y ∈ F , where X is an LCTVS. Of course, an example of this situation is when E, F are two vector spaces in duality ; we give other examples in the applications which follow. Let w(E, F ) [w(F, E)] be the weakest topology on E [F ] such that the linear maps x → x · y [y → x · y] from E into X [F into X] are continuous for all y ∈ F [x ∈ E]. If E, F are 2 vector spaces in duality, then w(E, F ) [w(F, E)] is just the weak topology σ(E, F ) [σ(F, E)]. A subset K ⊂ F is said to be conditionally w(F, E) sequentially compact if for every sequence {yj } ⊂ K, there is a subsequence {ynj } such that limj x·ynj exists for every x ∈ E. In this setting we have the analogue of Corollary 4.10 for the topology γ(X, X 0 ). Again, if E, F are 2 vector spaces in duality, this agrees with previous terminology.

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We establish a version of Corollary 4.10 for the topology γ(E, F ) in this setting. P Theorem 4.62. Let λ have signed-WGHP. If the series j xj is λ multiplier convergent in E with respect to w(E, F ), then for each t ∈ λ and each conditionally w(F, E) sequentially compact subset K ⊂ F , the series P∞ j=1 tj xj · y converge uniformly for y ∈ K.

Proof: If the conclusion fails to hold, there exists a neighborhood of 0, W , in X, yk ∈ K and an increasing X sequence of intervals {Ik } such that (#) t l xl · y k ∈ /W l∈Ik

for every k. We may assume, by passing to a subsequence if necessary, that limk x · yk exists for every x ∈ E. Consider X the matrix tl xl · yi ]. M = [mij ] = [ l∈Ij

We claim that M is a signed K-matrix (Appendix D). First, the columns of M converge. Next, given an increasing sequence of positive integers there is a subsequence {nj } and a sequence of signs {sj } such that u = P∞ j χInj t ∈ λ. Then j=1 s (∞ ) ∞ ∞ X X X X = t l xl · y i u l xl · y i = sj sj minj j=1

i

j=1

l∈Inj

i

l=1

i

converges. Hence, M is a signed K-matrix so the diagonal of M converges to 0 by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3). But, this contradicts (#). We now derive an analogue of Corollary 4.10 for λ multiplier convergent series and the topology λ(X, X 0 ). P Theorem 4.63. Let λ have signed-WGHP. If the series j xj is λ multiplier convergent in E with respect to w(E, F ), then for each w(F, E) compact (countably compact) subset K ⊂ F and each t ∈ λ, the series P∞ j=1 tj xj · y are convergent uniformly for y ∈ K.

Proof: Let p be a continuous semi-norm on X. We need to show that P the series j tj xj · y converge uniformly for y ∈ K with respect to p. This will follow if we can show that this property holds in the quotient space X/p. Hence, we may assume that p is actually a norm. Define an equivalence relation ∼ on F by y ∼ z iff xj · y = xj · z for all j. If

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P P∞ E0 = { ∞ j=1 sj xj : s ∈ λ, where j=1 sj xj is the w(E, F ) sum of the series}, then x · y = x · z for every x ∈ E0 when y ∼ z. Let y − be the equivalence class of y ∈ F and set F − = {f − : f ∈ F }. Define a metric d on F − by d(y − , z − ) =

∞ X j=1

p(xj · (y − z))/2j (1 + p(xj · (y − z)));

note that d is a metric since p is a norm. Define a bilinear mapping · : E0 × F − → (X, p) by x · y − = x · y so we may consider the triple E0 , F − , (X, p) as above. The quotient map F → F − is w(F, E)−w(F − , E0 ) continuous and the inclusion (F − , w(F − , E0 )) ⊂ (F − , d) is continuous so K − is compact (countably compact) with respect to w(F − , E0 ) and d and, therefore, w(F − , E0 ) = d on K − and K − is w(F − , E0 ) sequentially compact. Since the series P P j xj j xj is λ multiplier convergent with respect to w(E, F ), the series is λ multiplier convergent with respect to w(E0 , F − ) in the abstract triple E0, F − , (X, p). Since K − is sequentially compact in w(F − , E0 ), by Theorem P∞ P∞ 4.62 the series j=1 tj xj · y − = j=1 tj xj · y converge uniformly for y − ∈ K − with respect to p. Theorems 4.62 and 4.63 have as immediate corollaries the results in Corollary 4.10. We now present another corollary related to Theorem 4.9. Corollary 4.64. Let λ have signed-WGHP. Then (λ, γ(λ, λβ )) and (λ, λ(λ, λβ )) are AK-spaces. P j Proof: The series j e is λ multiplier convergent with respect to β σ(λ, λ ) and, therefore, is λ multiplier convergent with respect to γ(λ, λβ ) and λ(λ, λβ ) by Theorems 4.62 and 4.63. The result is now immediate. Note that the results above were derived in the other order previously. We now give several applications of Theorems 4.62 and 4.63. Example 4.65. Let Σ be a σ-algebra of subsets of a set S and let ca(Σ, X) be the space of all X valued countably additive R set functions from Σ into X. If E = S(Σ) and F = ca(Σ, X), then f · µ = S f dµ, f ∈ E, µ ∈ F , defines a bilinear map from E × F into X (note that we are only integrating simple functions so no elaborate integration theory is involved). If {Ej } ⊂ Σ P is pairwise disjoint, then the series j χEj is w(S(Σ), ca(Σ, X)) subseries

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P convergent. By Theorem 4.62 above, the series ∞ j=1 µ(Ej ) converge uniformly for µ belonging to any conditionally w(ca(Σ, X), S(Σ)) sequentially compact subset of ca(Σ, X). In particular, we have as a special case the Nikodym Convergence Theorem. Theorem 4.66. Let {µj } ⊂ ca(Σ, X) be such that limj µj (E) = µ(E) exists for every E ∈ Σ. Then {µj } is uniformly countably additive and µ ∈ ca(Σ, X). Proof: By the observation above, since {µj } is conditionally w(ca(Σ, X), S(Σ)) sequentially compact, {µj } is uniformly countably additive. That µ ∈ ca(Σ, X) then follows. From Theorem 4.63, we can also derive a result of Graves and Ruess ([GR]) Lemma 6). Theorem 4.67. If K ⊂ ca(Σ, X) is w(ca(Σ, X), S(Σ)) compact, then K is uniformly countably additive. Next, we derive a version of a theorem of Thomas ([Th]). Example 4.68. Let S be a sequentially compact Hausdorff space. Let E = SC(S, X) be the space of sequentially continuous functions from S into X and let F = span{δt : t ∈ S}, where δt is the Dirac measure concentrated at t. Then f · t = f (t) defines a bilinear mapping from E × F into X. Note that S is conditionally w(span{δt : t ∈ S}, SC(S, X)) sequentially compact since S is sequentially compact [here we are identifying t with δt ]. Thus, P from Theorem 4.62 above if λ has signed-WGHP and j fj is λ multiplier convergent in SC(S, X)with respect the topology of pointwise convergence P on S, then for each t ∈ λ the series j tj fj converges uniformly on S. Similarly, if S is compact, then S is w(span{δt : t ∈ S}, C(S, X)) compact P so from Theorem 4.63 if λ has signed-WGHP and the series j fj is λ multiplier convergent in C(S, X) with respect to the topology of pointwise P convergence on S, then for each t ∈ λ the series j tj fj converges uniformly on S. The subseries version of this result is due to Thomas ([Th]). We can also use Theorems 4.62 and 4.63 above to derive a version of the Orlicz-Pettis Theorem for continuous linear operators. Example 4.69. Let Z be an LCTVS. Set E = L(Z, X) and F = Z and define a bilinear mapping from E × F into X by T · x = T x. Then w(E, F ) is just the topology of pointwise convergence on Z or Ls (Z, X). If K ⊂ Z is sequentially compact (compact), then K is conditionally w(Z, L(Z, X))

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sequentially compact (w(Z, L(Z, X)) compact) so if K (C) denotes the set of all sequentially compact (compact) subsets of Z, from Theorem 4.62 (Theorem 4.63) above, we have P Theorem 4.70. Let λ have signed-WGHP. If j Tj is λ multiplier conP vergent in Ls (Z, X), then j Tj is λ multiplier convergent in LK (Z, X) (LC (Z, X)). We will obtain some similar results for operator valued series later in Chapter 6. An operator T ∈ L(Z, X) is completely continuous if T carries weakly convergent sequences into convergent sequences; denote all such operators by CC(Z, X). Note that if T is completely continuous, then T carries weak Cauchy sequences into Cauchy sequences. Now consider the abstract triple E = CC(Z, X), F = Z and the bilinear map · : E × F → X defined by · : (T, z) → T · z = T z. If a subset K ⊂ Z is conditionally weakly sequentially compact, then K is conditionally w(CC(Z, X), Z) sequentially compact. If CW denotes the set of all conditionally weakly sequentially compact subsets of Z, then from Theorem 4.62 we have P Theorem 4.71. Let λ have signed-WGHP. If the series j Tj is λ mulP tiplier convergent in CCs (Z, X), then j Tj is λ multiplier convergent in CCCW (Z, X). An operator T ∈ L(Z, X) is weakly compact if T carries bounded sets to relatively weakly compact sets; denote all such operators by W (Z, X). The space Z has the Dunford-Pettis property if every weakly compact operator from Z into any locally convex space X carries weak Cauchy sequences into convergent sequences. Consider the abstract triple E = W (Z, X), F = Z and the bilinear map · : E × F → X defined by · : (T, z) → T · z = T z. If K ⊂ Z is conditionally weakly sequentially compact and Z has the Dunford-Pettis property, then K is conditionally w(W (Z, X), Z) sequentially compact. If CW denotes the set of all conditionally weakly compact subsets of Z, then from Theorem 4.62 we have Theorem 4.72. Let λ have signed-WGHP and assume that Z has the P Dunford-Pettis property. If the series j Tj is λ multiplier convergent in P Ws (Z, X), then j Tj is λ multiplier convergent in WCW (Z, X).

A space Z is almost reflexive if every bounded sequence contains a weak Cauchy subsequence ([LW]). For example, Banach spaces with separable

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duals, quasi-reflexive Banach spaces and c0 (S) are almost reflexive ([LW]). If Z is almost reflexive and has the Dunford-Pettis property, then every bounded set is conditionally w(W (Z, X), Z) sequentially compact so from Theorem 4.62, we have Theorem 4.73. Let λ have signed-WGHP and assume that Z is almost P reflexive with the Dunford-Pettis property. If the series j Tj is λ mulP tiplier convergent in Ws (Z, X), then j Tj is λ multiplier convergent in Wb (Z, X). As another application of Theorem 4.63, we derive an Orlicz-Pettis result of Stiles for a locally convex TVS with a Schauder basis ([Sti]). Stiles’ version of the Orlicz-Pettis Theorem is for subseries convergent series with values in an F-space with a Schauder basis and his proof uses the metric properties of the space. Other proofs of Stiles’ result have been given in [Bs] and [Sw5]. We will establish a version of Stiles’ result for multiplier convergent series which requires no metrizability assumptions. Later in Chapter 9 we will establish a version of the result for non-locally convex spaces using the Antosik Interchange Theorem. Let X be an LCTVS with a Schauder basis {bj } and associated coordinate functionals {fj }. That is, every x ∈ X has a unique series repreP sentation x = ∞ j=1 tj bj and fj : X → R is defined by hfj , xi = tj . We do not assume that the coordinate functionals are continuous although this is the case when X is an F-space ([Sw2] 10.1.13). Define Pi : X → X by Pi Pi x = j=1 hfj , xi bj . Let E = X, F = span{Pi : i ∈ N} and let the bilinear mapping from E × F into X be the extension to F of the mapping x · Pi = Pi x. Let G = span{fi : i ∈ N}. P Theorem 4.74. Let λ have signed-WGHP. If j xj is λ multiplier conP vergent with respect to σ(X, G), then j xj is λ multiplier convergent with respect to the original topology of X. Proof: Since a sequence in X is σ(X, G) convergent iff the sequence is P w(E, F ) convergent, the series j xj is λ multiplier convergent with respect to w(E, F ). Now {Pi : i ∈ N} is conditionally w(F, E) sequentially compact since P x → x for every x ∈ X. By Theorem 4.62, for every t ∈ λ the series P∞ i N. Let U be a closed neighborhood j=1 tj Pi xj converge uniformly for i ∈ P∞ P∞ of 0 in X. There exists N such that j=m tj Pi xj = Pi ( j=m tj xj ) ∈ U P∞ for m ≥ N, i ∈ N. Let i → ∞ gives j=m tj xj ∈ U for m ≥ N . Note that we did not use the continuity of the coordinate functionals in the proof so the topology of X may not even be comparable to σ(X, G).

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We next consider a more general situation than that encountered in Stiles’ result. Assume that there exists a sequence of linear operators Pi : P∞ X → X such that for each x ∈ X, x = i=1 Pi x [convergence in X]. When each Pi is continuous, {Pi } is called a Schauder decomposition ([LT]). If X has a Schauder basis {bi } with coordinate functionals {fi }, then Pi x = hfi , xi bi is an example of this situation. Let E = X, F = span{Pi : i ∈ N} and let the bilinear mapping from E × F into X be the extension of the map x · Pi = Pi x. Theorem 4.75. Let λ have signed-WGHP and assume that each Pi is P w(E, F ) − X continuous. If the series x is λ multiplier convergent P j j with respect to w(E, F ), then the series j xj is λ multiplier convergent in X with respect to the original topology. Pn Proof: Define Sn : X → X by Sn = i=1 Pi . Then {Sn : n ∈ N} is conditionally w(F, E) sequentially compact so by Theorem 4.62 for each P∞ t ∈ λ the series j=1 tj Sn xj converge uniformly for n ∈ N. Let U be a P∞ closed neighborhood of 0 in X. There exists N such that j=m tj Sn xj = P∞ P∞ Sn ( j=m tj xj ) ∈ U for m ≥ N, n ∈ N. Letting n → ∞ gives j=m tj xj ∈ U for m ≥ N . We give an example where the theorem above is applicable. Example 4.76. Let Y be an LCTVS and let X be a vector space of Y valued sequences containing the space of sequences which are eventually 0. Then X is an AK-space if the coordinate functionals fj : X → Y , fj ({xj }) = xj are continuous for every j and each x = {xj } has a repreP j j sentation x = ∞ j=1 e ⊗ xj [Appendix C; here e ⊗ x denotes the sequence with x in the j th coordinate and 0 in the other coordinates]. The space X has the property (I) if the injections x → ej ⊗ x are continuous from Y into X. If Pj : X → X is defined by Pj ({xj }) = ej ⊗ xj , then {Pj } is a Schauder decomposition for X. If X has property (I), then the topology of coordinatewise convergence is equal to w(E, F ) so the result above applies and if λ has signed-WGHP, then any series which is λ multiplier convergent in the topology of coordinatewise convergence converges in the topology of X. For examples where the result above applies let Y be a normed space. If 1≤ p < ∞, then l p (Y ) and c0 (Y ) are AK-spaces satisfying the conditions in the example above. We will consider non-locally convex versions of these results later in Chapter 9.

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Finally, we show that the result in Theorem 2.26 can be obtained from Theorem 4.62. P Theorem 4.77. Let λ have signed-WGHP. Assume that j xij is λ mulP∞ tiplier convergent for every i ∈ N and that lim i j=1 tj xij exists for every t ∈ λ with xj = limi xij for every j. Then for every t ∈ λ the series P∞ j=1 tj xij converge uniformly for i ∈ N.

Proof: For every i ∈ N define a linear map fi : λ → X by fi (t) = P∞ j=1 tj xij and set F = span{fi : i ∈ N}. Consider the abstract triple E = λ, F and X and let the bilinear mapping from E × F into X be the extension of the map (t, fi ) → t · fi = fi (t). We first claim that the P j convergent with respect to w(E, F ). For if series j e is λ multiplier P P∞ P∞ ∞ j j t ∈ λ, t e · f = j i j=1 j=1 tj fi (e ) = j=1 tj xij converges for every i. Now {fi } is conditionally w(F, E) sequentially compact since {t · fi } = P∞ { j=1 tj xij } converges for every t ∈ λ. Theorem 4.62 implies that the P∞ P∞ series j=1 tj fi (ej ) = j=1 tj xij converge uniformly for i ∈ N. Recall that Theorem 4.77 (Theorem 2.26) and the convergence result in Lemma 2.27 were used to derive Stuart’s completeness result in Theorem 2.28. We can also derive a version of Kalton’s Theorem on subseries convergence in the space of compact operators. Let X and Y be normed spaces and let K(X, Y ) be the space of all compact operators from X into Y (an operator T ∈ L(X, Y ) is compact if T carries bounded sets into relatively compact sets). The space X has the DF property if every weak* subseries convergent series in X 0 is k·k subseries convergent ([DF]; Diestel and Faires have shown that for B-spaces this is equivalent to X 0 containing no subspace isomorphic to l∞ ). Theorem 4.78. Let X and Y be normed spaces and let X have the DF P property. If the series j Tj is subseries convergent in the weak operator topology of K(X, Y ), then the series is subseries convergent in the norm topology of K(X, Y ).

Proof: Each Tj has separable range so we may assume that Y is separable by replacing Y with span ∪∞ j=1 Tj X. By Lemma A.6 of Appendix

0 A or

T → 0. Lemma 4.4, it suffices to show that kT k → 0 or, equivalently, j

j

Pick yj0 ∈ Y 0 , yj0 = 1, such that Tj0 ≤ Tj0 yj0 + 1/j. By the separability of Y there exists a subsequence {yn0 j } which is weak* convergent to some y 0 ∈ Y 0 ; for convenience assume that the sequence {yj0 } is weak* convergent to y 0 . Consider the abstract triple E = {T 0 : T ∈

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K(X, Y )}, F = Y 0 and (X 0 , k·k) with the bilinear map E × F → (X 0 , k·k) P defined by (T 0 , y 0 ) → T 0 · y 0 = T 0 y 0 . For each z 0 ∈ Y 0 , the series j Tj0 z 0 is weak* subseries convergent in X 0 and is, therefore, subseries convergent in P (X 0 , k·k) by the DF property. Hence, the series j Tj0 is w(E, F ) subseries convergent. The sequence {yj0 } is w(F, E) relatively sequentially compact since k·k − lim T 0 yj0 = T 0 y 0 for every T ∈ K(X, Y ) ([DS] VI.5.6). By P∞ Theorem 4.62 the series j=1 Tj0 yi0 converge uniformly for i ∈ N. In particular, Tj0 yj0 → 0 so Tj0 = kTj k → 0 as desired. Kalton’s Theorem will also be considered in Chapter 6. We next show that the conclusions in Theorems 4.62 and 4.63 can be strengthened if the multiplier space has signed-SGHP instead of signedWGHP. P Theorem 4.79. Let Λ ⊂ λ have signed-SGHP. If j xj is λ multiplier convergent with respect to w(E, F ), then for each conditionally w(F, E) sequentially compact (w(F, E) compact, w(F, E) countably compact) subset P∞ K ⊂ F and each bounded subset B ⊂ Λ, the series j=1 tj xj · y converge uniformly for y ∈ K, t ∈ B. Proof: If the conclusion fails to hold, there exist a neighborhood, W , in X ,yk ∈ K, tk ∈ B and an increasing sequence of intervals {Ik } such that X (#) tkl xl · yk ∈ /W l∈Ik

for every k. We may assume, by passing to a subsequence if necessary, that limk x · yk exists for every x ∈ E. Consider the matrix X j M = [mij ] = [ tl xl · yi ]. l∈Ij

We claim that M is a signed K matrix as in Theorem 4.62 (Appendix D.3). First, the columns of M converge. Next given an increasing sequence of positive integers, there exist a sequence of signs {sj } and a subsequence P nk {nj } such that u = ∞ ∈ Λ. Then k=1 sk χInk t {

∞ X j=1

sj minj }i = {

∞ X j=1

sj

X

l∈Inj

n

t l j xl · y i } i = {

∞ X l=1

u l xl · y i } i

converges. Hence, M is a signed K matrix so the diagonal of M converges to 0 by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3). But, this contradicts (#).

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The proof of the statements in parentheses follow as in the proof of Theorem 4.63. If the multiplier space Λ ⊂ λ has signed-SGHP, then the conclusion of Corollary 4.10 can be improved (see Theorem 2.16). Corollary 4.80. Let Λ have signed-SGHP and let E, F be in duality. If P the series j xj is λ multiplier convergent with respect to σ(E, F ), then P∞ the series j=1 tj xj converge uniformly for t belonging to bounded subsets of Λ with respect to both λ(E, F ) and γ(E, F ). Corollary 4.80 covers the case of subseries convergent series (Λ = {χ σ : σ ⊂ N} ⊂ m0 = λ) and bounded multiplier convergent series (Λ the unit ball of l∞ ). Using Theorem 4.79 we can also obtain an improved conclusion in Theorem 4.66. In particular, if {Ej } is a pairwise disjoint sequence from Σ, then P the series ∞ χ (j)µi (Ej ) converge uniformly for i ∈ N, E ∈ Σ. That is, Pj=1 E the series ∞ j=1 µi (Ej ) are uniformly unordered convergent for i ∈ N. Similarly, we can obtain an improvement to the statements in Example 4.68 if the multiplier space λ has signed-SGHP. If λ has signed-SGHP and P the series j fj is λ multiplier convergent in SC(S, X) (C(S, X)) with respect to the topology of pointwise convergence on S, then the series P∞ j=1 tj fj (s) converge uniformly for s ∈ S and t belonging to bounded subsets of λ (Theorem 4.79). We can also obtain a strengthened version of the result given in Theorem 4.77. P Theorem 4.81. Let λ have signed-SGHP. Assume that j xij is λ mulP∞ tiplier convergent for each i ∈ N and that limi j=1 tj xij exists for each P∞ t ∈ λ with xj = limi xij for every j. Then the series j=1 tj xij converge uniformly for t belonging to bounded subsets of λ and i ∈ N, and the series P j xj is λ multiplier convergent. The proof of Theorem 4.77 carries forward using Theorem 4.79 in place of Theorem 4.62.

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Chapter 5

Orlicz-Pettis Theorems for the Strong Topology

In this chapter we consider Orlicz-Pettis Theorems for the strong topology. As the following example shows, in general, an Orlicz-Pettis Theorem does not hold for the strong topology. Recall that if X, X 0 is a pair of vector spaces in duality, the strong topology β(X, X 0 ) is the polar topology of uniform convergence on the family of σ(X 0 , X) bounded subsets of X 0 (Appendix A, Example A.2). As before, throughout this chapter λ will denote a scalar sequence space which contains c00 , the space of sequences which are eventually 0. P Example 5.1. The series j ej is subseries convergent in l ∞ with respect to the weak topology σ(l ∞ , l1 ) but is not subseries convergent in the strong topology β(l∞ , l1 ) = k·k∞ . In order to obtain an Orlicz-Pettis Theorem for the strong topology, we will impose stronger conditions on the multiplier space λ. Before proceeding in this direction, we use the Nikodym Boundedness Theorem to show that although a weak subseries convergent series may fail to be subseries convergent in the strong topology, the partial sums of the series are strongly bounded. P Theorem 5.2. Let X be an LCTVS. If j xj is σ(X, X 0 ) subseries conP P vergent, then P = { j∈σ xj : σ ⊂ N} is β(X, X 0 ) bounded, where j∈σ xj is the σ(X, X 0 ) sum of the series. Proof: Let P be the power set of N and define µ : P → X by µ(σ) = 0 0 0 j∈σ xj [ σ(X, X ) sum of the series ]. Let B ⊂ X be σ(X , X) bounded. 0 0 The family M = {x µ : x ∈ B} is a family of scalar valued, signed measures which is pointwise bounded on P since B is σ(X 0 , X) bounded. By the

P

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Nikodym Boundedness Theorem [Theorem 4.59], M is uniformly bounded on P. Since the range of µ is P , P is β(X, X 0 ) bounded. A similar result for barrelled AB-spaces λ is given in Corollary 2.6. First, we recall an Orlicz-Pettis result relative to the strong topology which was established in Corollary 4.6. Corollary 5.3. Let λ have signed-WGHP. Let X, X 0 be in duality. If P (X, β(X, X 0 )) is separable, then any series j xj which is λ multiplier convergent with respect to the weak topology σ(X, X 0 ) is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Example 5.1, where X = l ∞ and X 0 = l1 , shows that the separability condition in Corollary 5.3 is important. We next establish an Orlicz-Pettis Theorem for the strong topology which requires strong topological assumptions on the multiplier space. Theorem 5.4. Assume that λ is a barrelled AK-space and X is an LCTVS. P multiplier convergent with respect to the weak topology If j xj is λ P σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Proof: By Proposition 2.5, λ0 = λβ so the original topology of λ is β(λ, λβ ) and λ is an AK-space with respect to β(λ, λβ ). Since the strong topology is a Hellinger-Toeplitz topology, the result follows from Theorem 4.5. Example 5.1 where λ = m0 shows that the AK assumption in Theorem 5.4 is important. Although the assumptions on the multiplier space in Theorem 5.4 are quite restrictive, the result covers a large number of multiplier spaces. Example 5.5. If λ is a Banach [Frechet] AK-space, Theorem 5.4 applies. For example, λ = c0 , lp (1 ≤ p < ∞), cs or bv0 are Banach AK-spaces. Likewise, if λ = s or if λ is a K¨ othe echelon space ([K1] 30.8), then λ is a Frechet AK-space. The spaces (l p , k·k1 ), 0 < p < 1, are barrelled AKspaces ([Be]) so Theorem 5.4 applies. Examples of barrelled subspaces of (l1 , k·k1 ) are given in [RS] and more examples of barrelled AK-spaces are given in [BK].

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Whereas the assumptions on the multiplier space in Theorem 5.4 are topological in nature, we now consider a gliding hump assumption which is purely algebraic. We recall a gliding hump property which will be employed (Appendix B.36) Definition 5.6. The space λ has the infinite gliding hump property (∞-GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinate P sum ∞ j=1 apj χIpj t ∈ λ. The term ”infinite gliding hump” is used to suggest that the ”humps”, χIpj t, are multiplied by a sequence of scalars which converges to ∞; there are other gliding hump properties where the humps are multiplied by elements of classical sequence spaces [see Appendix B for the µ-gliding hump property]. Examples of multiplier spaces with ∞-GHP are given in Appendix B. For example, λ = l p , 0 < p < ∞, or λ = cs have ∞-GHP. We now establish an Orlicz-Pettis Theorem for spaces with ∞-GHP.

P Theorem 5.7. Let λ have ∞-GHP and let X be an LCTVS. If j xj is λ P multiplier convergent with respect to the weak topology σ(X, X 0 ), then j xj is λ multiplier convergent with respect to the strong topology β(X, X 0 ). Proof: If the conclusion fails to hold, there exist t ∈ λ, a σ(X 0 , X) bounded subset B ⊂ X 0 and > 0 such that for every k there exist P ∞ x0k ∈ B, mk > k such that j=mk hx0k , tj xj i > 2. For k = 1, let 0 x condition. There exists n1 > m1 such that 1 , m1 satisfy the previous P ∞ 0 j=n1 +1 hx1 , tj xj i < . Then

X X ∞ ∞ X n1 0 0 0 hx1 , tj xj i − hx1 , tj xj i > . hx1 , tj xj i = j=m1 j=m1 j=n1 +1

Continuing this construction produces {x0k } ⊂ B, increasing sequences {mk }, {nk } with mk < nk < mk+1 satisfying nk X (∗) tj hx0k , xj i > . j=mk

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Set Ik = [mk , nk ]. Since λ has ∞-GHP, there exists {pj }, apj > 0, apj → ∞ such that every subsequence of {pj } has a further subsequence {qj } such P∞ that the coordinate sum j=1 aqj χIqj t ∈ λ. Define matrix M = [mij ] = [

X

l∈Ipj

apj tl hx0i /api , xl i].

We claim that M is a K-matrix [Appendix D.2]. First, the columns of M converge to 0 since {x0i } is σ(X 0 , X) bounded and api → ∞. Next, given a subsequence there exists a further subsequence {qj } such that s = P∞ j=1 aqj χIqj t ∈ λ. Then + * ∞ ∞ ∞ X X X X 0 0 sl hxi /api , xl i = xi /aqi , sl xl → 0, miqj = j=1

j=1 l∈Iqj

l=1

P∞ where l=1 sl xl is the σ(X, X 0 ) sum of the series. Thus, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem [Appendix D.2], the diagonal of M converges to 0. But, this contradicts (∗) and establishes the result. 1/k

1/k

The spaces d = {t : supk |tk | < ∞} and δ = {t : lim |tk | = 0} furnish examples of spaces to which Theorem 5.7 applies but Theorem 5.4 does not [the natural metric on d does not give a vector topology ([KG] p.68)]. Diestel and Faires have established an interesting Orlicz-Pettis Theorem for the weak* topology on the dual of a Banach space. They have shown that if X is a Banach space such that X 0 contains no subspace isomorphic to P P l∞ and if j x0j is weak* subseries convergent, then j x0j is norm subseries convergent ([DU] I.4.7). We give statements of Theorems 5.4 and 5.7 for the weak* topology where the emphasis is on the multiplier space λ instead of topological assumptions on the dual space as in the Diestel-Faires Theorem. Corollary 5.8. Assume that λ either has ∞-GHP or is a barrelled AKP 0 xj is λ multiplier convergent in the space. Let X be an LCTVS. If jP weak* topology σ(X 0 , X) of X 0 , then j x0j is λ multiplier convergent with respect to the strong topology β(X 0 , X). Recall that if X is a Banach space, then the strong topology β(X 0 , X) is just the dual norm topology so Corollary 5.8 can be compared to the Diestel-Faires result in this case.

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Finally, we observe that the proof of Theorem 5.7 shows directly that the strong topology β(X, X 0 ) is weaker than the Dierolf topology Dλ (X, X 0 ) [Definition 4.38]. Theorem 5.7 will then follow from Theorem 4.41. Theorem 5.9. Let λ have ∞-GHP and let X be an LCTVS. Then the strong topology β(X, X 0 ) is weaker than the Dierolf topology Dλ (X, X 0 ). Proof: Let B be σ(X 0 , X) bounded. If B does not belong to Dλ , there P exists t ∈ λ and a λ multiplier convergent series j xj such that the series P∞ 0 0 j=1 tj hx , xj i do not converge uniformly for x ∈ B [Remark 4.40]. The proof of Theorem 5.7 then yields the result. Theorem 5.9 and Theorem 4.41 then yield Theorem 5.7 as a corollary and furnishes an alternate proof of Theorem 5.7.

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Chapter 6

Orlicz-Pettis Theorems for Linear Operators

In this chapter we consider multiplier convergent series of continuous linear operators and establish Orlicz-Pettis Theorems for such series. Throughout this chapter let X and Y 6= {0} be LCTVS and L(X, Y ) the space of continuous linear operators from X into Y . We first describe the topologies on L(X, Y ) which will be considered. Let A be a family of bounded subsets of X whose union is all of X and let Y be the family of all continuous semi-norms on Y . The pair (A, Y) generate a locally convex topology on L(X, Y ) defined by the family of semi-norms (1) pA,q (T ) = sup{q(T x) : x ∈ A}, q ∈ Y, A ∈ A. We denote by LA (X, Y ) the locally convex topology on L(X, Y ) generated by the semi-norms in (1); this notation suppresses the dependence of the topology on the semi-norms in Y . A net {Tδ } in L(X, Y ) converges to 0 in LA (X, Y ) iff for every A ∈ A lim Tδ x = 0 uniformly for x ∈ A; for this reason the topology LA (X, Y ) is called the topology of uniform convergence on A (Appendix A). If A is the family of all bounded subsets of X, the topology LA (X, Y ) is denoted by Lb (X, Y ). In the case when X and Y are normed spaces, the topology Lb (X, Y ) is called the uniform operator topology and is generated by the semi-norm kT k = sup{kT xk : kxk ≤ 1}. If A is the family of all finite subsets of X, the topology LA (X, Y ) is denoted by Ls (X, Y ) and is just the topology of pointwise convergence on X. When Y has its original topology, the topology Ls (X, Y ) is called the strong operator topology of L(X, Y ). When Y has the weak topology 89

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σ(Y, Y 0 ), the topology Ls (X, Y ) is called the weak operator topology of L(X, Y ). Thus, a net {Tδ } converges to 0 in the strong operator topology (weak operator topology) iff Tδ x → 0 in Y for every x ∈ X ( hy 0 , Tδ xi → 0 for every x ∈ X, y 0 ∈ Y 0 ). See Appendix A for more details. Throughout this chapter let λ be a sequence space containing c00 , the space of all sequences which are eventually 0. We begin by establishing an Orlicz-Pettis Theorem for the weak and strong operator topologies. Recall that λ has the signed weak gliding hump property (signed-WGHP) if for every t ∈ λ and every increasing sequence of intervals {Ij } there exist a sequence of signs {sj } and a subsequence {nj } such that the coordinate P sum ∞ j=1 sj χInj t ∈ λ; if the signs can all be chosen equal to 1, then λ has the weak gliding hump property (WGHP). Examples of spaces with these properties are given in Appendix B. For example, any monotone space has WGHP while bs has signed-WGHP but not WGHP. P Theorem 6.1. Let λ have signed-WGHP. If the series j Tj is λ multiplier convergent in L(X, Y ) with respect to the weak operator topology, then P the series j Tj is λ multiplier convergent in L(X, Y ) with respect to the strong operator topology. P Proof: For every x ∈ X the series j Tj x is λ multiplier convergent in Y with respect to σ(Y, Y 0 ). By the Orlicz-Pettis Theorem in Corollary P 4.10, the series j Tj x is λ multiplier convergent in Y with respect to the P original topology of Y . That is, the series j Tj is λ multiplier convergent in the strong operator topology. Since subseries convergence is just m0 multiplier convergence as a special case of Theorem 6.1, we have P Corollary 6.2. If the series j Tj is subseries convergent with respect to the weak operator topology of L(X, Y ), then the series is subseries convergent with respect to the strong operator topology. If the multiplier space λ does not satisfy the signed-WGHP, then the conclusion of Theorem 6.1 may fail. Example 6.3. Let λ = cc , the space of all sequences which are eventually P constant (Appendix B). Then a series j zj is λ multiplier convergent in P a TVS Z iff the series j zj converges in Z. Define continuous linear operators Tj : R → c0 by Tj s = s(ej+1 − ej ) = −s ej − ej+1 . Then P∞ for every s ∈ R, the series j=1 Tj s converges in σ(c0 , l1 ) but does not

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P converge in (c0 , k·k∞ ). That is, the series j Tj is λ multiplier convergent in the weak operator topology of L(R, c0 ) but is not λ multiplier convergent in the strong operator topology of L(R, c0 ). First, we have the analogue of Lemma A.6 of Appendix A for λ multiplier convergent series. Lemma 6.4. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. Let λ have signed-WGHP. Suppose that P every series j xj which is λ multiplier convergent with respect to σ is such that for every t ∈ λ and increasing sequence of intervals {Ik } there is P a subsequence {nk } such that τ − lim j∈In tj xj = 0. Then every series k P j xj which is λ multiplier convergent with respect to σ is also λ multiplier convergent with respect to τ . Proof: By Lemma A.4, it suffices to show that if t ∈ λ, then the partial P sums of the series j tj xj are τ Cauchy. Suppose the partial sums of P the series j tj xj are not τ Cauchy. Then there exist a symmetric τ neighborhood of 0,U, and an increasing sequence of intervals {Ik } such that X (∗) t j xj ∈ / U. j∈Ik

Since λ has signed-WGHP, there exist a sequence of signs {sk } and a P∞ subsequence {nk } such that u = k=1 sk χInk t [coordinate sum] belongs P∞ P P∞ to λ. The series j=1 uj xj = k=1 sk j∈In tj xj is σ convergent so by k P hypothesis, {nk } has a subsequence {pk } such that τ −lim j∈Ip tj xj = 0. k But, this contradicts (∗). We now use Lemma 6.4 to establish a general Orlicz-Pettis Theorem for linear operators. Let C = {C ⊂ X : if {xj } ⊂ C, then lim T xj exists for every T ∈ L(X, Y )}. P Theorem 6.5. Let λ have signed-WGHP. If j Tj is λ multiplier converP gent in L(X, Y ) with respect to the weak operator topology, then j Tj is λ multiplier convergent in LC (X, Y ). Proof: By Lemma 6.4 it suffices to show that if t ∈ λ and {Ik } is an increasing sequence of intervals and {xi } ∈ C, then (#) lim( k

X

l∈Ik

tl Tl )(xi ) = 0 uniformly for i ∈ N.

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Define the matrix M = [mij ] = [

X

(tl Tl )(xi )].

l∈Ij

We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge since {xi } ∈ C. Next, if {pj } is an increasing sequence of integers, then there exist a subsequence {qk } of {pk } and a sequence of signs {sk } such that the coordinate sum of the series u = P∞ k=1 sk χIqk t belongs to λ. Then ∞ X j=1

sj miqj =

∞ X j=1

sj

X

t l T l xi = (

l∈Iqj

∞ X

ul Tl )xi

l=1

P∞ converges as i → ∞, where l=1 ul Tl ∈ L(X, Y ) is the weak operator sum of the series. Hence, M is a signed K-matrix. By the signed version of the Antosik-Mikusinski Matrix Theorem the condition (#) is satisfied [Appendix D.3]. We use Theorem 6.5 to establish Orlicz-Pettis Theorems for two of the common topologies employed on L(X, Y ). If A is the family of all sequences in X which converge to 0, the topology LA (X, Y ) is denoted by L→0 (X, Y )[Appendix A]; this topology is studied in [GDS]. From Theorem 6.5, we have P Corollary 6.6. Let λ have signed-WGHP. If conj Tj is λ multiplier P vergent in L(X, Y ) with respect to the weak operator topology, then j Tj is λ multiplier convergent in L→0 (X, Y ). Proof: If xj → 0 in X, then {xj } ∈ C. We next consider the topology of uniform convergence on precompact subsets. If A is the family of all precompact subsets of X, the topology LA (X, Y ) is denoted by Lpc (X, Y ). In order to establish an Orlicz-Pettis Theorem for Lpc (X, Y ), we require the following representation theorem for precompact sets. Proposition 6.7. Let Z be a dense subspace of X and assume that X is metrizable. If K is a precompact subset of X, then there exists a null seP∞ quence {xk } ⊂ Z such that every x ∈ K has a representation x = j=1 tj xj P∞ with j=1 |tj | ≤ 1.

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Proof: Let k·k1 ≤ k·k2 ≤ ... be a sequence of semi-norms which generate the topology of X. Since Z is dense and K is precompact, there is a finite set F1 ⊂ Z such that for every x ∈ K there exists z 1 (x) ∈ F1 such that

x − z 1 (x) ≤ (1/2)(1/23). 1

Now K − F1 is precompact so there is a finite set F2 ⊂ Z such that for every x ∈ K there exists z 2 (x) ∈ F2 such that

x − z 1 (x) − z 2 (x) ≤ (1/3)(1/24). 2

Continuing this construction produces a sequence of finite subsets F1 , F2 , ... of Z such that every x ∈ K there exist z i (x) ∈ Fi satisfying

(1) x − z 1 (x) − ... − z i (x) i ≤ (1/(i + 1))(1/2i+2 ).

Therefore,

(2) z i (x) i−1

≤ x − z 1 (x) − ... − z i (x) i−1 + x − z 1 (x) − ... − z i−1 (x) i−1 ≤ (1/i)(1/2i )

Set y i (x) = 2i z i (x) for x ∈ K, i ∈ N. Arrange the elements of 2F1 , 22 F2 , ... in a sequence with the elements of F1 first, those of F2 second and so on. By (2) this sequence converges to 0 and by (1) x = z 1 (x) + z 2 (x) + ... = (1/2)y 1 (x) + (1/22 )y 2 (x) + ... This gives the desired representation. Theorem 6.8. Let λ have signed-WGHP and let X be metrizable or the P regular strict inductive limit of a sequence of metrizable LCTVS. If j Tj is λ multiplier convergent in L(X, Y ) with respect to the weak operator P topology, then j Tj is λ multiplier convergent in Lpc (X, Y ).

Proof: First suppose that X is metrizable and let K ⊂ X be precompact. By Proposition 6.7 there exists {xj } ⊂ K, xj → 0 such that every P∞ P∞ x ∈ K has a representation x = j=1 tj xj with j=1 |tj | ≤ 1. Let U be a closed, absolutely convex neighborhood of 0 in Y and let s ∈ λ. By P Corollary 6.6 there exists N such that nj=m sj Tj xk ∈ U for n > m ≥ N and k ∈ N. If x ∈ K has the representation above, then for n > m ≥ N we have n ∞ ∞ n n X X X X X s j T j xk ∈ U tk t k xk = s j Tj s j Tj x = j=m

j=m

k=1

k=1

j=m

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P since U is closed and absolutely convex. By Theorem 6.1, ∞ j=m sj Tj x ∈ U for m ≥ N and x ∈ K. This establishes the result in the metrizable case. If X is a regular strict inductive limit of metrizable spaces {Xk } and K ⊂ X is precompact, then K ⊂ Xk for some k and K is precompact in Xk . Thus, the first part gives the result in this case. A similar result was derived in Theorem 4.70. We next consider Orlicz-Pettis Theorems for the topology Lb (X, Y ). As was the case for the strong topology β(X, X 0 ) considered in Chapter 5 such results require strong hypotheses on the multiplier space λ. We present an example which shows that, in general, a series which is subseries convergent in the strong operator topology may not be subseries convergent in Lb (X, Y ). The example also suggests that to obtain Orlicz-Pettis theorems connecting the weak (strong) operator topology to the topology of Lb (X, Y ), one should consider the space K(X, Y ) of compact operators from X to Y . Example 6.9. Let X be a Banach space with an unconditional Schauder basis {bj }, i.e., every x ∈ X has a unique expansion ∞ X tj b j , x= j=1

where the series is unconditionally (subseries) convergent. Let fj be the j th coordinate functional associated with the basis {bj } defined by hfj , xi = tj in the expansion above. Each fj is linear and continuous ([Sw2] 10.1.13). Let Pj x = hfj , xi bj . If T ∈ L(X, Y ), where Y is a Banach space, then P∞ P∞ T x = j=1 hfj , xi T bj = j=1 T Pj x, where the series is norm convergent P∞ in Y . That is, the series j=1 T Pj is subseries convergent in the strong operator topology of L(X, Y ) to T . If L(X, Y ) has the property that any series which is subseries convergent in the strong operator topology (or weak operator topology) is subseries convergent in the norm topology of L(X, Y ), it follows that every T ∈ L(X, Y ) is compact, being the norm limit Pn of a sequence of compact operators, { j=1 T Pj }, with finite dimensional range. That is, if L(X, Y ) has this property, then L(X, Y ) = K(X, Y ), where K(X, Y ) is the space of all compact operators from X into Y . In particular, if X = Y , then the identity operator is compact and X must be finite dimensional ([Sw2] 7.8).

Before considering Orlicz-Pettis theorems for Lb (X, Y ), we observe a necessary condition for such a result to hold for subseries convergent series of operators.

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P Definition 6.10. The space X has the DF property iff every series j x0j in X 0 which is subseries convergent in σ(X 0 , X) is also subseries convergent in β(X 0 , X). For Banach spaces, Diestel and Faires have shown that X 0 has the DF property iff X 0 contains no subspace isomorphic to l ∞ ([DF]). P Theorem 6.11. If every series j Tj in L(X, Y ) which is subseries convergent in the weak operator topology is also subseries convergent in Lb (X, Y ), then X has the DF property. P 0 0 Proof: Let j xj be subseries convergent in X with respect

0 to 0 σ(X , X). Pick y ∈ Y, y 6= 0. Define Tj ∈ L(X, Y ) by Tj x = xj , x y. P Then so by j Tj is subseries convergent in the strong operator topology P∞

hypothesis the series converges in Lb (X, Y ). Thus, the series j=1 x0j , x converge uniformly for x belonging to bounded subsets of X. That is, the P series j x0j converges in (X 0 , β(X 0 , X)).

For our first Orlicz-Pettis Theorem for Lb (X, Y ), we establish the analogue of Theorem 5.4 for operators. P Theorem 6.12. Let λ be a barrelled AK space. If the series j Tj is λ P multiplier convergent in the weak operator topology of L(X, Y ), then j Tj is λ multiplier convergent in Lb (X, Y ). Proof: If x ∈ X and y 0 ∈ Y 0 , define a continuous linear functional x ⊗ y 0 on L(X, Y ) by hx ⊗ y 0 , T i = hy 0 , T xi. Let X ⊗ Y 0 = span{x ⊗ y 0 : x ∈ X, y 0 ∈ Y 0 }. Note that the weak operator topology on L(X, Y ) is just P σ(L(X, Y ), X ⊗ Y 0 ). From Theorem 5.4 it follows that the series j Tj is λ multiplier convergent in the strong topology β(L(X, Y ), X ⊗ Y 0 ). Thus, it suffices to show that the strong topology β(L(X, Y ), X ⊗ Y 0 ) is stronger than Lb (X, Y ). Let {Sδ } be a net in L(X, Y ) which converges to 0 in β(L(X, Y ), X ⊗ Y 0 ). Let A ⊂ X be bounded, B ⊂ Y 0 be equicontinuous and set C = {x ⊗ y 0 : x ∈ A, y 0 ∈ B}. Since A is β(X, X 0 ) bounded from the barrelledness assumption, sup{|hx ⊗ y 0 , T i| : x ∈ A, y 0 ∈ B} < ∞

for every T ∈ L(X, Y ); that is, C is σ(X ⊗ Y 0 , L(X, Y )) bounded. Thus, sup{|hx ⊗ y 0 , Sδ i| : x ∈ A, y 0 ∈ B} → 0

so Sδ → 0 in Y uniformly for x ∈ A or Sδ → 0 in Lb (X, Y ). Thus, β(L(X, Y ), X ⊗ Y 0 ) is stronger than Lb (X, Y ) as desired.

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An extensive list of multiplier spaces satisfying the assumptions of Theorem 6.12 is given in Remark 4.7. We next establish the analogue of Theorem 5.7 for linear operators. P Theorem 6.13. Let λ have ∞-GHP. If the series j Tj is λ multiplier P convergent in the strong operator topology, then j Tj is λ multiplier convergent in Lb (X, Y ). Proof: If the conclusion fails to hold, there exist > 0, A ⊂ X bounded,t ∈ λ, a continuous semi-norm p on Y and sequences {mk }, {nk } with m1 < n1 < m2 < ... and nk nk X X sup p( tl Tl x) = pA ( tl Tl ) > . x∈A

l=mk

l=mk

For every k there exists xk ∈ A such that (∗) p(

nk X

tl Tl xk ) > .

l=mk

Set Ik = [mk , nk ]. Since λ has ∞-GHP, there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such P∞ that s = k=1 aqk χIqk t ∈ λ. Define a matrix X (tl apj )Tl (xi /api )]. M = [mij ] = [ l∈Ij

We claim that M is a K-matrix [Appendix D.2]. First, the columns of M converge to 0 since xi /api → 0 and each Tl is continuous. Next, given a subsequence there is a further subsequence {qk } such that s = P∞ P∞ l=1 sl Tl converges in the strong operator k=1 aqk χIqk t ∈ λ. The series topology to an operator T ∈ L(X, Y ). Hence, ∞ X j=1

miqj =

∞ X X

j=1 l∈Iqj

sl Tl (xi /api ) = T (xi /api ) → 0.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem [Appendix D.2], the diagonal of M converges to 0. But, this contradicts (∗). Remark 6.14. If λ also has signed-WGHP in Theorem 6.13, we may reP place the assumption that the series j Tj is λ multiplier convergent in the strong operator topology with the assumption that the series is λ multiplier convergent in the weak operator topology (Theorem 6.1).

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Example 6.9 suggests that if one wishes to establish Orlicz-Pettis theorems for λ multiplier convergent series with respect to Lb (X, Y ) without imposing strong conditions on the multiplier space λ, one should consider the space K(X, Y ) of compact operators. The major result in this area is a result of Kalton. Kalton has shown that if X has the DF property (or, P if X 0 contains no subspace isomorphic to l ∞ ) and if j Tj is a series of compact operators from a Banach space X into a Banach space Y which is subseries convergent in the weak operator topology of K(X, Y ), then the P series j Tj is subseries convergent in the uniform operator topology of K(X, Y ) ([Ka]). We first establish a result of Wu and Lu which characterizes the OrliczPettis property for the space of compact operators ([WL]). Their result contains Kalton’s result as a special case. Let Kb (X, Y ) be the topology on K(X, Y ) induced by Lb (X, Y ). Theorem 6.15. Let λ have signed-WGHP. The following are equivalent: P (i) Every series j Tj which is λ multiplier convergent in the weak operator topology of K(X, Y ) is λ multiplier convergent in Kb (X, Y ). (ii) Every continuous linear operator S : X → (λβ , σ(λβ , λ)) is sequentially compact (an operator is sequentially compact if it carries bounded sets into relatively sequentially compact sets). P Proof: Suppose (ii) holds. Let j Tj be λ multiplier convergent in the weak operator topology of K(X, Y ). By Theorem 6.1 the series is λ multiplier convergent in the strong operator topology of K(X, Y ). SupP pose there exists t ∈ λ such that the series j tj Tj is not convergent in Kb (X, Y ). Then there exist T ∈ K(X, Y ) and a bounded set A ⊂ X P∞ such that j=1 tj Tj x = T x for every x ∈ X but the series do not converge uniformly for x ∈ A. Thus, there exist a continuous semi-norm p on Y , increasing sequences {mk } and {nk } with mk < nk < mk+1 ,xk ∈ A and > 0 such that ! nk X p t l T l xk > l=mk

for all k. By the Hahn-Banach Theorem there is a sequence {yk0 } ⊂ Y 0 such that + * nk X t l T l xk > (∗) yk0 , l=mk

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and sup{|hyk0 , yi| : p(y) ≤ 1} ≤ 1. Let Y0 be the closure in Y of span{Ti xj : i, j ∈ N}. Then (Y0 , p) is a separable semi-norm space. By the Banach-Alaoglu Theorem for separable semi-norm spaces, {yk0 } has a subsequence {yn0 k } and y 0 ∈ Y 0 such that

lim yn0 k , y = hy 0 , yi for every y ∈ Y0 and

sup{|hy 0 , yi| : p(y) ≤ 1} ≤ 1.

For notational convenience, assume that nk = k. Define a semi-norm q on X 0 by q(x0 ) = sup{|hx0 , xk i| : k ∈ N}. We claim that if U ∈ K(X, Y ) satisfies U xk ∈ Y0 , then (∗∗)

lim q(U 0 yk0 − U 0 y 0 ) = 0.

If (∗∗) fails to hold, there exist δ > 0, a subsequence {yn0 k } and a subsequence {xnk } such that

(∗ ∗ ∗) U 0 yn0 k − U 0 y 0 , xnk > δ.

Since U is compact, {U xnk } is a relatively compact subset of Y and, therefore, a relatively compact subset of (Y0 , p). Without loss of generality, we may assume that there exists y ∈ Y0 such that p(U xnk − y) → 0. Then 0

yn − y 0 , U xn ≤ yn0 − y 0 , U xn − y + yn0 − y 0 , y k k k k k

≤ sup{ yn0 k − y 0 , z : p(z) ≤ 1}p(U xnk − y)

+ yn0 k − y 0 , y

≤ 2p(U xn − y) + y 0 − y 0 , y → 0. nk

k

This contradicts (∗ ∗ ∗) and establishes the claim. P∞ If s ∈ λ, x ∈ X and z 0 ∈ Y 0 , the series j=1 sj hz 0 , Tj xi converges so we define a linear operator S(= Sz0 ) : X → (λβ , σ(λβ , λ)) by Sx = {hz 0 , Tj xi}. Since S is obviously continuous, S is sequentially compact by condition (ii). Thus, SA is sequentially compact with respect to σ(λβ , λ). By Corollary P 2.29, if s ∈ λ, then the series ∞ s hz 0 , T xi converge uniformly for x ∈ A P∞j=1 j 0 0 j or, equivalently, the series j=1 sj Tj z , x converge uniformly for x ∈ A. Now consider the matrix nj X M = [mij ] = tl Tl0 yi0 . l=mj

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We show that M is a signed K-matrix with values in the semi-norm space (X 0 , q) [Appendix D.3]. First, the columns of M converge by condition (∗∗). Next, given any subsequence {pj } there is a further subsequence P∞ Pnql {qj } and a sequence of signs {j } such that s = l=1 l j=m tj ∈ λ. P∞ P∞ Pnqqll There exist U ∈ K(X, Y ) such that s T = j=mql tj Tj j=1 j j l=1 l converges to U in the strong operator topology. By the paragraph above P∞ P n ql 0 0 0 0 j=mql tj Tj yk converges to U yk uniformly for x ∈ A. In particul=1 l lar, n ql ∞ X X q l tj Tj0 yk0 − U 0 yk0 → 0. l=1

j=mql

Thus, M is a signed K-matrix (with respect to (X 0 , q). By the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0 in (X 0 , q). This contradicts (∗) and establishes that (ii) implies (i). Suppose that (i) holds. Let S : X → (λβ , σ(λβ , λ)) be linear and continuous. So Sx = {Sx · ej }. Let y ∈ Y, y 6= 0. Define Tj ∈ K(X, Y ) by Tj x = (Sx · ej )y. Let t ∈ λ. Define T (= Tt ) ∈ K(X, Y ) by T x = (Sx · t)y. P∞ P Then j=1 tj Tj x = T x for every x ∈ X, i.e., the series j tj Tj converges to T in the strong operator topology of K(X, Y ). By (i) the series ∞ X j=1

t j Tj x =

∞ X j=1

tj (Sx · ej )y = T x = (Sx · t)y

converge uniformly for x belonging to bounded subsets of X or the series P∞ j j=1 tj (Sx · e ) = Sx · t converge uniformly for x belonging to bounded subsets of X. Now to show S is sequentially compact, let {xk } be a bounded sequence in X. Then {Sxk } is coordinatewise bounded in λβ since S is bounded. By the diagonal method ([DeS] 26.10), there is a subsequence {nk } such that limk Sxnk · ej exists for every j and since the series P∞ j j=1 tj (Sxnk · e ) converge uniformly for k ∈ N, limk t · Sxnk exists. Thus, β {Sxnk } is σ(λ , λ) Cauchy. By Corollary 2.28 there exists u ∈ λβ such that Sxnk → u in σ(λβ , λ). Therefore, {Sxnk } is relatively sequentially compact in (λβ , σ(λβ , λ)). Remark 6.16. Wu Junde has shown that subsets of λβ are σ(λβ , λ) sequentially compact iff they are σ(λβ , λ) compact so condition (ii) of Theorem 6.15 can be replaced with the hypothesis that the operator S is compact ([Wu]).

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For the case of subseries convergent series, that is, when λ = m0 , we have Theorem 6.17. Let X be a barrelled LCTVS. The following are equivalent: P (i) Every series j Tj which is subseries convergent in the weak operator topology of K(X, Y ) is subseries convergent in Kb (X, Y ). (ii) Every continuous linear operator S : X → (l 1 , σ(l1 , m0 )) is compact. (iii) Every continuous linear operator S : X → (l 1 , k·k1 ) is compact. (iv) (X 0 , β(X 0 , X)) contains no subspace isomorphic to c0 . (v) X has the DF property. Proof: Since subsets of l 1 are σ(l1 , m0 ) [k· k1 ] sequentially compact iff they are compact (Proposition 4.18), (i), (ii) and (iii) are equivalent by Theorem 6.15. Since X is barrelled (iii), (iv) and (v) are equivalent by Theorem 3.20. Remark 6.18. If X and Y are Banach spaces, the equivalence of (i) and (v) is Kalton’s result except that Kalton uses the hypothesis that X 0 contains no subspace isomorphic to l ∞ which is equivalent to the DF property by the Diestel/Faires result ([DF]). For Banach spaces the equivalence of (i) and (iv) was established by Bu and Wu ([BW]). We used the abstract set-up preceding Theorem 4.62 to establish a version of Kalton’s theorem for normed spaces (Theorem 4.78) which we restate. Theorem 6.19. Let X and Y be normed spaces and let X have the DF P property. If the series j Tj is subseries convergent in the weak operator topology of K(X, Y ), then the series is subseries convergent in the norm topology of K(X, Y ).

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Chapter 7

The Hahn-Schur Theorem

In this chapter we establish vector versions of the classical Hahn-Schur Theorem for multiplier convergent series. For later reference, we first give a statement of one version of the Schur Theorem for absolutely convergent scalar valued series. Theorem 7.1. (Schur) For each i ∈ N, let vergent series of scalars. Assume

P

j tij

be an absolutely con-

P ∞ (h) limi ∞ and let limi tij = tj for every j=1 sj tij exists for every {sj } ∈ l j ∈ N. Then P∞ P∞ is absolutely convergent and limi j=1 sj tij = j=1 sj tj for every {sj } ∈ l∞ , P∞ (ii) limi j=1 |tij − tj | = 0, P∞ (iii) the series j=1 |tij | converge uniformly for i ∈ N. (i)

P

j tj

The statement in Theorem 7.1 is often referred to as the Schur Lemma ([Sr]). In particular, Theorem 7.1 implies that a sequence in l 1 which is weak (σ(l1 , l∞ )) Cauchy is norm (k·k1 ) convergent. The Hahn version of Theorem 7.1 relaxes the hypothesis (h) and retains conclusions (ii) and (iii) with a slight restatement of condition (i). Again for later reference, we give a statement of Hahn’s theorem ([Ha]). Theorem 7.2. (Hahn) For each i ∈ N, vergent series of scalars. Assume 101

let

P

j tij

be an absolutely con-

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P (h’) limi j∈σ tij exists for every σ ⊂ N and let limi tij = tj for every j ∈ N. Then P P P (i) j tj is absolutely convergent and limi j∈σ tij = j∈σ tj for every σ ⊂ N, (ii) and (iii). In particular, the Hahn version in Theorem 7.2 implies that a sequence in l1 which is σ(l1 , m0 ) Cauchy is k·k1 convergent ([Ha]). Hahn’s theorem can also be used to establish an important result from summability. Namely, (S) A matrix T = [tij ] maps the sequence space m0 into the space c of convergent sequences iff the matrix T maps l ∞ into c. ([Sw2] 9.5.3; see Theorem 7.29 for a vector version of statement (S)). Both the Schur and Hahn theorems have numerous applications to various topics in analysis; in particular, Schur’s theorem was used in the proofs of the versions of the Orlicz-Pettis Theorem given by both Orlicz and Pettis ([Or], [Pe]). The conclusions in (ii) and (iii) (as well as those in the hypothesis and condition (i)) involve absolutely convergent series and, therefore, do not represent interesting suggestions for vector valued generalizations of either Theorem 7.1 or Theorem 7.2. However, conditions (ii) and (iii) can be restated in forms which do not involve absolute convergence and which do suggest possible vector valued generalizations. We first give a restatement of conditions (ii) and (iii) for hypothesis (h). P Proposition 7.3. Let j tij be absolutely convergent for every i ∈ N. Condition (ii) is equivalent to: P P∞ (ii)’ limi ∞ j=1 sj tij = j=1 sj tj uniformly for k{sj }k∞ ≤ 1.

Condition (iii) is equivalent to: P∞ (iii)’ the series j=1 sj tij converge uniformly for i ∈ N and k{sj }k∞ ≤ 1. Proof: Assume (ii). If {sj } ∈ l∞ , then ∞ ∞ X X s (t − t ) ≤ k{s }k |tij − tj | j ij j j ∞ j=1 j=1

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so (ii)’ follows immediately. P∞ Assume (ii)’. Fix i ∈ N. Define sj = sign(tij − tj ). Then j=1 sj (tij − P∞ tj ) = j=1 |tij − tj |. Since k{sj }k∞ ≤ 1, (ii) follows. The equivalence of (iii) and (iii)’ are established similarly. We next give a restatement of conditions (ii) and (iii) for hypothesis (h)’. For this we use Lemma 3.37. Proposition 7.4. Let

P

j tij

be absolutely convergent for every i ∈ N.

Condition (ii) is equivalent to: Proposition 7.5. P P (ii)” limi j∈σ tij = j∈σ tj uniformly for σ ⊂ N. Condition (iii) is equivalent to:

(iii)” for P every > 0 there exist N such that σ ⊂ N, min σ ≥ N implies j∈σ tij < for all i ∈ N.

Proof: Clearly (ii) implies (ii)” and (iii) implies (iii)”. that i ≥ N implies P Assume (ii)” holds. Let > 0. There exists N such P∞ (t − t ) < /2 for σ ⊂ N. By Lemma 3.37, j∈σ ij j j=1 |tij − tj | ≤ for i ≥ N . Thus, (ii) holds. Similarly, (iii)” implies (iii). The hypothesis (h) [(h)’] and conclusions (ii)’ and (iii)’ [(ii)” and (iii)”] suggest generalizations of the Schur and Hahn theorems for multiplier convergent series with values in a TVS. Hypothesis (h) and conditions (ii)’ and (iii)’ [(h)’ and conditions (ii)” and (iii)”] use bounded multiplier convergent series [subseries convergent series] or Λ multiplier convergent series where Λ = l∞ [where Λ = {χσ : σ ⊂ N}]. Thus, it would be natural to seek versions of Theorems 7.1 and 7.2 for Λ multiplier convergent series. We pursue these versions in this chapter. Theorems 7.1 and 7.2 will follow directly from our general results for multiplier convergent series. Let λ be a scalar sequence space which contains c00 , the space of sequences which are eventually 0 and let Λ ⊂ λ. Let X be a TVS. The analogue of hypotheses (h) and (h)’ for Λ multiplier convergent series would then be:

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P • (H) Let Λ multiplier convergent in X for every i ∈ N. j xij be P∞ Assume that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N. The analogues of conditions (ii)’ [(ii)”], and (iii)’ [(iii)”] for multiplier convergent series require topological assumptions on the multiplier space Λ. Assume that λ is a K-space. The analogues of conditions (i), (ii)’ [(ii)”], and (iii)’ [(iii)”] for multiplier convergent series would then be: P∞ P • (C1) the series j xj is Λ multiplier convergent and limi j=1 tj xij = P∞ every t ∈ Λ. j=1 tj xj for P∞ P∞ • (C2) limi j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ. P∞ • (C3) the series j=1 tj xij converge uniformly for t belonging to bounded subsets of Λ. We first consider conclusion (C1). Under the hypothesis (H) conclusion (C1) follows from Lemma 2.27 and Theorem 2.26 preceding Stuart’s weak completeness result. Recall that Λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ Λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence P∞ {nj } such that the coordinate sum of the series j=1 sj χInj t belongs to Λ [Appendix B.6]. Theorem 7.6. Assume that Λ has signed-WGHP. Then condition P (H) Let ij be Λ multiplier convergent in X for every i ∈ N. Assume jx P ∞ that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N implies the conclusion P P (C1) the series xj is Λ multiplier convergent and limi ∞ j j=1 tj xij = P∞ t x for every t ∈ Λ. j=1 j j The result in Theorem 7.5 may fail if λ does not have signed-WGHP.

Example 7.7. Let λ = c. Define xij by xij = 1 if i = j and xij = 0 P∞ otherwise. If t ∈ c, then limi j=1 tj xij = limi ti and xj = limi xij = 0. P∞ P∞ But, limi j=1 tj xij = limi ti 6= j=1 tj xj = 0 if limi ti 6= 0.

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This example shows that if {ej } ⊂ l1 , then {ej } is σ(l1 , c) Cauchy but does not have a σ(l 1 , c) limit. That is, σ(l1 , c) is not sequentially complete. We next consider conclusions (C2) and (C3). This will require stronger assumptions on the multiplier space λ. Let λ be a K-space and let Λ ⊂ λ. Recall that Λ has the signed strong gliding hump property (signed-SGHP) if whenever {tk } ⊂ Λ is bounded and {Ik } is an increasing sequence of intervals, there exist a sequence of signs {sk } and a subsequence {nk } such P∞ that the coordinate sum of the series k=1 sk χInk tnk belongs to Λ. If all of the signs can be chosen equal to 1, then Λ is said to have the strong gliding hump property (SGHP)(Appendix B.17). For example, the space l∞ has SGHP while the subset Λ = M0 = {χσ : σ ⊂ N} ⊂ m0 has SGHP but the space m0 does not have SGHP; the space bs of bounded series has signed-SGHP but not SGHP [see Appendix B for these and additional examples]. We first establish a lemma which is a special case of condition (C2). The proof of the lemma uses a property of TVS which we now establish. Lemma 7.8. Let X be a TVS. If lim xj = 0 in X, then lim txj = 0 uniformly for |t| ≤ 1. Proof: Let U be a balanced neighborhood of 0 in X. There exists N such that k ≥ N implies that xk ∈ U . Therefore, if k ≥ N and |t| ≤ 1, txk ∈ tU = U . Remark 7.9. There is another proof of Lemma 7 in Yosida ([Y] I.2.2) which uses Egoroff’s Theorem and properties of Lebesgue measure and one in [Sw1] 8.2.4 which uses the Banach-Steinhaus Theorem. Lemma 7.10. Assume that Λ ⊂ λ and Λ has signed-SGHP and B ⊂ Λ P is bounded. If j xij is Λ multiplier convergent for every i ∈ N, P∞ limi j=1 tj xij = 0 for every {tj } ∈ Λ and limi xij = 0 for every j, then P∞ limi j=1 tj xij = 0 uniformly for t ∈ B.

P∞ Proof: It suffices to show that limi j=1 tij xij = 0 for any sequence {ti } ⊂ B. Let U be a neighborhood of 0 in X and pick a symmetric neighborhood of 0,V , in X such that V +V +V ⊂ U . Set n1 = 1 and pick N1 P n1 i such that ∞ j=N1 tj xn1 j ∈ V . Since limi xij = 0 for every j and {tj : i ∈ N} i is bounded from the K-space assumption, limi tj xij = 0 for every j by PN1 −1 i Lemma 7.7. Therefore, there exists n2 > n1 such that j=1 tj xij ∈ V for P∞ every i ≥ n2 . Pick N2 > N1 such that j=N2 tnj 2 xn2 j ∈ V . Continuing

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this construction produces increasing sequences {nk }, {Nk } such that ∞ X

l=Nj

Nj −1 n

X

tl j xnj l ∈ V and

l=1

ti xil ∈ V f or i ≥ nj .

Set Ij = {l : Nj−1 ≤ l < Nj }. Define a matrix X n M = [mij ] = [ tl j xni l ]. l∈Ij

We show that M is a signed K-matrix (Appendix D.3). First, the columns of M converge to 0 since limi xil = 0 for every l. Given an increasing sequence {pj }, there is a sequence of signs {sj } and a subsequence P∞ q {qj } of {pj } such that t = j=1 sj χIqj tj j ∈ Λ (coordinate sum). Then ∞ X

sj miqj =

j=1

∞ X

sj

j=1

X

q

t l j xn i l =

∞ X

t j xn i j

j=1

l∈Iqj

P∞ and j=1 tj xni j → 0 by hypothesis. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.3). Thus, there exists N such that mii ∈ V for i ≥ N . If i ≥ N, then ∞ X

Ni−1 −1

tnl i xni l =

l=1

X l=1

P∞

tnl i xni l +

X

l∈Ii

tnl i xni l +

∞ X

l=Ni

tnl i xni l ∈ V + V + V ⊂ U

so limi l=1 tnl i xni l = 0. Since the same argument can be applied to any P∞ subsequence, it follows that limi j=1 tij xij = 0.

We can now establish the result with conclusions (C2) and (C3) under hypothesis (H).

Theorem 7.11. Assume that Λ ⊂ λ and Λ has signed-SGHP. If P (H) Let ij be Λ multiplier convergent in X for every i ∈ N. Assume jx P ∞ that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N, then the following conclusions hold: P P∞ (C1) the series j xj is Λ multiplier convergent and limi j=1 tj xij = P∞ t x for every t ∈ Λ, j j j=1

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P P∞ (C2) limi ∞ j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ and P∞ (C3) the series j=1 tj xij converge uniformly for t belonging to bounded subsets of Λ Proof: Since Λ has signed-WGHP, conclusion (C1) holds by Theorem 7.5. P∞ Since limi j=1 tj (xij − xj ) = 0 by conclusion (C1), Lemma 7.9 now applies and gives conclusion (C2) immediately. Suppose that (C3) fails to hold. Then there exist a closed, symmetric neighborhood of 0,U , in X and a bounded set B ⊂ Λ such that for every P i i there exist ki > i, ni > i, ti ∈ B with ∞ / U . There exists j=ni tj xki j ∈ Pmi i P mi > ni such that j=ni tj xki j ∈ / U . Set I = [ni , mi ] so j∈Ii tij xki j ∈ / U. By the condition above for i1 = 1, there exist k1 , a finite interval I1 P with min I1 > i1 , t1 ∈ B with j∈I1 t1j xk1 j ∈ / U . By Theorem 2.35, there P∞ exists j1 such that k=j tk xik ∈ U for every t ∈ B, 1 ≤ i ≤ k1 , j ≥ j1 . Set i2 = max{I1 + 1, j1 }. Again by the condition above there exist k2 > i2 , P / U. a finite interval I2 with min I2 > i2 , t2 ∈ B such that k∈I2 t2k xk2 k ∈ Note that k2 > k1 by the definition of i2 . Continuing this construction produces an increasing sequence {ki }, an increasing sequence of intervals {Ii } and {ti } ⊂ B such that X tik xki k ∈ / U. (∗) k∈Ii

Define a matrix M = [mij ] = [

X

tjk xki k ].

k∈Ij

We claim that M is a signed K-matrix (Appendix D.3). First, the columns of M converge by hypothesis. Next, given any increasing sequence {pj }, there exist a sequence of signs {sj } and a subsequence {qj } of {pj } P∞ q such that the coordinate sum t = j=1 sj χIqj tj j ∈ Λ. Then the sequence ∞ X j=1

sj miqj =

∞ X j=1

sj

X

l∈Iqj

q

t l j xk i l =

∞ X

t j xk i j

j=1

converges by hypothesis. Hence, M is a signed K-matrix so by the signed version of the Antosik-Mikusinski Matrix Theorem (Appendix D.3), the diagonal of M converges to 0. But, this contradicts (∗).

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If the multiplier space Λ does not have signed-SGHP, then the hypothesis (H) may not imply (C2) and (C3) even if Λ has WGHP or 0-GHP. Example 7.12. Let 1 ≤ p < ∞ and λ = Λ = l p . Define xij = ej if P 1 ≤ j ≤ i and xij = 0 if j > i. Then j xij is λ multiplier convergent for every i, limi xij = ej = xj for every j and lim i

∞ X

tj xij = lim i

j=1

i X

tj e j =

j=1

∞ X

tj e j =

j=1

∞ X

t j xj

j=1

in lp for every t ∈ l p so (H) holds and (C1) holds. However, both (C2) and P∞ (C3) fail to hold. [Take tk = ek so {tk } is bounded in lp but j=1 tkj xij = Pi k j k j=1 tj e = e if i ≥ k so (C2) and (C3) fail.]

We next consider the hypothesis (H) as a conclusion. P Proposition 7.13. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. P (1) If for every t ∈ Λ the series ∞ j xij converge uniformly for i ∈ N, j=1 tP then for every t ∈ Λ the sequence { ∞ j=1 tj xij }i is Cauchy. P∞ (2) If the series j=1 tj xij converge uniformly for i ∈ N and t belonging P∞ to bounded subsets of Λ, then the sequences { j=1 tj xij }i satisfy a Cauchy condition uniformly for t belonging to bounded subsets of Λ. Proof: (1): Let t ∈ Λ. Let U be a neighborhood of 0 in X. Pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists P∞ N such that n ≥ N implies j=n tj xij ∈ V for every i ∈ N. There exists PN n > N such that i, k ≥ n implies j=1 tj (xij − xkj ) ∈ V . If i, k ≥ n, then (∗)

∞ X j=1

N X j=1

tj (xkj − xij ) +

∞ X

j=N +1

tj xkj −

tj xkj −

∞ X

tj xij =

j=1

∞ X

j=N +1

tj xij ∈ V + V + V ⊂ U.

(2): Let B ⊂ Λ be bounded. Let U be a neighborhood of 0 in X. Pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists P∞ N such that n ≥ N implies j=n tj xij ∈ V for every i ∈ N, t ∈ B. By the K-space assumption {tj : t ∈ B} is bounded for every j. By Lemma 7.7 PN there exists n > N such that j=1 tj (xij − xkj ) ∈ V for every t ∈ B. If i, k ≥ n, then (∗) holds.

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Proposition 7.14. Assume that λ is an AB-space (Appendix B.3). Let P be Λ multiplier convergent for every i ∈ N and assume that j xij P∞ limi xij = xj exists for every j ∈ N. If limi j=1 tj xij exists for every P∞ t ∈ Λ and the series j=1 tj xij converge uniformly for i ∈ N and t beP longing to bounded subsets of Λ, then j xj is Λ multiplier convergent and P∞ P∞ limi j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ. P Proof: Let t ∈ Λ. Put z = limi ∞ j=1 tj xij . Let U be a neighborhood of 0 in X. Pick a closed, symmetric neighborhood of 0, V , such that V + V + P V ⊂ U . There exists k such that ∞ j=1 tj xkj − z ∈ V and by Proposition 7.12(2) since V is closed and {Pm t : m ∈ N} is bounded by the ABPm Pm assumption (Pm is the section map Pm t = j=1 tj ej ), j=1 tj (xj − xkj ) ∈ P∞ V for every m. There exists M such that m ≥ M implies j=m+1 tj xkj ∈ V . Then if m ≥ M , we have m X j=1

tj xj −z =

∞ X j=1

tj xkj −z +

m X j=1

tj (xj −xkj )−

∞ X

j=m+1

tj xkj ∈ V +V +V ⊂ U.

P∞ Pm Hence, z = limm j=1 tj xj = j=1 tj xj . P∞ Let B ⊂ Λ be bounded. Since j=1 tj xij converges uniformly for i ∈ P∞ N, t ∈ B, the series j=1 tj xij satisfy a Cauchy condition uniformly for i ∈ N, t ∈ B. Therefore, there exists N such that n > m ≥ N implies that Pn P∞ j=m tj xij ∈ V for i ∈ N, t ∈ B. Hence, j=m tj xj ∈ V for m ≥ N, t ∈ B. Since {tj : t ∈ B} is bounded for every j, by Lemma 7.7 there exists M P such that N j=1 tj (xij − xj ) ∈ V for i ≥ M, t ∈ B. If i ≥ M , then ∞ X j=1

tj (xij −xj ) =

N X j=1

tj (xij −xj )+

∞ X

j=N +1

tj xij −

∞ X

j=N +1

tj xj ∈ V +V +V ⊂ U

for every t ∈ B. Corollary 7.15. Let Λ have signed-WGHP and let X be sequentially comP plete. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. The following are equivalent: P∞ (i) limi j=1 tj xij exists for every t ∈ Λ, P∞ P∞ P and limi j=1 tj xij = j=1 tj xj , (ii) j xj is Λ multiplier convergent P∞ (iii) for every t ∈ Λ the series j=1 tj xij converge uniformly for i ∈ N.

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Proof: That (ii) implies (i) is clear; (iii) implies (i) by Proposition 7.12(1); (i) implies (ii) by Lemma 2.27; (ii) implies (iii) by Theorem 2.26. Corollary 7.16. Let Λ have signed-SGHP and let X be sequentially comP plete. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. The following are equivalent: P (I) limi ∞ j=1 tj xij exists for every t ∈ Λ, P P P∞ x is Λ multiplier convergent and limi ∞ (II) j j j=1 tj xij = j=1 tj xj uniformly for t belonging to bounded subsets of Λ, P∞ (III) the series converge uniformly for i ∈ N, t belonging to j=1 tj xij bounded subsets of Λ. P∞ (IV) for every t ∈ Λ the series j=1 tj xij converge uniformly for i ∈ N. Proof: Clearly (II) implies (I); (I) implies (II) and (III) by the HahnSchur Theorem 7.10; that (III) implies (IV) is clear; (IV) implies (I) by Proposition 7.12(2).

We also obtain a boundedness result. P Proposition 7.17. Let j xij be Λ multiplier convergent for every i ∈ N and assume that limi xij = xj exists for every j ∈ N. If B ⊂ Λ is bounded P∞ and the series j=1 tj xij converge uniformly for i ∈ N, t ∈ B, then ∞ X S= tj xij : i ∈ N, t ∈ B j=1

is bounded.

Proof: Let U be a balanced neighborhood of 0 in X and pick a balanced neighborhood of 0, V , such that V + V ⊂ U . There exists N such that P∞ for i ∈ N, t ∈ B. Since {xij : i ∈ N} and {tj : t ∈ B} j=N +1 tj xij ∈ V P are bounded for every j, there exists t > 1 such that { N j=1 tj xij : i ∈ N, t ∈ B} ⊂ tV . Hence, S ⊂ V + tV ⊂ tU and S is bounded. Since both M0 = {χσ : σ ⊂ N} ⊂ m0 = spanM0 and l∞ have SGHP, the previous results hold for both M0 multiplier (=subseries) convergent series and l∞ multiplier (=bounded multiplier) convergent series. We record these special cases for the previous results. We begin with the subseries case. From Theorem 7.10, we have

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P Theorem 7.18. Let j xij be subseries convergent for every i ∈ N. AsP sume that limi j∈σ xij exists for every σ ⊂ N and limi xij = xj for every j. Then P (1) j is subseries convergent, j xP P (2) limi j∈σ xij = j∈σ xj uniformly for σ ⊂ N and P (3) the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N. Note that the scalar case of Theorem 7.17 gives the scalar version of the Hahn Theorem stated in Theorem 7.2. From Proposition 7.12, we have P Proposition 7.19. Let j xij be subseries convergent for every i ∈ N and limi xij = xj for every j. P (1) If for every σ ⊂ N the series j∈σ xij converge uniformly for i ∈ N, P then for every σ ⊂ N the sequence { j∈σ xij }i is Cauchy. P (2) If the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, then the P sequences { j∈σ xij }i satisfy a Cauchy condition uniformly for σ ⊂ N. From Corollary 7.15, we have P Corollary 7.20. Let X be sequentially complete. Let j xij be subseries convergent for every i ∈ N and limi xij = xj for every j. The following are equivalent: P (1) limi j∈σ xij exists for every σ ⊂ N, P P P (2) j xj is subseries convergent and lim i j∈σ xij = j∈σ xj uniformly for σ ⊂ N, P (3) the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, P (4) for every σ ⊂ N the series j∈σ xij converge uniformly for i ∈ N. From Proposition 7.16, we have P Proposition 7.21. Let j xij be subseries convergent for every i ∈ N and P limi xij = xj for every j. If the series j∈σ xij converge uniformly for i ∈ N, σ ⊂ N, then X S= xij : i ∈ N, σ ⊂ N j∈σ

is bounded.

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We now show that Theorem 7.17 can be improved for sequentially complete spaces by replacing the family of all subsets of N by a smaller family of subsets. Recall that a family of subsets F of N is an FQσ family if F contains the finite subsets and whenever {Ik } is a pairwise disjoint sequence of finite subsets there is a subsequence {Ink } such that ∪∞ k=1 Ink ∈ F (see Appendix B for examples). If Λ = {χσ : σ ∈ F} ⊂ m0 where F is an FQσ family, then Λ has SGHP (Appendix B) so Theorem 7.10 applies. We first establish a lemma. Lemma 7.22. Let X be sequentially complete and F be an FQσ family P with Λ = {χσ : σ ∈ F} ⊂ m0 . If j xj is Λ multiplier convergent in X, P then j xj is subseries convergent. P∞ Proof: Let {nj } be a subsequence and σ = {nj : j ∈ N}. If j=1 xnj does not converge, there exist a neighborhood of 0, U , in X and an increasP P ing sequence of intervals {Ik } such that j∈Ik xnj = j∈Ik ∩σ xj ∈ / U . By the FQσ property there is a subsequence {mk } such that I = ∪∞ I k=1 mk ∩σ ∈ P F. But, then j∈I xj does not converge since the series fails the Cauchy condition. We now give the improvement of Theorem 7.17 for sequentially complete spaces. Theorem 7.23. Let X be sequentially complete and F be an FQσ famP ily with Λ = {χσ : σ ∈ F} ⊂ m0 . Assume that j xij is Λ multiplier P convergent for every i and that limi j∈σ xij exists for every σ ∈ F with P xj = limi xij for every j. Then limi j∈σ xij exists for every σ ⊂ N so conclusions (1),(2) and (3) of Theorem 7.17 hold. P Proof: We first claim that for each σ ⊂ N, the series j∈σ xj converges. P∞ We show that j=1 xj converges; the same argument can be applied to any P subseries j∈σ xj . Let U be a neighborhood of 0 in X and pick a closed, symmetric neighborhood of 0, V , such that V +V ⊂ U . By (C2) of Theorem P 7.10 there exists n such that j∈σ (xij − xj ) ∈ V for i ≥ n, σ ∈ F. By Pl Lemma 7.21 there exists m such that j=k xnj ∈ V for l ≥ k ≥ m. If l ≥ k ≥ m, then l X j=1

so

P∞

j=1

xj −

k X j=1

xj =

l X

j=k+1

(xj − xnj ) +

l X

j=k+1

xnj ∈ V + V ⊂ U

xj converges by the sequential completeness of X.

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P Since j∈σ (xij − xj ) ∈ V for i ≥ n and finite σ, Lemma 7.21 gives that j∈σ (xij − xj ) ∈ V for i ≥ n and any σ ⊂ N. Therefore, the hypothesis of Theorem 7.17 holds and (1), (2) and (3) of Theorem 7.17 follow.

P

We next give a scalar corollary of Theorem 7.22 due to Samaratanga and Sember ([SaSe]) which will be used later. Corollary 7.24. Let F be an FQσ family with Λ = {χσ : σ ∈ F} ⊂ m0 and set λ = spanΛ. Assume that ti ∈ l1 and limi t i · s exists for each s ∈ λ with tj = limi tij . Then t = {tj } ∈ l1 and ti − t 1 → 0. In particular, if ti → 0 in σ(l1 , λ), then ti → 0. 1

We next show that we can relax the hypothesis in Theorem 7.22 and retain part of the conclusion of the theorem. Recall that a family F of subsets of N is an IQσ family if F contains the finite subsets of N and whenever {Ik } is an increasing sequence of intervals there is a subsequence {Ink } such that ∪∞ k=1 Ink ∈ F (see Appendix B for examples). If F is an IQσ family and Λ = {χσ : σ ∈ F} ⊂ m0 , then Λ has SGHP so Theorem 7.10 applies and gives Theorem 7.25. Let F be an IQσ family which contains N, let Λ = {χσ : P σ ∈ F} ⊂ m0 and assume that j xij is Λ multiplier convergent for every P P i. If limi j∈σ xij exists for every σ ∈ F and xj = limi xij , then j xj is P∞ P∞ P∞ Λ multiplier convergent and limi j=1 xij = j=1 xj . [Note that j=1 xij converges since N ∈ F.] We use this result later in Chapter 9 when we consider iterated series. For the bounded multiplier case we have the following result as a special case of Theorem 7.10. P Theorem 7.26. Let be bounded multiplier convergent for every j xij P ∞ i ∈ N. Assume that limi ∞ t and limi xij = xj j=1 j xij exists for every t ∈ l for every j. Then P (1) j is bounded multiplier convergent, j xP P∞ ∞ (2) limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k∞ ≤ 1 and P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1.

Note that the scalar version of Theorem 7.25 gives the scalar version of the Schur Theorem stated in Theorem 7.1. From Proposition 7.12, we have

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P Proposition 7.27. Let j xij be bounded multiplier convergent for every i ∈ N and limi xij = xj for every j. P (1) If for every t ∈ l ∞ , the series ∞ j xij converge uniformly for i ∈ N, j=1 tP ∞ then for every t ∈ l the sequence { ∞ j=1 tj xij }i is Cauchy. P∞ t x converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, (2) If the series j ij j=1 P ∞ then the sequences { j=1 tj xij }i satisfy a Cauchy condition uniformly for k{tj }k∞ ≤ 1. From Corollary 7.15, we have P Corollary 7.28. Let X be sequentially complete. Let j xij be bounded multiplier convergent for every i ∈ N and limi xij = xj for every j. The following are equivalent: P ∞ (1) limi ∞ j=1 tj xij exists for every t ∈ l , P∞ P (2) = j xj is bounded multiplier convergent and limi j=1 tj xij P∞ t x uniformly for k{t }k ≤ 1, j j j j=1 ∞ P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, P ∞ (4) for every t ∈ l ∞ the series j=1 tj xij converge uniformly for i ∈ N. From Proposition 7.16, we have P Proposition 7.29. Let j xij be bounded multiplier convergent for every P∞ i ∈ N and limi xij = xj for every j. If the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1, then ∞ X S= tj xij : i ∈ N, k{tj }k∞ ≤ 1 j=1

is bounded.

In a sequentially complete LCTVS we can obtain a stronger conclusion in Theorem 7.17 for the subseries convergent version of the Hahn-Schur Theorem. Recall that the inequality of McArthur/Rutherford (Lemma 2.53) implies that a series in a sequentially complete LCTVS is subseries convergent iff the series is bounded multiplier convergent (Theorem 2.54). Also, from the inequality, we obtain Theorem 7.30. Let X be a sequentially complete LCTVS. Assume that P subseries convergent for every j and limi xij = xj exists for every j xij is P j. If limi j∈σ xij exists for every σ ⊂ N, then

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P (1) j is bounded multiplier convergent, j xP P∞ ∞ (2) limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k∞ ≤ 1 and P∞ (3) the series j=1 tj xij converge uniformly for i ∈ N, k{tj }k∞ ≤ 1.

Proof: The first statement in (1) follows from Theorems 7.17(1) and 2.54. For (2) let p be a continuous semi-norm on X. By Theorem 7.17(2), for P > 0 there exists n such that p( j∈σ (xij − xj )) < for i ≥ N, σ ⊂ N. If {tj } ∈ l∞ and i ≥ n, then from Lemma 2.53 for σ ⊂ N finite X X p( tj (xij − xj )) ≤ 2 k{tj }k∞ sup p( (xij − xj )) ≤ k{tj }k∞ . σ 0 ⊂σ

j∈σ

j∈σ 0

Therefore, if i ≥ n, then p(

∞ X j=1

tj (xij − xj )) ≤ k{tj }k∞

and (2) follows. (3) follows from the McArthur/Rutherford inequality in a similar fashion. Theorem 7.29 gives a generalization of the summability result stated in (S) following Theorem 7.2. Namely, we have: (S)’ If X is sequentially complete, the vector valued matrix [xij ] maps m0 into c(X), the space of X valued convergent sequences, iff [xij ] maps l∞ into c(X). Corollaries 7.19 and 7.27 give necessary and sufficient conditions for a vector valued matrix [xij ] to map m0 or l∞ into c(X) analogous to the scalar case (see [Sw2] 9.5.3). We next consider a generalization of the compactness result in Theorem 2.45. For this we require a preliminary lemma. Lemma 7.31. Let S be a compact Hausdorff space and gi : S → X continuous functions for i = 0, 1, 2, .... Suppose that lim gi (t) = g0 (t) uniformly for t ∈ S. Then R = ∪∞ i=0 Rgi is compact, where Rgi is the range of gi . Proof: Let G be an open cover of R. For each x ∈ R there exists Ux ∈ G such that x ∈ Ux . Then −x + Ux is an open neighborhood of 0 so there is an open neighborhood of 0, Vx , such that Vx + Vx ⊂ −x + Ux . Then G 0 = {x + Vx : x ∈ R} is an open cover of R.

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Since Rg0 is compact, there exist finite x1 + Vx1 , ..., xk + Vxk covering Rg0 . Put V = ∩kj=1 Vxj so V is an open neighborhood of 0. There exists n such that gi (t) − g0 (t) ∈ V for i ≥ n, t ∈ S. For t ∈ S there exists j such that g0 (t) ∈ xj + Vxj so gi (t) ∈ g0 (t) + V ⊂ xj + Vxj + Vxj ⊂ Uxj for i ≥ n. Hence, Ux1 , ..., Uxk covers ∪∞ i=n Rgi . Since Rgi , i = 0, ...n − 1, are compact, a finite subcover of G covers the union of these sets, and, hence, G has a finite subcover covering R. From Theorem 7.10, Corollary 2.43 and Lemma 7.30, we have Theorem 7.32. Assume that Λ ⊂ λ is bounded and has signed-SGHP and is compact with respect to p, the topology of pointwise convergence on Λ. If P x be Λ multiplier convergent in X for every i ∈ N. Assume (H) Let P∞ j ij that limi j=1 tj xij exists for every t ∈ Λ and assume that limi xij = xj exists for every j ∈ N, P∞ P∞ then B = { j=1 tj xij : i ∈ N, t ∈ Λ} ∪ { j=1 tj xj : t ∈ Λ} is compact.

P Proof: Let Si (S0 ) be the summing operator with respect to j xij ( j xj ). If (C2) holds, then Si and S0 are continuous with respect to p and the topology of X (Corollary 2.43). If (C3) holds, then Si → S0 uniformly on Λ so it follows from Lemma 7.30 that B is compact. P

In particular, if Λ = {t ∈ l ∞ : k{tj }k∞ ≤ 1} ⊂ λ = l∞ or if Λ = {χσ : σ ⊂ N} ⊂ λ = m0 , Theorem 7.31 applies. We next consider another property of the multiplier space which implies vector versions of the Hahn-Schur Theorem. Definition 7.33. Let λ be a K-space. The multiplier space λ has the Hahn-Schur property if si ∈ λβ and si · t → 0 for every t ∈ λ implies that lim si · t = 0 uniformly for t belonging to bounded subsets of λ. From Corollary 7.23 we have the following example of a multiplier space with the Hahn-Schur property. Example 7.34. Let F be an FQσ family with Λ = {χσ : σ ∈ F} ⊂ m0 and set λ = spanΛ. Then λ with the sup-norm has the Hahn-Schur property. For multiplier spaces with the Hahn-Schur property we have a vector Hahn-Schur Theorem.

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Theorem 7.35. Assume that λ has the Hahn-Schur property. If {xk } ⊂ λβX is such that lim xk · t = 0 for every t ∈ λ, then lim xk · t = 0 uniformly for t belonging to bounded subsets of λ. Proof: Let B ⊂ λ be bounded. Suppose that the conclusion of the theorem fails for B. Then there exist δ > 0, a continuous semi-norm p on X, tk ∈ B and an increasing sequence {nk } such that p(xnk · tk ) > δ.

By the Hahn-Banach Theorem, for every k there exist x0k ∈ X 0 such that sup{|hx0k , xi| : p(x) ≤ 1} ≤ 1 and * + ∞ X 0 k nk (∗) xk , t j xj > δ. j=1

Since p(

1,

P∞

nk j=1 tj xj )

→ 0 for every t ∈ λ and sup{|hx0k , xi| : p(x) ≤ 1} ≤ lim k

∞ X j=1

tj x0k , xnj k = 0

uniformly for t belonging to bounded subsets of λ by the Hahn-Schur property. This contradicts (∗). Corollary 7.36. Assume that λ has the Hahn-Schur property, (λβX , ω(λβX , λ)) is sequentially complete and {xk } ⊂ λβX . If lim xk · t exists for every t ∈ λ and xj = limk xkj , then x ∈ λβX and lim xk · t = x · t uniformly for t belonging to bounded subsets of λ. Proof: By the sequential completeness assumption, there exists y ∈ λβX such that xk → y in ω(λβX , λ). Since xkj → yj for every j, y = {yj } = {xj } = x. If t ∈ λ, then xk · t → x · t so the result follows from Theorem 7.34. We now consider some Hahn-Schur type results for normed spaces due to the Spanish school in Cadiz ([AP1])). Let X be a Banach space. The space X is a Grothendieck space if every weak* convergent sequence in X 0 is weakly convergent. For example, the space l∞ is a Grothendieck space by Phillips’ Lemma (Lemma 7.52 or [Sw2] 15.16). Definition 7.37. Let M be a subspace of X 0 such that X ⊂ M ⊂ X 00 . Then X is an M Grothendieck space if every σ(X 0 , X) convergent sequence in X 0 is σ(X 0 , M ) convergent.

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Thus, X is a Grothendieck space iff X is an X 00 Grothendieck space. There are closed subspaces λ of l ∞ with λ 6= l∞ such that λ is a Grothendieck space not containing a subspace isomorphic to l ∞ . For example, if H is the algebra of Hayden (Appendix B.21), and if H is the Stone space of H , then C(H) can be isometrically identified with a closed subspace of l∞ such that c0 ⊂ C(H) and which is a Grothendieck space not containing a copy of l ∞ (see [AP1]). When X is a Banach space and c0 ⊂ λ ⊂ l∞ , we define a norm on λβX by 0

kxk = sup{kt · xk : t ∈ λ, ktk∞ ≤ 1}. 0

Note that k·k is finite by Proposition 3.8 since for any x = {xi } ∈ λβX , P the series j xj is c0 multiplier convergent. Note that we also have 0

kxk = sup{kt · xk : t ∈ c0 , ktk∞ ≤ 1}

= sup{kt · xk : t ∈ c00 , ktk∞ ≤ 1} = sup{

∞ X j=1

|hx0 , xj i| : kx0 k ≤ 1}.

Theorem 7.38. Let λ be a subspace of l ∞ containing c0 which is an l∞ P Grothendieck space. Let j xij be λ multiplier convergent for every i. If P∞ lim t x exists for every t ∈ λ and xj = limi xij for every j, then P i j=1 j ij P∞ P∞ x is λ multiplier convergent and limi j=1 tj xij = j=1 tj xj for every j j t ∈ λ. 0

Proof: Put xi = {xij }j ∈ λβX and x = {xj }. We claim that {xi } is k·k Cauchy in λβX . If not, there exist a subsequence

{n

0k } and δ > 0 such that 0 kxnk+1 − xnk k > δ . Put z k = xnk+1 − xnk so z k > δ and z k · t → 0 for every t ∈ λ. For each k pick x0k ∈ X 0 such that kx0k k ≤ 1 and (∗)

∞ X 0 k xk , zj > δ. j=1

The series j zjk is λ multiplier convergent so let Sk be the summing opP∞ P erator with respect to the series j zjk , Sk t = j=1 tj zjk for t ∈ λ (Theorem E D P∞ 2.2). Then x0k Sk ∈ λ0 and x0k Sk (t) = x0k , j=1 tj zjk → 0 for t ∈ λ. That P

space, is, {x0k Sk } is σ(λ0 , λ) convergent to 0. Since λ is an l ∞DGrothendieck E P∞ 0 0 ∞ 0 0 k {xk Sk } is σ(λ , l ) convergent to 0. Thus, xk Sk (t) = xk , j=1 tj zj → 0

for t ∈ l∞ . By the classical Hahn-Schur Theorem 7.1, { x0k , zjk }j → 0 in P∞ 0 k k·k1 or j=1 xk , zj → 0. This contradicts (∗).

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xj is λ multiplier convergent and xi · t → x · t for

0 every t ∈ λ. It will then follow that xi − x → 0 and the result follows. Let > 0 and

t ∈ λ. There exists N such that k, l ≥ N implies

P n k l

j=m tj (xj − xj ) < for all n > m by the part above. Let k → ∞ to obtain We claim that

P

j

X

n l t (x − x ) (∗∗) j j j ≤ for all n > m, l ≥ N.

j=m P P This shows that the series j tj (xj −xN j ) is Cauchy so j tj xj is Cauchy and, therefore, convergent since X is complete. Condition (∗∗) also shows that xl · t → x · t as desired. Example 7.11 shows that even when the multiplier space λ has the 0GHP and the signed-WGHP, the summing operators in Theorem 7.37 may not converge uniformly on bounded subsets of λ. However, we show that in this case we do have uniform convergence on null sequences. P Theorem 7.39. Assume that λ has 0-GHP and signed-WGHP. Let j xij P∞ be λ multiplier convergent for every i. If limi j=1 tj xij exists for every t ∈ λ and limi xij = xj for every j, then P∞ P (C1) the series j xj is λ multiplier convergent and limi j=1 tj xij = P∞ t x and j j j=1 P∞ P∞ (C4) if tk → 0 in λ, then limi j=1 tkj xij = j=1 tkj xj uniformly for k ∈ N. Proof: (C1) follows from Theorem 7.5. For (C4) let U be a neighborhood of 0 in X and pick a neighborhood of 0, V , such that V + V ⊂ U . By Theorem 2.39 there exists n such that ∞ X

j=n+1

tkj (xij − xj ) ∈ V

for i, k ∈ N. Since limi (xij − xj ) = 0 for every j and {tkj : k ∈ N} is bounded for every j, then for every j limi tkj (xij − xj ) = 0 uniformly for k ∈ N by Lemma 7.7. Therefore, there exists m such that i ≥ m implies Pn k j=1 tj (xij − xj ) ∈ V for every k ∈ N. If i ≥ m, then ∞ X j=1

tkj (xij

− xj ) =

n X j=1

tkj (xij

− xj ) +

∞ X

j=n+1

tkj (xij − xj ) ∈ V + V ⊂ U

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and (C4) follows. As noted earlier Example 7.11 shows that conclusion (C4) cannot be improved to uniform convergence on bounded subsets of λ (conclusion (C2)). The following example shows that the signed-WGHP in Theorem 7.38 cannot be dropped even in the presence of 0-GHP. P Example 7.40. Let λ = c and X = R. Then is λ multij δij plier convergent for every i and limi δij = 0 for every j. If t ∈ c, then P∞ P∞ limi j=1 tj δij = limi ti exists. However, if tk = ek , then j=1 tkj δij = 1 so (C4) fails. Note that λ = c has 0-GHP but not signed-WGHP. We now establish a Hahn-Schur Theorem in the spirit of Li’s Lemma 3.29. These theorems are useful in treating operator valued series with vector valued multipliers. Let Ω be a non-empty set and G be an Abelian topological group. Let fij : Ω → G for i, j ∈ N and assume that Ω has a distinguished element w0 such that fij (w0 ) = 0 for every i, j. P∞ Theorem 7.41. Assume that the series j=1 fij (wj ) converges for every P∞ i and every sequence {wj } ⊂ Ω and that limi j=1 fij (wj ) exists for every sequence {wj } ⊂ Ω. Then (1) limi fij (w) = fj (w) exists for every w ∈ Ω, j ∈ N, P∞ (2) the series j=1 fij (wj ) converge uniformly for i ∈ N and all sequences {wj } ⊂ Ω, P∞ P∞ (3) limi j=1 fij (wj ) = j=1 fj (wj ) for every sequence {wj } ⊂ Ω.

Proof: Let w ∈ Ω and j ∈ N. Define a sequence in Ω by wj = w and P∞ wi = w0 for i 6= j. Then fij (wj ) = 0 if i 6= j so limi j=1 fij (wj ) = limi fij (w) exists by hypothesis and (1) holds. P We first show that for each {wj } the series ∞ j=1 fij (wj ) converge uniformly for i ∈ N. If this fails to hold, there exists a neighborhood of 0, U , in G such that

(∗) for every k there exist p > k and q such that

∞ X j=p

Hence, there exist n1 > 1, i1 such that ∞ X

j=n1

fi1 j (wj ) ∈ / U.

fqj (wj ) ∈ / U.

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Pick a neighborhood of 0, V , such that V + V ⊂ U . There exists m1 > n1 such that ∞ X

j=m1 +1

Hence,

m1 X

fi1 j (wj ) ∈ V.

fi1 j (wj ) ∈ / V.

j=n1

P∞ Since j=1 fij (wj ) converge for i = 1, ..., i1 by (∗) there exist n2 > m1 , i2 > i1 such that ∞ X fi2 j (wj ) ∈ /U j=n2

and as above there exists m2 > n2 such that m2 X

j=n2

fi2 j (wj ) ∈ / V.

Continuing this construction produces increasing sequences {ip }, {mp } and {np } with np+1 > mp > np such that (∗∗)

mp X

j=np

fip j (wj ) ∈ / V.

Now consider the matrix M = [mpq ] = [

mq X

fip j (wj )].

j=nq

We claim that M is a K-matrix (Appendix D.2). The columns of M converge by (1). If {kq } is an increasing sequence, set vj = wj if nkq ≤ j ≤ mkq and vj = w0 otherwise. Then lim p

∞ X q=1

mpkq = lim p

∞ X

fip j (vj )

j=1

exists by hypothesis. Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗∗).

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If (2) fails to hold, then as above there exist increasing sequences {ik }, {mk } and {nk } with nk < mk < nk+1 , a matrix {wij } ⊂ Ω and a neighborhood, V , with mk X (∗ ∗ ∗) fik j (wkj ) ∈ / V. j=nk

Now define a sequence {wj } ⊂ Ω by wj = wkj if nk ≤ j ≤ mk and P∞ wj = w0 otherwise. But, then the series j=1 fij (wj ) do not satisfy the Cauchy condition uniformly for i ∈ N by (∗ ∗ ∗) and, therefore, violates the condition established above. For (3), let U be a neighborhood of 0 and {wj } ⊂ Ω. Pick a neighP borhood of 0,V , such that V + V + V ⊂ U . Put g = limi ∞ j=1 fij (wj ). P∞ We show that the series j=1 fj (wj ) converges to g. By (2) there exists P n such that ∞ ≥ n and i ∈ N. Suppose m > n. j=m fij (wj ) ∈ V for m Pn Then by (1) there exists i such that j=1 (fij (wj ) − fj (wj )) ∈ V and g− P∞ f (w ) ∈ V . So ij j j=1 g−

n X j=1

fj (wj ) = g −

∞ X

fij (wj ) +

∞ X

fij (wj ) +

j=1

j=n+1

j=1

∈ V + V + V ⊂ U.

n X

(fij (wj ) − fj (wj ))

Concerning the converse of Theorem 7.40, we have P∞ Theorem 7.42. Assume that the series j=1 fij (wj ) converges for every i and every sequence {wj } ⊂ Ω and that limi fij (w) = fj (w) exists for every P∞ j and w ∈ Ω. If for every {wj } ⊂ Ω the series j=1 fij (wj ) converge P∞ uniformly for i ∈ N, then { j=1 fij (wj )}i is Cauchy. If G is sequentially complete, then the stronger conclusion (2) of Theorem 7.40 holds. Proof: Let {wj } ⊂ Ω and let U be a neighborhood of 0. Pick a symmetric neighborhood of 0, V , such that V +V +V ⊂ U . By hypothesis there exP∞ ists n such that j=n fij (wj ) ∈ V for all i. Since limi fij (w) = fj (w) exists Pn−1 for every j and w ∈ Ω there exists m such that j=1 (fij (wj )−fkj (wj )) ∈ V for all i, k ≥ m. Then for all i, k ≥ m, ∞ X j=1

fij (wj ) − =

n−1 X j=1

∞ X

fkj (wj )

j=1

(fij (wj ) − fkj (wj )) +

∈ V + V + V ⊂ U.

∞ X

j=n

fij (wj ) −

∞ X

j=n

fkj (wj )

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The last statement follows from Theorem 7.40. Under stronger assumptions we establish a stronger convergence conclusion than condition (3) in Theorem 7.40. P∞ Theorem 7.43. Assume that the series j=1 fij (wj ) converges for every i and every sequence {wj } ⊂ Ω. If for each j ∈ N limi fij (w) = fj (w) P∞ converges uniformly for w ∈ Ω and if the series j=1 fij (wj ) converge uniformly for all sequences {wj } ⊂ Ω and i ∈ N, then the sequences P∞ { j=1 fij (wj )}i satisfy a Cauchy condition uniformly for all sequences P∞ {wj } ⊂ Ω. If G is sequentially complete, then limi j=1 fij (wj ) = P∞ j=1 fj (wj ) uniformly for all sequences {wj } ⊂ Ω.

Proof: Let U be a closed neighborhood of 0 in G and pick a symmetric neighborhood of 0,V , such that V + V + V ⊂ U . P∞ There exists n such that fij (wj ) ∈ V for all {wj } ⊂ Ω and i ∈ N. Pn−1j=n There exists m such that j=1 (fij (w) − fkj (w)) ∈ V for all i, k ≥ m and w ∈ Ω by the uniform convergence assumption. Hence, if i, k ≥ m and {wj } ⊂ Ω, we have (∗)

∞ X j=1

n−1 X j=1

(fij (wj ) − fkj (wj )) +

∞ X

j=n

fij (wj ) − fij (wj ) −

∞ X

fkj (wj ) =

j=1

∞ X

j=n

fkj (wj ) ∈ V + V + V ⊂ U

so the first part of the statement is established. P∞ If G is sequentially complete, then limi j=1 fij (wj ) exists by (∗). The last statement then follows from (3) of Theorem 7.40 and (∗) above. We can also obtain a version of Lemma 2.42. Proposition 7.44. Let Ω be a topological space with gj : Ω → G continuous P and assume that the series ∞ ) converges for every {wj } ⊂ Ω. If j=1 gj (wjP F : ΩN → G is defined by F ({wj }) = ∞ j=1 gj (wj ), then F is continuous with respect to the product topology. Proof: Let wk = {wjk } be a net in ΩN which converges to w = {wj } in the product topology. Let U be a neighborhood of 0 in G and pick a symmetric neighborhood,V , such that V + V + V ⊂ U . By Lemma 3.29

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P there exists n such that ∞ gj (vj ) ∈ V for all {vj } ⊂ Ω. There exists k0 Pj=n n−1 such that k ≥ k0 implies j=1 (gj (wjk ) − gj (wj )) ∈ V . If k ≥ k0 , then F (wk ) − F (w) =

n−1 X j=1

(gj (wjk ) − gj (wj )) +

Thus, F is continuous.

∞ X

j=n

gj (wjk ) −

∞ X

j=n

gj (wj ) ∈ V + V + V ⊂ U.

From Proposition 7.43, we have Corollary 7.45. Let Ω be a compact topological space with gj : Ω → G P∞ continuous and assume that the series j=1 gj (wj ) converges for every {wj } ⊂ Ω.Then ∞ X gj (wj ) : {wj } ⊂ Ω S= j=1

is compact.

From Theorem 7.42, Lemma 7.30 and Proposition 7.43, we also obtain

Corollary 7.46. Let Ω be a compact topological space. Assume that each P∞ fij is continuous, the series j=1 fij (wj ) converge uniformly for {wj } ⊂ Ω and i ∈ N and for each j ∈ N limi fij (w) = fj (w) converges uniformly for w ∈ Ω, then ∞ X S= fij (wj ) : {wj } ⊂ Ω, i ∈ N j=1

is compact.

Proof: As in Proposition 7.43 define Fi : ΩN → G (F0 : ΩN → G) by P∞ P∞ Fi ({wj }) = j=1 fij (wj ) (F0 ({wj }) = j=1 fj (wj )). By Proposition 7.43 each Fi is continuous and by Theorem 7.42, Fi → F0 uniformly on ΩN . The result follows from Lemma 7.30. The results above cover the cases of subseries convergent series and bounded multiplier convergent series given in Theorems 7.17 and 7.25. In the subseries case, we take Ω = {0, 1} and in the bounded multiplier conP vergent case, we take Ω = [0, 1]. If j xij are the series in these statements, we define fij (t) = txij and take for the distinguished element w0 = 0. That limi fij (w) = fj (w) converges uniformly for w ∈ Ω follows from Lemma 7.7.

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Throughout the remainder of this chapter we give applications of the Hahn-Schur results to various topics in functional analysis and measure theory. The original proofs of the Orlicz-Pettis Theorem for normed spaces given by both Orlicz and Pettis used the version of the Schur Theorem stated in Theorem 7.1. We indicate how this version of the Orlicz-Pettis Theorem can easily be obtained from Hahn’s Theorem 7.2. P Theorem 7.47. Let X be a normed space. If the series j xj is subseries convergent in the weak topology of X, then the series is subseries convergent in the norm topology. Proof: By replacing X by the span of {xj : j ∈ N}, we

may

assume 0 0

x0 = 1 and that X is separable. For every j pick x ∈ X such that j j

0 xj , xj = kxj k. Since X is separable, {x0j } has a subsequence {x0nj } which P is weak* convergent to some x0 ∈ X 0 . We have limi j∈σ x0ni , xnj = E D P P x0 , j∈σ xnj for every σ ⊂ N, where j∈σ xnj is the weak sum of the P∞

series. By Theorem 7.2, the series j=1 x0ni , xnj converge uniformly for D E i ∈ N. In particular, x0nj , xnj = xnj → 0. Since the same argument

can be applied to any subsequence of {xj }, it follows that kxj k → 0 so P j xj is norm subseries convergent by Lemma 4.4. We next indicate several applications of the Hahn-Schur results to topics in vector valued measure theory. Let Σ be a σ-algebra of subsets of a set S. P ∞ A set function µ : Σ → X is countably additive if ∞ j=1 µ(Aj ) = µ(∪j=1 Aj ) for every pairwise disjoint sequence {Aj } ⊂ Σ. A family of countably additive set functions {µa : a ∈ I} is uniformly countably additive if for P∞ every pairwise disjoint sequence {Aj } ⊂ Σ, the series j=1 µa (Aj ) converge uniformly for a ∈ I. We have the following result due to Nikodym. Theorem 7.48. (Nikodym Convergence Theorem) Let µj : Σ → X be countably additive for every j ∈ N. If lim µj (A) = µ(A) exists for every A ∈ Σ , then (1) µ is countably additive and (2) {µj } is uniformly countably additive. Proof: Let {Aj } ⊂ Σ be pairwise disjoint. For any σ ⊂ N, we have X µi (Aj ) = µi (∪j∈σ Aj ) → µ(∪j∈σ Aj ). j∈σ

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P By Theorem 7.17 it follows that the series ∞ j=1 µi (Aj ) converge uniformly for i ∈ N so {µj } is uniformly countably additive. Also, from Theorem 7.17, ∞ X

µ(Aj ) = lim i

j=1

∞ X j=1

∞ µi (Aj ) = lim µi (∪∞ j=1 Aj ) = µ(∪j=1 Aj ) i

so µ is countably additive. A theorem closely related to the Nikodym Convergence Theorem is the Vitali-Hahn-Saks Theorem which we now derive. Let ν : Σ → [0, ∞] be a measure. If µ : Σ → X is countably additive, then µ is ν continuous if limν(A)→0 µ(A) = 0. If {µj } is a sequence of countably additive set functions, then {µj } is uniformly ν continuous if limν(A)→0 µj (A) = 0 uniformly for j ∈ N. We have the following result which connects uniform countable additivity and uniform ν continuity. Theorem 7.49. Let {µj } be countably additive, µj : Σ → X, such that each µj is ν continuous. If {µj } is uniformly countably additive, then {µj } is uniformly ν continuous. Proof: If the conclusion fails to hold, there exists a neighborhood, U , of 0 such that for every δ > 0 there exist k ∈ N, E ∈ Σ such that µk (E) ∈ / U and ν(E) < δ. In particular, there exists E1 ∈ Σ, n1 such that µn1 (E1 ) ∈ / U and ν(E1 ) < 1. Pick a neighborhood of 0, V , such that V + V ⊂ U . There exists δ1 > 0 such that µn1 (E) ∈ V when ν(E) < δ1 . There exist / U and ν(E2 ) < δ1 /2. Continuing this E2 ∈ Σ, n2 > n1 such that µn2 (E2 ) ∈ construction produces sequences {Ek } ⊂ Σ, δk+1 < δk /2 , {nk } such that / U, ν(Ek+1 ) < δk /2 and µnk (E) ∈ V when ν(E) < δk . Note that µnk (Ek ) ∈ ν(∪∞ j=k+1 Ej ) ≤

∞ X

ν(Ej ) < δk /2 + δk+1 /2 + ... < δk /2 + δk /22 + ... = δk

j=k+1

so that µnk (Ek ∩ ∪∞ j=k+1 Ej ) ∈ V.

Now set Ak = Ek \ ∪∞ j=k+1 Ej . The {Ak } are pairwise disjoint and / V. µnk (Ak ) = µnk (Ek ) − µnk (Ek ∩∞ j=k+1 Ej ) ∈

However, by the uniform countable additivity of {µj } we have limk µj (Ak ) = 0 uniformly for j ∈ N. This gives the desired contradiction.

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From the Nikodym Convergence Theorem, we can now obtain the VitaliHahn-Saks Theorem. Theorem 7.50. (Vitali-Hahn-Saks) Let {µj } be countably additive, µj : Σ → X, such that each µj is ν continuous. If lim µj (A) = µ(A) exists for every A ∈ Σ, then (1) {µj } is uniformly ν continuous and (2) µ is countably additive and ν continuous. Proof: The result is an immediate consequence of Theorems 7.47 and 7.48. As noted earlier there is a notion between the concepts of finite additivity and countable additivity called strong boundedness. If µ : Σ → X is finitely additive, then µ is strongly bounded (strongly additive, exhaustive) if µ(Aj ) → 0 whenever {Aj } is a pairwise disjoint sequence from Σ (3.36). A family {µa : a ∈ A} of finitely additive set functions is uniformly strongly bounded (strongly additive, exhaustive) if whenever {Aj } is a pairwise disjoint sequence from Σ, limj µa (Aj ) = 0 uniformly for a ∈ A. We have the analogue of Theorem 3.42 for uniformly strong bounded set functions. Theorem 7.51. For a ∈ A let µa : Σ → X be finitely additive. The following are equivalent: (i) {µa : a ∈ A} is uniformly strongly bounded, P∞ (ii) for any pairwise disjoint sequence {Aj } ⊂ Σ the series j=1 µa (Aj ) satisfy a Cauchy condition uniformly for a ∈ A. Proof: Clearly (ii) implies (i). If (ii) fails to hold, there exist a neighborhood of 0,U , an increasing P / sequence of intervals {Ik } and a sequence ak ∈ A such that j∈Ik µak (Aj ) ∈ / U so U . Set Bk = ∪j∈Ik Aj . Then {Bk } is pairwise disjoint and µak (Bj ) ∈ (i) fails. We now establish a version of the Nikodym Convergence Theorem for strongly bounded set functions. Theorem 7.52. Let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (A) = µ(A) exists for every A ∈ Σ, then (1) µ is strongly bounded and (2) {µi } is uniformly strongly bounded.

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Proof: Suppose that (2) fails. Then there exist a pairwise disjoint sequence {Aj } and a neighborhood of 0, U , such that for every k there exist jk > k, ik such that µik (Ajk ) ∈ / U . For k = 1 there exist j1 > 1, i1 such that / U . There exist J1 > j1 such that µi (Aj ) ∈ U for 1≤ i ≤ i1 and µi1 (Aj1 ) ∈ j ≥ J1 . For k = J1 there exist j2 > J1 and i2 such that µi2 (Aj2 ) ∈ / U. Note that i2 > i1 . Continuing this construction produces increasing sequences {jk }, {ik } such that / U. (∗) µik (Ajk ) ∈

By Drewnowski’s Lemma (Appendix E.2) there is a subsequence {nk } such that each µik is countably additive on the σ-algebra Σ0 generated by the {Ajnk }. By the Nikodym Convergence Theorem 7.47, {µink } is uniformly countably additive on Σ0 . In particular, lim µink (Ajnk ) = 0. This contradicts (∗). (1) follows from (2) since if {Aj } ⊂ Σ is pairwise disjoint, lim µ(Aj ) = lim lim µi (Aj ) = lim lim µi (Aj ) = 0 j

j

i

i

j

by the uniform convergence of limj µi (Aj ) = 0. We next establish a vector version of a lemma due to Phillips which he used to show that there is no continuous projection from l ∞ onto c0 . We first state the scalar version of Phillips’ Lemma. Let ba be the space of all finitely additive, bounded real valued set functions defined on 2N equipped with the variation norm, kνk = var(ν)(N). ba equipped with this norm is the dual space of (l ∞ , k·k∞ ) ([DS] IV.5.1, [SW3] 6.3). If j ∈ N, we write ν({j}) = ν(j) for ν ∈ ba. Lemma 7.53. (Phillips) Let νk ∈ ba for every k ∈ N and suppose that P∞ lim νk (E) = 0 for every E ⊂ N. Then limk j=1 |νk (j)| = 0. Phillips’ Lemma has the following duality interpretation. Let J be the canonical imbedding of c0 into its bidual l∞ . Then the transpose operator J 0 : (l∞ )0 = ba → (c0 )0 = l1 is given by J 0 ν = {ν(j)}. Phillips’ Lemma asserts that if {νi } converges to 0 in the weak topology σ(ba, m0 ), then {J 0 νi } converges to 0 in k·k1 . In particular, if {νi } converges to 0 in the weak* topology σ(ba, l ∞ ), then kJ 0 νi k1 → 0. This also shows that l ∞ is a Grothendieck space. We next show how Phillips’ Lemma can be used to show that there is no continuous projection of l ∞ onto c0 . Theorem 7.54. There is no continuous projection of l ∞ onto c0 .

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Proof: If P were such a projection, then for y ∈ l ∞ , P y ∈ c0 so ek ·P y = P e · y → 0. Hence, P 0 ek → 0 weak* in ba. By the observation above,

0 0 k

J P e = sup{ J 0 P 0 ek · x : x ∈ c0 , kxk ≤ 1} ∞ 1 = sup{ ek · P Jx : x ∈ c0 , kxk∞ ≤ 1}

= sup{ ek · x : x ∈ c0 , kxk ≤ 1} = ek = 1 → 0 0 k

∞

1

an obvious contradiction.

We now establish a vector version of Phillips’ Lemma which yields Lemma 7.52 as a special case. Theorem 7.55. Let X be sequentially complete and let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (E) = 0 for every E ∈ Σ, then P for every pairwise disjoint sequence {Ej } from Σ, limi j∈σ µi (Ej ) = 0 uniformly for σ ⊂ N. P Proof: By Theorem 7.17 it suffices to show that lim i j∈σ µi (Ej ) = 0 for every σ ⊂ N. If this fails to hold, we may assume, by passing to a subsequence if necessary, that there exists a closed neighborhood of 0, U , P∞ such that j=1 µi (Ej ) ∈ / U for every i. Pick a symmetric neighborhood of P n1 0, V , such that V + V ⊂ U . There exists n1 such that j=1 µ1 (Ej ) ∈ / U. Pn1 µ (E ) ∈ V for i ≥ m . There exists There exists m1 such that i j 1 j=1 Pn2 n2 > n1 such that j=1 µm1 (Ej ) ∈ / U . Hence, n2 X

µm1 (Ej ) =

j=n1 +1

n2 X j=1

µm1 (Ej ) −

n1 X j=1

µm1 (Ej ) ∈ / V.

Continuing this construction produces increasing sequences {mi }, {ni } Pni+1 ni+1 such that j=n µmi (Ej ) ∈ / V . Set Fi = ∪j=n Ej so i +1 i +1 and {Fi } is pairwise disjoint. Consider the matrix

/V (∗) µmi (Fi ) ∈

M = [mij ] = [µmi (Fj )]. We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 by hypothesis. If {rj } is an increasing sequence, by Drewnowski’s Lemma (Appendix E.2), there is a subsequence {sj } of {rj } such that each µi is countably additive on the σ-algebra generated by {Fsj }. Thus, ∞ X µmi (Fsj ) = µmi (∪∞ j=1 Fsj ) → 0. j=1

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Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). We indicate how Theorem 7.54 yields Phillips’ Lemma 7.52 and, therefore, can be viewed as a vector version of Phillips’ Lemma. Let > 0 and let the notation be as in Lemma 7.52. By Theorem 7.54 there exists P N such that j∈σ νi (j) < for every σ ⊂ N, i ≥ N . By Lemma 3.37, P∞ j=1 |νi (j)| ≤ 2 for i ≥ N. We next give a generalization of Theorem 7.54 which in turn yields a generalization of Phillips’ Lemma 7.52. Theorem 7.56. Let X be sequentially complete and let µi : Σ → X be strongly bounded for every i ∈ N. If lim µi (E) = µ(E) exists for every E ∈ P Σ, then for every pairwise disjoint sequence {Ej } ⊂ Σ, lim j∈σ µi (Ej ) = P j∈σ µ(Ej ) uniformly for σ ⊂ N. [In particular, µ is strongly bounded.] P Proof: By Theorem 7.17 it suffices to show that limi j∈σ µi (Ej ) exists for every σ ⊂ N. Since this is trivial for finite σ, assume that σ is infinite P with σ = {m1 < m2 < ...}. We claim that { ∞ j=1 µi (Emj )} is a Cauchy sequence in X. For this, assume that {pi } and {qi } are increasing sequences with pi < qi < pi+1 . Then lim(µpi (E) − µqi (E)) = 0 for every E ∈ Σ so by P∞ P∞ Theorem 7.54, limi j=1 (µpi (Emj ) − µqi (Emj )) = 0 so { j=1 µi (Emj )} is a Cauchy sequence and the result follows. The scalar case of Theorem 7.55 gives an improvement to Phillips’ Lemma 7.52. In particular, as in the proof of Phillips’ Lemma from Theorem 7.54 indicated above, we have Corollary 7.57. Let νk ∈ ba for every k ∈ N and suppose that lim νk (E) = P∞ ν(E) exists for every E ⊂ N. Then ν ∈ ba and j=1 |νk (j) − ν(j)| → 0. We show that Theorem 7.55 can be used to derive a version of the Nikodym Boundedness Theorem for strongly bounded set functions.

Theorem 7.58. (Nikodym Boundedness Theorem) Let νi : Σ → R be bounded and finitely additive. If {νi (E) : i ∈ N} is bounded for every E ∈ Σ, then {νi (E) : i ∈ N, E ∈ Σ} is bounded. Proof: Let {Ej } ⊂ Σ be pairwise disjoint. By Lemma 4.58 it suffices to show that {νi (Ei ) : i ∈ N} is bounded.

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Let ti → 0 in R. Then limi ti νi (E) = 0 for every E ∈ Σ. Since each ν ∈ ba is strongly bounded (Proposition 3.38) Theorem 7.55 implies that {ti νi } is uniformly strongly bounded so limi ti νi (Ei ) = 0. Therefore, {νi (Ei ) : i ∈ N} is bounded. From Theorem 7.57 and the Uniform Boundedness Principle, we can immediately obtain a version of the Nikodym Boundedness Theorem for LCTVS (see the proof of Corollary 4.60). Theorem 7.59. (Nikodym Boundedness Theorem) Let X be an LCTVS. Let νi : Σ → X be strongly bounded. If {νi (E) : i ∈ N} is bounded for every E ∈ Σ, then {νi (E) : i ∈ N, E ∈ Σ} is bounded. Recall the local convex assumption in Theorem 7.58 cannot be dropped; see Remark 4.61. We can use the version of the Nikodym Boundedness Theorem in Theorem 7.57 to show that (m0 , k·k∞ ) is barrelled. More generally, let S(Σ) be the space of all real valued Σ simple functions equipped with the sup-norm. Then the dual of S(Σ) is the space ba(Σ) of all bounded, finitely additive, real valued set functions ν defined on Σ with the variation 0 norm, kνk = var(ν)(S); the pairing R between f ∈ S(Σ) and ν ∈ ba(Σ) is given by integration, hf, gi = S gdν, g ∈ S(Σ) [no elaborate integration theory is used since we are only integrating simple functions] (see [DS], [Sw3] 6.3). A norm equivalent to the variation norm is given by 0 kνk = sup{|ν(E)| : E ∈ Σ} ([DS], [Sw3] 2.2.1.7). From the Nikodym Boundedness Theorem, we have Theorem 7.60. (S(Σ), k·k∞ ) is barrelled. That is, if {νj } ⊂ ba(Σ) is σ(ba(Σ), S(Σ)) bounded, then {νj } is norm bounded. In particular, (m0 , k·k∞ ) is barrelled. Proof: Since {νj } is pointwise bounded on S(Σ), {νj (E)} is bounded for each E ∈ Σ. Theorem 7.57 then implies that {kνj k0 : j ∈ N} is bounded so {kνj k : j ∈ N} is bounded.

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Chapter 8

Spaces of Multiplier Convergent Series and Multipliers

In this chapter we consider topological properties of the space of λ multiplier convergent series. Throughout this chapter let λ be a sequence space containing c00 and let X be an LCTVS. Recall that λβX is the space of all X valued λ convergent series. If x = {xj } ∈ λβX and t = {tj } ∈ λ, P∞ we write x · t = j=1 tj xj . We define a locally convex topology on λβX induced by X and λ when λ is a K-space. Assume that λ is a K-space and let B = {B ⊂ λ : B is bounded and {x · t : t ∈ B} is bounded in X ∀ t ∈ λ}. Let X be the family of all continuous semi-norms on X. For B ∈ B and p ∈ X , define a semi-norm on λβX by pB (x) = sup{p(x · t) : t ∈ B}.

Let τB be the locally convex topology on λβX generated by the semi-norms pB for B ∈ B and p ∈ X . Remark 8.1. The family B is equal to the family of all bounded subsets of λ if either λ has 0-GHP (Corollary 2.12) or if λβ ⊂ λ0 (Corollary 2.4). If the maps t → x · t from λ into X are continuous for all x ∈ λβX , then λβX ⊂ L(λ, X) and the topology τB is just the relative topology from Lb (λ, X). We consider the sequential completeness of τB . For this let Pi be the Pi sectional operator on λ defined by Pi t = j=1 tj ej . Recall the following property from Appendix B.4. Definition 8.2. The space λ has the sections uniformly bounded property (SUB) if {Pi t : i ∈ N, t ∈ B} is bounded for every bounded subset B of λ. 133

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Theorem 8.3. Assume that X is sequentially complete, λ has SUB and for each y ∈ λβX the operator t → y · t from λ into X is bounded. Then τB is sequentially complete. Proof: Let {xk } be τB Cauchy, B ∈ B , p ∈ X and > 0. First, put t = ej . Then p{t} (xk − xl ) = p(xkj − xlj ) so {xkj }k is Cauchy in X for each j. Let xj = limk xkj . Set x = {xj }. Set B 0 = B ∪ {Pi B : i ∈ N} so B 0 is bounded by the SUB assumption. Also, B 0 ∈ B by the boundedness assumption of the maps t → y · t for y ∈ λβX . There exists N such that k, l ≥ N implies pB 0 (xk − xl ) < . Pn Thus, if k, l ≥ N and n > m, then p( i=m ti (xki − xli )) < for t ∈ B. Pn Hence, p( i=m ti (xki − xi )) ≤ for t ∈ B, k ≥ N, m > n. This implies that x ∈ λβX and pB (xk − x) ≤ for k ≥ N . Remark 8.4. The boundedness assumption on the maps t → y · t for y ∈ λβX means that B is equal to the family of all bounded subsets of λ. This condition is satisfied, for example, if λ has 0-GHP (Corollary 2.12) or if λβ ⊂ λ0 (Corollary 2.4). The completeness assumption in Theorem 8.3 is necessary. Proposition 8.5. If λβX is τB sequentially complete, then X is sequentially complete. Proof: Let {xk } be Cauchy in X. Put xkj = xk if j = 1 and xkj = 0 otherwise and set xk = {xkj }j . Then {xk } is τB Cauchy. Therefore, there exists y = {yj } ∈ λβX such that xk → y in τB . In particular, xk1 = xk → y1 in X. From the uniform convergence result in Theorem 2.16, we have Theorem 8.6. Let λ have signed-SGHP. Then for every x ∈ λβX , n X τB − lim xj ej = x, j=1

i.e., λ

βX

is a vector valued AK-space (Appendix C).

Recall that ω(λβX , λ) is the weakest topology on λβX such that the maps x → x · t from λβX into X are continuous for all t ∈ λ. From Corollary 2.28, we have Theorem 8.7. If λ has signed-WGHP and X is sequentially complete, then ω(λβX , λ) is sequentially complete.

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We now give conditions which guarantee sequential convergence in ω(λβX , λ). Proposition 8.8. Let {xk } ⊂ λβX . (1) If xk → 0 in ω(λβX , λ), then limk xkj = 0 for every j. P k (2) If limk xkj = 0 for every j and if for every t ∈ λ the series j t j xj k βX converge uniformly for k ∈ N, then x → 0 in ω(λ , λ). (3) If λ has signed-WGHP, the converse of (2) holds. Proof: (1) follows since limk xk · ej = limk xkj = 0. For (2), let t ∈ λ and consider (∗) xk · t =

n X j=1

tj xkj +

∞ X

tj xkj .

j=n+1

Let U be a neighborhood of 0 in X and pick a neighborhood of 0,V , such that V + V ⊂ U . By hypothesis, there exists n such that the last term in (∗) belongs to V for every k. Since limk xkj = 0 for every j, for large k the first term on the right hand side of (∗) belongs to V . Therefore, for large k, xk · t ∈ U . The statement in (3) follows from (1) and Theorem 2.26. One of the scalar versions of the Hahn-Schur theorem asserts that if the sequence {ti } in l1 is σ(l1 , l∞ ) convergent, then the sequence {ti } is k·k1 convergent. The vector version of the Hahn-Schur Theorem given in Theorem 7.10 can be given a similar interpretation. Theorem 8.9. Let λ have signed-SGHP and let X be sequentially complete. If {xk } is ω(λβX , λ) Cauchy, then there exists x ∈ λβX such that lim xk ·t = x · t uniformly for t belonging to bounded subsets of λ. The scalar version of Theorem 8.9 gives a generalization of the classical scalar version of the Hahn-Schur Theorem for l 1 described above. Corollary 8.10. Let λ have signed-SGHP. If {tk } is σ(λβ , λ) Cauchy, then there exists t ∈ λβ such that β(λβ , λ) − lim tk = t. For λ = l∞ , this is the Hahn-Schur result for l 1 described above. We next establish a Banach-Steinhaus equicontinuity type result for λβX . This will lead to another sequential completeness result for ω(λβX , λ).

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Theorem 8.11. Let λ have 0-GHP and {xk } ⊂ λβX . If lim xi · t = Lt exists for every t ∈ λ, then {xi } is sequentially equicontinuous, i.e., if tj → 0 in λ, then limj xi · tj = 0 uniformly for i ∈ N. Proof: If the conclusion fails to hold, we may assume that there exist > 0, p ∈ X and tj → 0 in λ such that p(xj · tj ) > for all j. Set m1 = 1 and pick n1 such that n1 X 1 m1 p( tm k xk ) > . k=1

limi xik

i

Note that = limi x · ek exists for every k so {xik }i is bounded for each k. Thus, since limi tik = 0 for every k, limi tik xik = 0 (Lemma 7.7) so there exists m2 > m1 such that n1 X 2 m2 p( tm k xk ) < /2. k=1

There exists n2 > n1 such that n2 X 2 m2 p( tm k xk ) > . k=1

Hence,

p(

n2 X

2 m2 tm k xk ) > /2.

k=n1 +1

Continuing this construction produces increasing sequences {mk }, {nk } such that X m m (∗) p( tk j xk j ) > /2 k∈Ij

for all j, where Ij = [nj−1 + 1, nj ]. Define a matrix M = [mij ] = [

X

m

i t k j xm k ].

k∈Ij

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge by the observation above. Next, if {pj } is an increasing sequence, by 0-GHP, there is a further subsequence, still denoted by {pj }, such that P∞ s = j=1 χIpj tmpj ∈ λ. Then ∞ X j=1

mipj =

∞ X X

j=1 k∈Ipj

mp j

tk

mi i xm ·s k =x

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so limi xmi · s = Ls exists. Thus, M is a K-matrix and by the AntosikMikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). We can now use Theorem 8.11 to obtain another sequential completeness result for ω(λβX , λ). Corollary 8.12. Let λ be an AK-space with 0-GHP and X be sequentially complete. If {xi } is ω(λβX , λ) Cauchy, then there exists x ∈ λβX such that lim xi · t = x · t for every t ∈ λ, i.e., ω(λβX , λ) is sequentially complete. Proof: If Lt = lim xi · t for t ∈ λ, then by Theorem 8.11, L : λ → λ is linear and sequentially continuous. Set xk = limi xi ·ek and x = {xk }. If t ∈ P P∞ P∞ j j j λ, by the AK-property, t = ∞ j=1 tj e so Lt = j=1 tj Le = j=1 tj x = x · t. We next establish a uniform boundedness result for λβX . Theorem 8.13. Let λ have 0-GHP. If Γ ⊂ λβX is pointwise bounded on λ, then Γ is uniformly bounded on bounded subsets of λ. Proof: Suppose the conclusion fails to hold. Then there exist p ∈ X , > 0, {xk } ⊂ Γ, a bounded sequence {tk } ⊂ λ and sk → 0, sk > 0, such that p(sk xk · tk ) >

for all k. Put k1 = 1 and pick m1 such that m1 X p(sk1 tkj 1 xkj 1 ) > . j=1

{tkj

Since λ is a K-space, : k ∈ N} is bounded for each j and {xkj : k ∈ N} is bounded for each j by hypothesis. Therefore, limk sk tkj xkj = 0 for every j (Lemma 7.7) so there exists k2 > k1 such that m1 X tkj 2 xkj 2 ) < /2. p(sk2 j=1

Pick m2 > m1 such that

p(sk2

m2 X

tkj 2 xkj 2 ) > .

j=1

Set I2 = [m1 + 1, m2 ] and note p(sk2

X

j∈I2

tkj 2 xkj 2 ) > /2.

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Continuing this construction produces an increasing sequence {kp } and an increasing sequence of intervals {Ip } such that X k k (∗) p(skp tj p xj p ) > /2 j∈Ip

for all p. Define a matrix

√ √ M = [mpq ] = [ skp xkp · skq χIq tkq ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M √ converge to 0 since {xk } is pointwise bounded on λ. Next, since sk tk → 0, by 0-GHP if {rq } is any subsequence, there is a further subsequence, still P∞ denoted by {rq }, such that t = q=1 √skrq χIrq tkrq ∈ λ. Hence, ∞ X q=1

mprq =

√ skp xkp · t → 0.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2) the diagonal of M converges to 0. But, this contradicts (∗). Recall that a pair of vector spaces X, X 0 in duality is called a Banach Mackey pair if σ(X, X 0 ) bounded subsets are β(X, X 0 ) bounded. An LCTVS X is a Banach-Mackey space if X, X 0 form a Banach-Mackey pair ([Wi] 10.4.3). The scalar version of Theorem 8.13 has the following corollary. Corollary 8.14. Let λ have 0-GHP. (i) λ, λβ is a Banach-Mackey pair. (ii) If λ0 ⊂ λβ , then λ, λ0 is a Banach-Mackey pair. (iii) If λ0 = λβ and λ is quasi-barrelled, then λ is barrelled. Conditions which guarantee that the hypotheses in (i) and (ii) are satisfied are given in Proposition 2.5. We consider another uniform boundedness result which requires another type of gliding hump property (Appendix B.30 and B.31). Let µ be a sequence space containing c00 . Definition 8.15. The K-space λ has the strong µ gliding hump property (strong µ-GHP) if whenever {Ik } is an increasing sequence of intervals and {tk } is a bounded sequence in λ, then for every s ∈ µ the coordinate sum P∞ of the series j=1 sj χIj tj ∈ λ.

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Definition 8.16. The K-space λ has the weak µ gliding hump property (weak µ-GHP) if whenever {Ik } is an increasing sequence of intervals and {tk } is a bounded sequence in λ, there is a subsequence {nk } such that for P∞ every s ∈ µ the coordinate sum j=1 sj χInj tnj ∈ λ.

The elements s ∈ µ are called multipliers since their coordinates multiply the blocks {χIj tj } determined by the {Ik } and the {tk }. The signedWGHP and signed -SGHP are somewhat similar in that the ”humps” are multiplied by ±1 in these cases. Examples of spaces with the strong µ-GHP and weak µ-GHP are given in Appendix B. For example, any locally complete LCTVS has strong l 1 GHP, l∞ and c0 have strong c0 -GHP and (l2 , σ(l2 , l2 )) has strong l1 -GHP but not 0-GHP. We next establish a basic lemma. If A ⊂ λ and B ⊂ λβ , we write |B · A| = sup{|s · t| : s ∈ B, t ∈ A}. Lemma 8.17. Suppose A ⊂ λ is coordinate bounded and B ⊂ λβ is coordinate bounded. If |B · A| = ∞, then there exist an increasing sequence of intervals {Ik }, {tk } ⊂ A and {sk } ⊂ B such that k s · χ I t k > k 2 k for all k.

Proof: There exist sk ∈ B, tk ∈ A such that sk · tk > k 2 + k. Set Pn1 k1 k1 k1 = 1 and pick n1 such that j=1 sj tj > k12 + 1. By hypothesis for

every j, {skj : k ∈ N} and {tkj : k ∈ N} are bounded so there exists k2 > k1 Pn1 k2 k2 such that k12 j=1 sj tj < 1. Hence, n1 ∞ ∞ X k k X k k X k2 k2 2 2 ≥ s t s t sj 2 tj 2 > k22 . j j − j j j=1 j=1 j=n1 +1 P k2 k2 2 Pick n2 > n1 such that nj=n s t > k22 . Set I2 = [n1 + 1, n2 ] so j j +1 1 k s 2 · χI2 tk2 > k22 .

Now just continue and relabel.

Theorem 8.18. Let λ have weak µ-GHP. Assume (N) {ek : k ∈ N} is β(λ, λβ ) bounded in λ.

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If A ⊂ λ is bounded and B ⊂ λβ is σ(λβ , λ) bounded, then |B · A| < ∞. Proof: Suppose the conclusion fails and let the notation be as in Lemma 8.17. Let {nj } be as in Definition 8.16. Define a linear map T : µ → λ P∞ by T s = j=1 sj χInj tnj [coordinate sum]. We claim that T is σ(µ, µβ ) − σ(λ, λβ ) continuous. Let s ∈ µ, t ∈ λβ . Then (∗) t · T s =

∞ X j=1

sj (t · χInj tnj ).

Equation (∗) implies that {t · χInj tnj } ∈ λβ and t · T s = s · {t · χInj tnj } so T is σ(µ, µβ ) − σ(λ, λβ ) continuous. Hence, T is β(µ, µβ ) − β(λ, λβ ) continuous ([Wi] 11.2.6, [Sw2] 26.15). Thus, by (N), {T ej } = {χInj tnj } is β(λ, λβ ) bounded. But, this contradicts the conclusion of Lemma 8.17. Corollary 8.19. Under the hypothesis of Theorem 8.18, if λ0 ⊂ λβ , then λ is a Banach-Mackey space. Conditions for λ0 ⊂ λβ are given in Proposition 2.5. We next consider some results of the Spanish school in Cadiz which use series to characterize completeness and barrelledness of normed linear spaces ([AP2], [PBA]). For the remainder of this chapter let X be a normed space. Let {xj } ⊂ P X and let M ∞ ( xj )be the space of bounded multipliers for the series P xj : ∞ X X M ∞( xj ) = {tj } ∈ l∞ : tj xj converges . j=1

∞

We equip M ( xj )with the sup-norm from l ∞ . When X is a Banach space, we give necessary and sufficient conditions P P for the space M ∞ ( xj ) to be complete. Recall a series j xj in a normed P∞ 0 0 0 space is wuc iff j=1 |hx , xj i| < ∞ for all x ∈ X (see Definition 3.7). P Theorem 8.20. Let X be a Banach space and j xj be a series in X. P P Then M ∞ ( xj ) is complete iff x is wuc. j j Pn P Proof: Suppose j xj is wuc. Then E = { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded (Proposition 3.8) so let M > 0 be such that kxk ≤ M for all x ∈ E. P Let {tk } be a Cauchy sequence in M ∞ ( xj ) and let t ∈ l∞ be such that

tk − t → 0. Let > 0 and pick n such that ktn − tk < /2M . Since ∞ P∞ n ∞ j=1 tj xj is convergent, there exists N > n such that q > p ≥ N implies P

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P

P Pq

q

q n n (t − t )x ∈ E, (t − t )x

<

j=p tnj xj < /2. Since 2M j j j j j=p j j=p j

P

P

q

/2 so j=p tj xj < for q > p ≥ N . Hence, j tj xj converges since X P is a Banach space and t ∈ M ∞ ( xj ). P Assume that M ∞ ( xj ) is complete. Let t ∈ c0 . It suffices to show that P P P t ∈ M ∞ ( xj ) (Proposition 3.8). For each n, tn = nj=1 tj ej ∈ M ∞ ( xj ) P and ktn − tk∞ → 0 so t ∈ M ∞ ( xj ).

We can use Theorem 8.20 and the multiplier space M ∞ ( acterize completeness of X.

P

xj ) to char-

Theorem 8.21. The normed space X is complete iff for every wuc series P P ∞ xj ) is complete. j xj in X the space M (

Proof: Suppose that X is not complete. Then there exists a nonP∞ P = jxj . Then convergent series j xj with j=1 j kxj k < ∞. Set zjP P 0 0 0 0 does z is wuc since |hx , z i| ≤ kx k kz k for x ∈ X but j j (1/j)zjP j j Pj not converge. That is, {i/j} ∈ / M ∞ ( xj ). Since {i/j} ∈ c0 , M ∞ ( xj ) is not complete. The converse follows from Theorem 8.20. We can also characterize wuc series in terms of the ”summing operator” P P∞ P T : M ∞ ( xj ) → X defined by T t = T {tj } = j=1 tj xj , t ∈ M ∞ ( xj ) (Theorem 2.2). P Theorem 8.22. The summing operator T is continuous iff j xj is wuc. In this case,

n

X

: |tj | ≤ 1, n ∈ N . t x kT k = sup j j

j=1 P ∞ Proof: Suppose that T is continuous. Now c00 ⊂

PM ( x j ) and if

t ∈ c00 , ktk∞ ≤ 1 and ti = 0 for i ≥ n, then kT tk = nj=1 tj xj ≤ kT k .

P P

n Hence, { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded so j xj is wuc by Proposition 3.8.

P P

n Suppose that j xj is wuc and set M = sup{ j=1 tj xj : |tj | ≤ 1, n ∈ P N}. Let t ∈ M ∞ ( xj ), ktk∞ ≤ 1. For every n,

n

n X

X

j

T (

t e ) = t x j j j ≤ M

j=1

j=1

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P

∞ so j=1 tj xj = kT tk ≤ M and T is continuous with kT k ≤ M . The last statement follows from the computations above. We next consider weakly convergent series and the associated multiplier P spaces. Let j xj be a series in X. Define ∞ X X tj xj is weakly convergent . Mw∞ ( xj ) = {tj } ∈ l∞ : j=1

Again we supply Mw∞ ( xj ) with the sup-norm topology from l ∞ . From Proposition 3.8, we have P Lemma 8.23. Let X be a Banach space and j xj a series in X. Then P P ∞ xj ). j xj is wuc iff c0 ⊂ Mw ( P

We have the analogue of Theorem 8.20.

P Theorem 8.24. Let X be a Banach space and j xj a series in X. The P P space Mw∞ ( xj ) is complete iff j xj is wuc. P P Proof: Suppose that j xj is wuc. Let {tk } be a sequence in Mw∞ ( xj ) P∞ which converges to t ∈ l ∞ . Let zk = j=1 tkj xj , where this is the σ(X, X 0 ) Pn n ∈ N} and sup{kxk : sum of the series. Let E = { j=1 aj xj : |aj | ≤ 1, x ∈ E} < M . Let > 0. There exists n such that tk − t ∞ < /3M for k ≥ n. Then

m

X k

(∗) (t − t )x j j ≤ /3 for k ≥ n, m ∈ N j

j=1

so

X

m k

(t − tl )xj ≤ 2/3 for k, l ≥ n, m ∈ N. j j

j=1

Thus, kzk − zl k ≤ for k, l ≥ n. Let z = k·k − lim zk . P We claim that j tj xj is weakly convergent to z. There exists N > n 0 0 0 such P that kzk − zk < /3 for k ≥ N . If x ∈ X , kx k ≤ 1, then by (∗) m k j=1 (tj − tj ) hx0 , xj i ≤ /3 for k ≥ N, m ∈ N. If m ∈ N, then * * + + m m 0 X 0 X x, tj xj − z ≤ 2/3 + x , tN j xj − z N . j=1 j=1

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D P E m − z There exists N1 such that x0 , j=1 tN < /3 for m ≥ N1 . Hence, N j D P E 0 m m ≥ N1 implies that x , j=1 tj xj − z ≤ and the claim is established. P Suppose that Mw∞ ( xj ) is complete. Let t ∈ c0 . For every n, tn = Pn P P j ∞ xj ) so t = k·k − lim tn ∈ Mw∞ ( xj ) by Lemma 8.23. j=1 tj e ∈ Mw (

The analogue of Theorem 8.22 also holds. P Theorem 8.25. Define T : Mw∞ ( xj ) → X by T t = σ(X, X 0 ) − Pn P lim j=1 tj xj . Then T is continuous iff j xj is wuc. In this case,

n

X

kT k = sup t j xj : |tj | ≤ 1, n ∈ N .

j=1

P Proof: Suppose that T is continuous. Now c00 ⊂ Mw∞ ( x j ) and if

P

t ∈ c00 , ktk∞ ≤ 1 and ti = 0 for i ≥ n, then kT tk = nj=1 tj xj ≤ kT k.

P

P

n

Hence, { j=1 tj xj : |tj | ≤ 1, n ∈ N} is bounded so j xj is wuc by Proposition 3.8.

P

P

n

Suppose that j xj is wuc and set M = sup{ j=1 tj xj : |tj | ≤ 1, n ∈

P

Pn N}. If t ∈ Mw∞ ( xj ), ktk∞ ≤ 1, then for every n we have T ( j=1 tj ej =

P

n

j=1 tj xj ≤ M . Therefore, if x0 ∈ X 0 , kx0 k ≤ 1, then * + ∞ ∞ X X 0 0 0 |hx , T ti| = x , t j xj = hx , tj xj i ≤ M. j=1 j=1 Hence, kT tk ≤ M and T is continuous with kT k ≤ M . The last statement follows from the computations above.

P We next consider the analogue of Theorem 8.21 for Mw∞ ( xj ). We say P P that a series j xj is bounded multiplier Cauchy if the series j tj xj is Cauchy for every t ∈ l ∞ . P Then Lemma 8.26. Suppose j xj is bounded multiplier Cauchy. P P ∞ ∞ Mw ( xj ) = M ( xj ). P P Proof: Let t ∈ Mw∞ ( xj ) and let x = ∞ j=1 tj xj [weak sum]. Since the P∞ partial sums of the series j=1 tj xj form a Cauchy sequence in X, there P∞ exists x00 ∈ X 00 such that x00 = j=1 tj xj [norm limit in X 00 ]. If x0 ∈ X 0 , D P E P ∞ then x0 , j=1 tj xj = hx0 , xi = hx00 , x0 i so x00 = x and t ∈ M ∞ ( xj ).

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Theorem 8.27. X is a Banach space iff for every wuc series P space Mw∞ ( xj ) is complete.

P

j

xj the

Proof: Suppose that X is not complete. By the proof of Theorem P 8.21, there is an absolutely convergent series j xj which is wuc but for P P which M ∞ ( xj ) is not complete. The series j xj is bounded multiplier P P Cauchy since it is absolutely convergent so Mw∞ ( xj ) = M ∞ ( xj ) by P Lemma 8.26 and Mw∞ ( xj ) is not complete. The converse follows from Theorem 8.24.

We next consider series with values in the dual space X 0 and use the P series to characterize barrelled spaces. If j x0j is a series in X 0 , define X ∞ Mw∗ ( x0j ) = {{tj } ∈ l∞ :

∞ We supply Mw∗ (

∞ X

tj x0j converges in X 0 with respect to σ(X 0 , X)}.

j=1

x0j ) with the sup-norm topology from l ∞ . P Theorem 8.28. Let j x0j be a series in X 0 . Consider the following conditions: P 0 is wuc (i) j xjP ∞ l∞ (ii) Mw∗ ( x0j ) = P∞ 0 < ∞ for every x ∈ X. (iii) j=1 xj , x P

Then (i) implies (ii) implies (iii). The conditions (i), (ii) and (iii) are equivalent iff X is barrelled. P P Proof: (i) implies (ii): If j x0j is wuc and t ∈ l∞ , then j tj x0j is also Pn wuc. Hence, { j=1 tj x0j : n ∈ N} is a bounded (equicontinuous) sequence in X 0 that is also σ(X 0 , X) Cauchy and, therefore, σ(X 0 , X) convergent. P ∞ Hence, t ∈ Mw∗ ( xj ).

0 P∞ ∞ (ii) implies

0 (iii): 1For every t ∈ l and x ∈ X, j=1 tj xj , x converges. Hence, { xj , x } ∈ l and (iii) holds. P Suppose that X is barrelled and (iii) is satisfied for the series j x0j . P The set { j∈σ x0j : σ finite} is pointwise bounded on X and is, therefore, P norm bounded since X is barrelled. Thus, j x0j is wuc by Proposition 3.8. Suppose (i),(ii) and (iii) are equivalent and X is not barrelled. Then there exists a subset F ⊂ X 0 which bounded but not norm

0 is weak* 0

y > 22j and set x0 = y 0 /2j . Then bounded. Pick y ∈ F such that j j j

j0 P∞ 0

x > 2j so x , x < ∞ for every x ∈ X so (iii) holds. But, j j Pj=1 P 0 ∞ 0 j j xj is not wuc and (i) fails j=1 xj /2 does not converge in norm and to hold.

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Chapter 9

The Antosik Interchange Theorem

A problem often encountered in analysis is the interchange of two limiting processes. For example, the Lebesgue Dominated Convergence Theorem gives sufficient conditions to interchange the pointwise limit of a sequence of integrable functions with the Lebesgue integral, i.e., to take the ”limit under the integral sign”. In this chapter we consider sufficient conditions for the equality of two iterated series. For real valued series one of the most useP∞ P∞ ful criterion for interchanging the limit of an iterated series i=1 j=1 tij is the absolute convergence of the iterated series. However, absolute convergence for series with values in an LCTVS is a very strong condition and is, therefore, not appropriate. Antosik has given a sufficient condition involving subseries convergence of an iterated series with values in a topological group which has proven to be useful in a number of applications ([A]). We begin this chapter with a presentation of Antosik’s result for series with values in a TVS. We then give generalizations of Antosik’s result to multiplier convergent series. Throughout this chapter let X be a TVS and let λ be a sequence space P containing c00 . Let xij ∈ X for i, j ∈ N. The double series i,j xij converges to x ∈ X if for every neighborhood, U, of 0 in X, there exists Pp Pq N such that i=1 j=1 xij − x ∈ U for p, q ≥ N . We have the following familiar properties of double series. P Proposition 9.1. Let i,j xij be a double series.

P (i) If the double series if the series i,j xij converges to x ∈ X and P∞ P∞ P∞ j=1 xij i=1 j=1 xij converge for each i, then the iterated series converges to x. Pm P∞ (ii) If the series { i=1 j=1 xij : m ∈ N} converge uniformly and if the 145

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P∞ P∞ iterated series i=1 j=1 xij converges to x, then the double series P x converges to x. i,j ij

Proof: (i): Let U be a neighborhood of 0 in X and let V be a symmetric neighborhood such that V + V ⊂ U . There exists N1 such that p, q ≥ N1 Pp Pq implies that i=1 j=1 xij − x ∈ V . For each p there exists N2 (p) such Pp P∞ Pp Pq that i=1 j=1 xij − i=1 j=1 xij ∈ V for q ≥ N2 (p). Let p ≥ N1 and fix q ≥ max{N1 , N2 (p)}. Then p X ∞ X i=1 j=1

xij − x =

p X ∞ X i=1 j=1

xij −

q p X X

xij +

q p X X i=1 j=1

i=1 j=1

xij − x ∈ V + V ⊂ U.

P P (ii): There exists N such that pi=1 ∞ x ∈ V for q > N and for j=q+1 Pp ij P∞ every p ∈ N. There exists M > N such that i=1 j=1 xij − x ∈ V for p ≥ M . If p, q ≥ M , then q p X X i=1 j=1

xij − x =

p X ∞ X i=1 j=1

xij − x −

p ∞ X X

i=1 j=q+1

xij ∈ V + V ⊂ U.

9.2. (Antosik) Let {xij } ⊂ X. Suppose the series converges for every increasing sequence {mj }. Then the i=1 j=1 xim Pj double series i,j xij converges and

Theorem P∞ P∞

(∗)

X i,j

xij =

∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X

xij .

j=1 i=1

P∞ Proof: Note that the series x converges for every k [consider i=1 P∞ P∞ P∞ik P∞ the difference between the two series i=1 j=1 xinj and i=1 j=1 ximj , where nj = j for every j and {mj } is the sequence {1, ..., k−1, k+1, ...}]. Set P P Pm P zmj = m i=1 xij . Then for σ ⊂ N, j∈σ zmj = i=1 j∈σ xij converges P∞ P to i=1 j∈σ xij as m → ∞ by hypothesis. By the Hahn-Schur Theorem P P∞ 7.17, the series ∞ j=1 ( i=1 xij ) is subseries convergent and lim m

m X X i=1 j∈σ

xij =

∞ XX

xij

j∈σ i=1

P∞ P∞ P∞ P∞ uniformly for σ ⊂ N. In particular, i=1 xij . j=1 j=1 xij = i=1 By Proposition 9.1 the uniform convergence implies that the double series P i,j xij converges and (∗) holds. We give applications of Theorem 9.2 later in the chapter.

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Although Antosik’s Theorem is easy to apply in many concrete situations, it is only a necessary condition for the equality of the two iterated series. For example, suppose that aj , bj ∈ R and xij = ai bj . If both series P P j bj converge, then j aj and ∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X j=1 i=1

xij =

∞ X

ai

i=1

∞ X

bj .

j=1

P However, if the ”inner” series, ∞ j=1 bj is conditionally convergent, the hypothesis in Theorem 9.2 is not satisfied. Stuart has given a result which covers this case ([St3]). The result uses Stuart’s weaker form of the HahnSchur Theorem 7.24. Theorem 9.3. Let F be an IQσ family which contains N (Appendix B.23). P P If the series ∞ i=1 j∈σ xij converges for every σ ∈ F, then ∞ X ∞ X i=1 j=1

xij =

∞ X ∞ X

xij .

j=1 i=1

P∞ Proof: As before in Theorem 9.2, the series i=1 xij converges for evPm ery j. As in the proof of Theorem 9.2, set zmj = i=1 xij . For any P P∞ P σ ∈ F, limm j∈σ zmj exists and equals x i=1 j∈σ ij . By the version of the Hahn-Schur Theorem for IQσ families given in Theorem 7.24, P∞ P∞ P∞ P∞ j=1 xij . i=1 i=1 xij ) converges and equals j=1 (

Note that by employing the weaker version of the Hahn-Schur Theorem in Theorem 7.24, we cannot assert the uniform convergence of the limit, P limm j∈σ zmj , and the existence of the double series. However, Theorem 9.3 is sufficiently strong to cover the case mentioned prior to Theorem 9.3 P since if j bj is a conditionally convergent series, the set F={σ ⊂ N : P j∈σ bj converges} is an IQσ family containing N (Appendix B.24). As has been done before, Antosik’s Interchange Theorem can be viewed as a result concerning m0 multiplier convergent series and this suggests generalizations to more general λ multiplier convergent series. The hypothesis P P∞ in Theorem 9.2 that the series ∞ every increasi=1 j=1 ximj converges for P P∞ ing sequence {mj } can be restated to assert that the series ∞ i=1 j=1 tj xij converges for every t = {tj } ∈ m0 . This suggests that we might generalize Antosik’s theorem by replacing m0 by other sequence spaces λ. We now give such a generalization. Recall that λ has the signed weak gliding hump property (signed-WGHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence

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P {nj } such that the coordinate sum of the series ∞ j=1 sj χInj t belongs to λ; if the signs can all be chosen equal to 1, then λ has the weak gliding hump property (WGHP) [see Appendix B for examples]. Theorem 9.4. Let λ have signed-WGHP. Let {xij } ⊂ X. Suppose that the P series j xij is λ multiplier convergent for every i and the iterated series P∞ P∞ t = {t } ∈ λ. Then for every t ∈ λ, i=1 j=1 tj xij converges for every Pm P∞ j the sequence of iterated series { i=1 j=1 tj xij } converge uniformly for m ∈ N. Proof: If the conclusion fails to hold, then there exists a neighborhood of 0, U , in X such that for every k there exist jk > k and mk such that mk X ∞ X tj xij ∈ / U. i=1 j=jk

Pick a balanced neighborhood of 0, V , such that V + V ⊂ U . There exists lk > jk such that mk X ∞ X tj xij ∈ V i=1 j=lk +1

so

(∗)

lk mk X X

i=1 j=jk

tj xij ∈ / V.

By the condition in (∗) for k = 1, there exist j1 < l1 and m1 such that P m1 P l 1 / V. There exists J1 > j1 such that j=j1 tj xij ∈ i=1 (∗∗)

m n+p X X i=1 j=n

tj xij ∈ V for 1 ≤ m ≤ m1 , n > J1 and p > 0.

P m2 P l 2 By (∗) there exist l2 > j2 > J1 and m2 such that / V. i=1 j=j2 tj xij ∈ By (∗∗), m2 > m1 . We can continue this construction to produce increasing sequences mk , lk , jk with lk−1 < jk < lk and (∗ ∗ ∗)

lk mk X X

i=1 j=jk

tj xij ∈ / V.

Put Ik = [jk , lk ] so {Ik } is an increasing sequence of intervals. Define a matrix mp X X M = [mpq ] = [ tj xij ]. i=1 j∈Iq

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We claim that M is a signed K-matrix (Appendix D.3). First, note P∞ that the series i=1 tj xij converges for every j by setting t = ej in the hypothesis. Thus, each column of M converges. Next, given an increasing sequence of positive integers there is a subsequence {nq } and a sequence of P∞ P signs {sq } such that the coordinate sum u = q=1 sq j∈Inq tj ∈ λ. Then ∞ X q=1

sq mpnq =

∞ X

sq

q=1

mp X X

tj xij =

i=1 j∈Inq

mp ∞ X X i=1 q=1

sq

X

tj xij =

j∈Inq

mp ∞ X X

uj xij

i=1 j=1

P P∞ P∞ so limp ∞ q=1 sq mpnq = i=1 j=1 uj xij exists. Hence, M is a signed K-matrix and by the signed version of the Antosik-Mikusinski Matrix Theorem, the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗ ∗ ∗). From Proposition 9.1 and Theorem 9.4, we obtain Corollary 9.5. Under the hypothesis of Theorem 9.4, for every t ∈ λ the P double series i,j tj xij converges and X i,j

tj xij =

∞ X ∞ X i=1 j=1

tj xij =

∞ X ∞ X

tj xij .

j=1 i=1

By strengthening the hypothesis on the multiplier space λ, we can also strengthen the conclusions of Theorem 9.4 and Corollary 9.5 to uniform convergence over bounded sets in the multiplier space. Recall that the K-space λ has the signed strong gliding hump property (signed-SGHP) if whenever {tj } is a bounded sequence in λ and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence {nj } P nj such that the coordinate sum of the series ∞ belongs to λ; j=1 sj χInj t if all of the signs can be chosen equal to 1, then λ has the strong gliding hump property (SGHP) [see Appendix B for examples]. Theorem 9.6. Let λ have signed-SGHP. Let {xij } ⊂ X. Suppose that the P series j xij is λ multiplier convergent for every i and the iterated series P∞ P∞ i=1 j=1 tj xij converges for every t = {tj } ∈ λ. Then the family of iterated series m X ∞ X tj xij : m ∈ N, t ∈ B i=1 j=1

converge uniformly for every bounded subset B ⊂ λ.

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Proof: The proof is similar to the proof of Theorem 9.4 which we now sketch. If the conclusion fails, there exists a neighborhood U of 0 in X such that for every k there exist jk > k, tk ∈ B and mk such that mk X ∞ X

i=1 j=jk

tkj xij ∈ / U.

Pick a balanced neighborhood V such that V + V ⊂ U . Then there exists lk > jk such that mk X ∞ X

i=1 j=lk +1

so

(∗)

tkj xij ∈ V

lk mk X X

i=1 j=jk

tkj xij ∈ / V.

By the condition in (∗) for k = 1, there exist j1 < l1 , t1 ∈ B and m1 such that (∗∗)

m1 X l1 X

i=1 j=j1

t1j xij ∈ / V.

By Theorem 2.16, there exists J1 > j1 such that (∗ ∗ ∗)

m n+p X X i=1 j=n

tj xij ∈ V when 1 ≤ m ≤ m1 , n > J1 , p > 0 and t ∈ B

(this is where signed-SGHP is used to guarantee the uniform convergence over B). By (∗∗) there exist l2 > j2 > J1 , m2 and t2 ∈ B such that P m2 P l 2 2 / V . By (∗ ∗ ∗) m2 > m1 . Continuing this construction j=j2 tj xij ∈ i=1 produces increasing sequences mk , lk , jk with lk−1 < jk < lk and tk ∈ B such that (∗ ∗ ∗∗)

lk mk X X

i=1 j=jk

tkj xij ∈ / V.

Put Ik = [jk , lk ] so {Ik } is an increasing sequence of intervals. Define a matrix M = [mpq ] = [

mp X X

i=1 j∈Iq

tqj xij ].

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Using the signed-SGHP, the proof of Theorem 9.4 can now be employed to show that M is a signed K-matrix. By the signed version of the AntosikMikusinski Matrix Theorem, the diagonal of M converges to 0 (Appendix D.3). But, this contradicts (∗ ∗ ∗∗). We can now use Theorem 9.6 to give a generalization of Corollary 9.5. P Let xij (a) ∈ X for i, j ∈ N and a ∈ A. The double series i,j xij (a) converge uniformly for a ∈ A if for every neighborhood, U , of 0 in X, there P∞ P∞ exists N such that i=p j=q xij (a) ∈ U for p, q ≥ N and a ∈ A. The Pm P∞ proof of Proposition 9.1 shows that if the series i=1 j=1 xij (a) converge P uniformly for m ∈ N and a ∈ A, then the double series i,j xij (a) converge uniformly for a ∈ A. Thus, from Theorem 9.6, we have Corollary 9.7. Under the hypothesis of Theorem 9.6, the double series P i,j tj xij converge uniformly for t ∈ B.

Using the proof of Theorem 9.6 and Theorem 2.22 on the uniform convergence over null sequences for multiplier spaces with 0-GHP in place of Theorem 2.16 on the uniform convergence of multiplier convergent series over bounded sets with signed-SGHP, we can obtain the following results. Theorem 9.8. Let λ have 0-GHP. Let {xij } ⊂ X. Suppose that the seP ries j xij is λ multiplier convergent for every i and the iterated series P∞ P ∞ i=1 j=1 tj xij converges for every t ∈ λ. Then the family of iterated series m X ∞ X tkj xij : m, k ∈ N i=1 j=1

converge uniformly whenever {tk } is a null sequence in λ.

Corollary 9.9. Under the hypothesis of Theorem 9.8, the double series P k k i,j tj xij converge uniformly for any null sequence {t } in λ.

As an application of Theorem 9.4, we establish a multiplier convergent version of an Orlicz-Pettis Theorem for non-locally convex TVS with a Schauder basis due to Stiles ([Sti]) which was considered for LCTVS in Chapter 4.74. Stiles’ result seems to be the first version of an OrliczPettis Theorem for non-locally convex spaces. Indeed, Kalton remarks that Stiles’ result motivated his far-reaching generalization of the OrliczPettis Theorem for series with values in a topological group ([Ka2]). Stiles’ version of the Orlicz-Pettis Theorem is for subseries convergent series with

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values in an F-space with a Schauder basis and his proof uses the metric properties of the space. Other proofs of Stiles’ result have been given in [Bs] and [Sw5]. We will establish a version of Stiles’ result for multiplier convergent series which requires no metrizability assumption. Let X be a TVS with a Schauder basis {bj } and associated coordinate functionals {fj }. That is, every x ∈ X has a unique series representation P∞ x = j=1 tj bj and fj : X → R is defined by hfj , xi = tj . We do not assume that the coordinate functionals {fj } are continuous although this is the case if X is a complete metric linear space ([Sw2] 10.1.13). Let F = {fj : j ∈ N}. We consider the weak topology σ(X, F ) on X and use the interchange theorem to establish a multiplier convergent version of the Orlicz-Pettis Theorem for σ(X, F ) and the original topology of X. P Theorem 9.10. Let λ have signed-WGHP. If j xj is λ multiplier conP vergent with respect to σ(X, F ), then j xj is λ multiplier convergent with respect to the original topology of X. P∞ P∞ Proof: Let t ∈ λ. Consider the iterated series i=1 j=1 tj hfi , xj i bi . P∞ σ(X, F ) sum of the series. Then for each i, Let x = j=1 tj xj be theP P∞ ∞ t hf , xi = hf , xi so j i i j=1 tj hfi , xi bi = hfi , xi bi , where the converj=1 gence is in the original topology of X. But, ∞ X ∞ X i=1 j=1

tj hfi , xj i bi =

∞ X i=1

hfi , xi bi = x

with convergence in the original topology of X since {bj } is a Schauder basis. By Corollary 9.5, ∞ X ∞ X i=1 j=1

tj hfi , xj i bi =

∞ X ∞ X j=1 i=1

tj hfi , xj i bi =

∞ X

t j xj

j=1

with convergence in the original topology of X.

The use of Corollary 9.5 in the proof of Theorem 9.10 removes the metrizability and completeness assumptions in Stiles’ Theorem. Note that we did not use the continuity of the coordinate functional {fj } in the proof so the topology of X and the topology σ(X, F ) may not even be compatible! Remark 9.11. Theorem 9.10 covers the non-locally convex case when λ = lp , 0 < p < 1. The case when λ = bs gives a generalization of Stiles’ result for subseries convergent series. If X is an AK-space, then {ej }

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is a Schauder basis and the weak topology σ(X, {ej }) is just the topology of coordinatewise convergence on X. Thus, from Theorem 9.10 a series in X which is λ multiplier convergent with respect to the topology of coordinatewise convergence is λ multiplier convergent with respect to the original topology of X. In particular, these remarks apply to the classical sequence spaces X = lp , 0 < p < ∞, c0 , c00 . We next consider a generalization of Theorem 9.10 for multiplier convergent series; see also Theorem 4.75 for a locally convex version. Let (X, τ ) be a TVS and assume that there exists a sequence of linear operP∞ ators Pj : X → X such that x = j=1 Pj x [convergence in X] for every x ∈ X. When each Pj is a continuous projection, {Pj } is called a Schauder decomposition ([LT]). If X has a Schauder basis {bj } with continuous coordinate functionals {fj } and Pj x = hfj , xi bj for x ∈ X, then {Pj } is a Schauder decomposition for X. Theorem 9.12. Let λ have signed-WGHP. Let (X, τ ) be a TVS and σ a Hausdorff topology on X. Assume that each Pj : X → X is σ − τ P continuous. If j xj is λ multiplier convergent with respect to σ, then P x is λ multiplier convergent with respect to τ . j j P∞ Proof: Let t ∈ λ and j=1 tj xj be the σ sum of the series. For each P∞ P∞ i, j=1 tj Pi xj = Pi ( j=1 tj xj ), where the series is τ convergent by the continuity of Pi . Hence, ∞ X ∞ X

Pi (tj xj ) =

i=1 j=1

∞ X

Pi (

i=1

∞ X

t j xj )

j=1

converges with respect to τ . By Antosik’s Interchange Theorem 9.4 (Corollary 9.5), ∞ X ∞ X i=1 j=1

Pi (tj xj ) =

∞ X

t j xj ,

j=1

where the convergence is with respect to τ . Note that Theorem 9.10 is a corollary of Theorem 9.12. For suppose that {bj } is a Schauder basis for X with coordinate functionals {fj } and define Pj : X → X by Pj x = hfj , xi bj . If F = {fj : j ∈ N} and σ = σ(X, F ), then each Pj is σ − τ continuous and Theorem 9.12 applies. We indicate applications to vector valued sequence spaces; see also Example 4.76.

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Example 9.13. Let E be a vector space of X valued sequences equipped with a vector topology τ . E is a K-space if the coordinate maps fj : E → X, fj ({xj }) = xj are continuous for every j. If E is a K-space and every P∞ x = {xj } ∈ E has a representation x = j=1 ej ⊗ xj , then E is an AKspace [Appendix C; here ej ⊗ x is the sequence with x in the j th coordinate and 0 in the other coordinates]. If E is an AK-space, define Pj : E → E by Pj ({xj }) = ej ⊗ xj . If E is an AK-space and σ is the topology of coordinatewise convergence on E, then each Pj is σ − τ continuous, where τ is the original topology of E. Thus, Theorem 9.12 is applicable and P any series j xj in E which is coordinatewise λ multiplier convergent is λ multiplier convergent in E. For situations where Example 9.13 is applicable, let X be a metric linear space whose topology is generated by the quasi-norm |·|. For 0 < p < ∞, P∞ p let lp (X) be all X-valued sequences such that j=1 |xj | < ∞. If 1 ≤ p 0 such that (∗) s · Atk > δ for every k. Set k1 = 1. Pick m1 , n1 such that X n1 m1 X k1 (∗∗) si aij tj = Pm1 s · APn1 tk1 > δ. i=1 j=1 From (b) and (c),

(∗ ∗ ∗) s · APn1 tk = AT s · Pn1 tk = limk tkj

n1 X

tkj (AT s)j for every k.

j=1

= 0 for every j so it follows from (∗ ∗ ∗) that Since λ is a K-space, s · APn1 tk → 0 as k → ∞. Therefore, there exists k2 > k1 such that (♣) s · APn1 tk2 < δ/2.

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From (∗) and (♣), s · A(tk2 − Pn1 tk2 > δ/2.

Pick m2 > m1 , n2 > n1 such that X n2 X m2 k2 si aij tj = Pm2 s · A(Pn2 tk2 − Pn1 tk2 ) > δ/2. i=1 j=n1 +1

Continuing this construction produces increasing sequences {kp }, {mp }, and {np } such that Pmp s · A(Pnp tkp − Pnp−1 tkp ) > δ/2. Let Ip = {j ∈ N : np−1 < j ≤ np }. So {Ip } is an increasing sequence of intervals with (♠) Pmp s · AχIp tkp > δ/2. Define a matrix

M = [mpq ] = [Pmp s · AχIq tkq ] = [AT Pmp s · χIq tkq ].

We claim that M is a K-matrix (Appendix D.2). First, by (a), the columns of M converge to s · AχIq tkq . Next, given any increasing sequence {rq }, by 0-GHP there is a further subsequence, still denoted by {rq }, such that P∞ t = q=1 χIrq tkrq ∈ λ. Therefore, the sequence {

∞ X q=1

mprq } = {AT Pmp s · t} = {Pmp s · At}

converges with limit s · At by (a) and (c). Hence, M is a K-matrix and the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (♠). We give several corollaries of Theorem 10.10. Corollary 10.11. Let (λ, τ ) be a K-space with 0-GHP. Let η be a vector topology on µ such that (µ, η)0 = µ0 ⊂ µβ . If A : λ → µ, then A is τ − η bounded. Proof: The hypothesis implies that σ(µ, µβ ) bounded sets are η bounded so the result follows from Theorem 10.10. For example, the hypothesis in Corollary 10.11 is satisfied if µ is an AK-space (Proposition 2.5). Corollary 10.11 is not applicable to l ∞ or its subspace m0 . We give a result which is applicable to these spaces.

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Corollary 10.12. Let (λ, τ ) be a K-space with 0-GHP. Let η be a vector topology on µ such that η and σ(µ, µβ ) have the same bounded sets. If A : λ → µ, then A is τ − η bounded. Since (l∞ )β = (m0 )β = cβ = l1 , Corollary 10.12 is applicable if µ = c, m0 or l∞ with the sup-norm. Even though bs, the space of bounded series, is not an AK-space, Corollary 10.12 is applicable to bs with its natural topology since bs = (bv0 )0 and (bs)β = bv0 (Appendix B or [KG] p. 69). We next establish another automatic continuity result which requires assumptions on the range space µ. Theorem 10.13. Let (λ, τ ) be a K-space with 0-GHP. Assume that (µ, η) is a separable K-space such that µ0 = (µ, η)0 ⊂ µβ and (1) the sectional projections {Pn } are η equicontinuous. If A : λ → µ, then A is τ − η sequentially continuous. Proof: Let xk → 0 in τ . It suffices to show that y k · Axk → 0 when {y k } is an equicontinuous subset of µ0 . If this fails we may assume that there exists δ > 0 such that k y · Axk > δ for every k. Set k1 = 1 and pick m1 , n1 such that Pm1 y k1 · Pn1 xk1 > δ.

By (b), AT y k ∈ s, the space of all sequences, and since AT y k · ej = y k · Aej = (AT y k )j

and {y k } ⊂ µ0 is σ(µ0 , µ) bounded, {(AT y k )j : k ∈ N} is bounded for every j. Since limk xkj = 0 for every j, by Lemma 7.7, limk (AT y k )j xkj = 0 for every j. Therefore, there exists k2 > k1 such that X n1 T k (A y 2 )j xk2 = y k2 · APn1 xk2 < δ/2. j j=1 Hence,

k y 2 · A(xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that Pm2 y k2 · A(Pn2 xk2 − Pn1 xk2 ) > δ/2.

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Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (z) Pmp y kp · AχIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Since {Pn } is equicontinuous, {Pn0 = Pn } will carry equicontinuous subsets of µ0 into equicontinuous subsets of µ0 . Therefore, {Pn y k : n, k ∈ N} ⊂ µ0 is equicontinuous. Since η is separable, {Pmp y kp } has a subsequence, still denoted by {Pmp y kp }, which is σ(µ0 , µ) convergent to some y ∈ µ0 ⊂ µβ ([Wi] 9.5.3, [Sw] 18.9). Define a matrix M = [mpq ] = [Pmp y kp · AχIq xkq ].

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to y ·AχIq xkq . Next, given any subsequence {rq } there is a further P∞ subsequence, still denoted by {rq }, such that t = q=1 χIrq xkrq ∈ λ. Then ∞ X q=1

mprq = Pmp y kp · At → y · At.

Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (z). Proposition 2.5 gives sufficient conditions for µ0 = (µ, η)0 = µβ to be satisfied. If µ is a barrelled AB-space (Appendix B.3), then condition (1) is satisfied. Thus, Theorem 10.13 in particular applies to the spaces l p , 1 ≤ p < ∞. Without some assumptions on the domain space λ the continuity conclusions in Theorems 10.10 and 10.13 may fail. Example 10.14. Let A = [aij ] be the matrix aij = 1 if i ≤ j and aij = 0 otherwise. Then A : c00 → c is not norm continuous with respect to the P sup-norm and c is separable (kA( nj=1 ej )k∞ = n).

The method of proof of Theorem 10.10 can also be used to establish another boundedness result with respect to the strong topology β(µ, µβ ). In contrast to Theorem 10.10 this result requires an assumption on the range space µ. Actually we are able to establish a Uniform Boundedness result for families of pointwise bounded matrix mappings. Theorem 10.15. Let (λ, τ ) be a K-space with 0-GHP. Let Ak = [akij ]:λ → µ for every k ∈ N. Assume that µ satisfies

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(2) The sectional projections Pn : µβ → µβ are uniformly bounded on σ(µβ , µ) bounded subsets with respect to σ(µβ , µ) (i.e., if B ⊂ µβ is σ(µβ , µ) bounded and t ∈ µ, then {Pn s · t : s ∈ B, n ∈ N} is bounded). If {Ak } is pointwise bounded on λ with respect to β(µ, µβ ), then {Ak } is uniformly bounded on bounded subsets of λ with respect to β(µ, µβ ). Proof: If the conclusion fails, we may assume, by passing to a subsequence if necessary, that there exist xk → 0 in λ, {y k } ⊂ µβ which is σ(µβ , µ) bounded, tk → 0 and δ > 0 such that (∗) tk y k · Ak xk > δ for all k. P P n1 m1 Ak1 xkj 1 > δ. Now Set k1 = 1. Pick m1 , n1 such that i=1 tk1 yik1 j=1 tk y k · Ak Pn1 xk = tk (Ak )T y k · Pn1 xk = limk tk xkj

n1 X

tk xkj ((Ak )T y k )j .

j=1

Since λ is a K-space, = 0 for each j, and the pointwise boundedness assumption implies that {((Ak )T y k )j : k ∈ N} is bounded for each j. Hence, limk tk y k · Ak Pn1 xk = 0 so there exist k2 > k1 such that tk2 y k2 · Ak2 Pn1 xk2 < δ/2. Therefore, from (∗), tk2 y k2 · Ak2 (xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that tk2 Pm2 y k2 · Ak2 (Pn2 xk2 − Pn1 xk2 ) > δ/2.

Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (♥) tkp Pmp y kp · Akp χIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Define a matrix

M = [mpq ] = [tkp Pmp y kp · Akp χIq xkq ].

We claim that M is a K-matrix (Appendix D.2). Since {Pn y k : n, k ∈ N} is σ(µβ , µ) bounded by (2) and {Akp χIq xkq : p ∈ N} is β(µ, µβ ) bounded, the columns of M converge to 0. Since xk → 0 in λ, by the 0-GHP, given a subsequence {rq } there is a further subsequence, still denoted by {rq }, P∞ such that x = q=1 χIrq xkrq ∈ λ. Therefore, ∞ X q=1

mprq = tkp (Akp )T Pmp y kp · x = tkp Pmp y kp · Akp x → 0

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as above. Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem the diagonal of M converges to 0 (Appendix D.2). But, this contradicts (♥). Of course, Theorem 10.15 is applicable to a single matrix A and gives a boundedness result. We now give sufficient conditions for (2) to hold. Proposition 10.16. Consider the following conditions: (α) If (µ, µβ ) is a Banach-Mackey pair [i.e., if σ(µβ , µ) bounded sets are β(µβ , µ) bounded], then (2) holds. (β) If (µ, β(µ, µβ )) is an AB-space (Appendix B.3), then (2) holds. Proof: Let B ⊂ µβ be σ(µβ , µ) bounded and x ∈ µ. (α) : By (d), {Pn x} is σ( µ, µβ ) bounded so

sup{|Pn y · x| : y ∈ B, n ∈ N} = sup{|y · Pn x| : y ∈ B, n ∈ N} < ∞

and (2) holds. (β) : {Pn x} is β(µ, µβ ) so (2) holds by the computation in part (α). We can use the methods of Theorem 10.15 to establish a BanachSteinhaus type result. Theorem 10.17. Let (λ, τ ) be a K-space with 0-GHP. Let Ak = [akij ]:λ → µ for every k ∈ N. Assume that µ satisfies (2) The sectional projections Pn : µβ → µβ are uniformly bounded on σ(µβ , µ) bounded subsets with respect to σ(µβ , µ) (i.e., if B ⊂ µβ is σ(µβ , µ) bounded and t ∈ µ, then {Pn s · t : s ∈ B, n ∈ N} is bounded). If limk Ak x exists with respect to the strong topology β(µ, µβ ), then {Ak } is τ −β(µ, µβ ) sequentially equicontinuous. Proof: If the conclusion fails to hold, we may assume, by passing to a subsequence if necessary, that there exist δ > 0, xj → 0 in λ and {y j } ⊂ µβ which is σ(µβ , µ) bounded such that j y · Aj xj > δ. Set k1 = 1 and pick m1 , n1 such that Pm1 y k1 · Ak1 Pn1 xk1 > δ. Since {Ak ej : k ∈ N} is strong bounded for every j, {(Ak )T y k · ej = y k · Ak ej = ((Ak )T y k )j : k ∈ N}

is bounded for every j. Since limk xkj = 0 for every j, by Lemma 7.7, lim((Ak )T y k )j xkj = 0 k

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for every j. Therefore, there exist k2 > k1 such that k y 2 · Ak2 Pn1 xk2 < δ/2. Therefore,

k y 2 · Ak2 (xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that Pm2 y k2 · Ak2 (Pn2 xk2 − Pn1 xk2 ) > δ/2.

Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that (∗) Pmp y kp · Akp χIp xkp > δ/2, where Ip = {j ∈ N : np−1 < j ≤ np }. Define a matrix

M = [mpq ] = [Pmp y kp · Akp χIq xkq ]. We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge since {Ak } converges pointwise with respect to the strong topology and (2) implies that {Pmp y kp } is σ(µβ , µ) bounded. Next, given an increasing sequence {rq } there exists a further subsequence, still denoted P∞ by {rq }, such that t = q=1 χIrq xkrq ∈ λ. Then ∞ X q=1

mprq = Pmp y kp · Akp t

and the sequence {Pmp y kp · Akp t} converges by the same argument that the columns converge. Hence, M is a K-matrix so by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗). The original version of the Hellinger-Toeplitz Theorem asserts that if P P∞ 2 the series a(x, y) = ∞ i=1 j=1 aij xj yi converges for every x, y ∈ l , then there exists M such that |a(x, y)| ≤ M for kxk2 ≤ 1, kyk2 ≤ 1 ([HT]). That is, the bilinear form a induced by the matrix A is a continuous bilinear form a : l2 × l2 → R. We can use the method of proof in Theorem 10.10 to establish the sequential continuity of bilinear forms between products of sequence spaces which are induced by matrices. If the series a(x, y) = P∞ P∞ i=1 j=1 aij xj yi = y · Ax converges for every x ∈ λ, y ∈ µ, where λ and µ are sequence spaces, then a is a bilinear form on λ × µ. We use the

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method of proof in Theorem 10.10 to establish a sequential continuity result for such bilinear forms. Theorem 10.18. Let (λ, τ ) be a K-space with 0-GHP and let (µ, η) be a K-space such that µβ ⊂ µ0 = (µ, η)0 . Assume (3) the sectional projections Pn : µ → µ are sequentially equicontinuous with respect to η. P P∞ If a(x, y) = ∞ i=1 j=1 aij xj yi = y · Ax converges for every x ∈ λ, y ∈ µ, then a : λ × µ → R is sequentially τ × η continuous.

Proof: If the conclusion fails, there exist sequences xk → 0 in τ , y k → 0 in η and δ > 0 such that k k k a(x , y ) = y · Axk > δ

for all k. Set k1 = 1 and pick m1 , n1 such that X n1 m1 X k1 k1 aij xj yj > δ. i=1 j=1

Note that Ax ∈ µβ ⊂ µ0 for each x ∈ λ by the convergence of the series P∞ P∞ i=1 j=1 aij xj yi for each y ∈ µ. For each j the series a(ej , y k ) =

∞ X i=1

aij yik = y k · Aej = (AT y k )j

converges, and since {y k } is η convergent to 0 and Aej ∈ µ0 by the observation above, for each j, {(AT y k )j : k ∈ N } is bounded. Since λ is a K-space, limk xkj = 0 for for each j. Therefore, by Lemma 7.7, lim k

n1 X j=1

xkj (AT y k )j = lim k

n1 X ∞ X

aij xkj yik = 0

j=1 i=1

so there exists k2 > k1 such that n1 ∞ X X k k2 k2 2 · APn1 xk2 < δ/2. a x y ij j i = y j=1 i=1 Hence,

k y 2 · A(xk2 − Pn1 xk2 ) > δ/2.

Pick m2 > m1 , n2 > n1 such that m2 n 2 X X k2 k2 a x y ij j j > δ/2. i=1 j=n1 +1

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Continuing this construction produces increasing sequences {kp }, {mp } and {np } such that Pmp y kp · A(Pnp − Pnp−1 )xkp > δ/2. Set Ip = {j ∈ N : np−1 < j ≤ np } so (♦) Pmp y kp · AχIp xkp > δ/2. Define a matrix

M = [mpq ] = [Pmp y kp · AχIq xkq ]. We claim that M is a K-matrix (Appendix D.2). First, since y k → 0 in η, from (3) Pk y k → 0 in η and AχIq xkq ∈ µ0 , the columns of M converge to 0. Next, given any subsequence {rq }, there is a further subsequence, P∞ kr q still denoted by {rq }, such that t = ∈ λ by the 0-GHP q=1 χIrq x assumption. Then ∞ X q=1

0

mprq = Pmp y kp · At → 0

since At ∈ µ by the observation above. Hence, M is a K-matrix and by the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (♦). Proposition 2.5 gives sufficient conditions for the condition µβ ⊂ µ0 = (µ, η)0 to be satisfied. Note that this condition is satisfied by the topology σ(µ, µβ ). If µ is a barrelled AB-space, then condition (3) is satisfied. Thus, Theorem 10.18 is applicable to a wide range of sequence spaces including lp , 0 < p < ∞, and includes the original Hellinger-Toeplitz result.

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Chapter 11

Operator Valued Series and Vector Valued Multipliers

In this chapter we will consider operator valued series but allow the space of multipliers to be vector valued with values in the domain space of the operators. Throughout this chapter let X, Y be LCTVS and L(X, Y ) the space of all continuous linear operators from X into Y . Let E be a vector valued sequence space with values in X which contains c00 (X), the space of all X valued sequences which are eventually 0. P Definition 11.1. A series j Tj in L(X, Y ) is E multiplier convergent if P∞ the series j=1 Tj xj converges in Y for every x = {xj } ∈ E. The series P P∞ j Tj is E multiplier Cauchy if the series j=1 Tj xj is Cauchy for every x = {xj } ∈ E. The elements of E are called multipliers. P If E = l∞ (X), a series j Tj which is l∞ (X) multiplier convergent is said to be bounded multiplier convergent. If E = m0 (X), then a series P j Tj which is m0 (X) multiplier convergent is subseries convergent in the strong operator topology. We now consider the basic properties of E multiplier convergent series. Many of these properties are the same as those for scalar valued multiplier convergent series and the proofs are essentially identical. When this phenomena occurs we will make references to the appropriate scalar results. However, due to the presence of continuous linear operators, the results for vector valued multipliers and operator valued series often require additional technical assumptions which we will indicate. P Let j Tj be E multiplier convergent. The summing operator S : E → P Y (with respect to j Tj and E) is defined by Sx =

∞ X j=1

Tj xj , x = {xj } ∈ E. 169

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Recall the β-dual of E (with respect to the scalar field) is defined to be E β = {{x0j } : x0j ∈ X 0 ,

∞ X

x0j , xj

j=1

converges for every {xj } ∈ E}

P∞ 0 and E, E β form a dual pair under the pairing x0 · x = j=1 xj , xj (Appendix C). Also, recall that a locally convex topology w defined for dual pairs is a Hellinger-Toeplitz topology if whenever X, X 0 and Y, Y 0 are dual pairs and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is a continuous linear operator, then T : (X, w(X, X 0 )) → (Y, w(Y, Y 0 )) is continuous (Appendix A.1). As in Theorem 2.2, we have Theorem 11.2. The summing operator S : E → Y is σ(E, E β ) − σ(Y, Y 0 ) continuous. Therefore, S is w(E, E β ) − w(Y, Y 0 ) continuous for any Hellinger-Toeplitz topology w. As in Corollaries 2.3 and 2.4, we have Corollary 11.3. If B is σ(E, E β ) bounded, then SB = { B} is bounded in Y.

P∞

j=1

T j xj : x ∈

Corollary 11.4. Let E be a K-space. If E β ⊂ E 0 and B ⊂ E is bounded, P∞ then SB = { j=1 Tj xj : x ∈ B} is bounded in Y.

For conditions which guarantee that E β ⊂ E 0 , we have the analogue of Proposition 2.5. For this we need the following property of vector valued sequence spaces. If z ∈ X and j ∈ N, recall that ej ⊗ z is the sequence with z in the j th coordinate and 0 in the other coordinates. The space E has the property (I) if the maps z → ej ⊗ z are continuous from X into E for every j. Proposition 11.5. We have the following conditions:

(i) If E is a barrelled K-space, then E β ⊂ E 0 . (ii) If E is an AK-space with property (I), then E 0 ⊂ E β . (iii) If E is a barrelled AK-space with property (I), then E 0 = E β .

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Proof: (i): Let y ∈ E β . For each n define Fn : E → R by Fn (x) = Pn j=1 hyj , xj i. Since E is a K-space, each Fn is continuous and linear. Since Fn (x) → y · x for every x ∈ E, the map x → y · x is continuous by the barrelledness assumption. Hence, y ∈ E 0 .

(ii): Let F ∈ E 0 . Define yj : X → R by hyj , xi = F, ej ⊗ x . Since E has property (I), yj ∈ X 0 . Set y = {yj }. If x ∈ E, then * ∞ + ∞ ∞ X X

X j hF, xi = F, e ⊗ xj = F, ej ⊗ xj = hyj , xj i . j=1

j=1

j=1

Therefore, y ∈ E β and hF, xi = y · x. (iii) follows from (i) and (ii). Corollary 11.6. Assume that E is a barrelled AB-space (Appendix C.3) P T is E multiplier convergent. If B ⊂ E is bounded, then and Pn j j { j=1 Tj xj : n ∈ N, x ∈ B} is β(Y, Y 0 ) bounded. P Proof: Let Pn : E → E be the section operator Pn (x) = nj=1 ej ⊗ xj . By the AB assumption {Pn : n} is pointwise bounded on E and, therefore, equicontinuous since E is barrelled. Since E is barrelled, E has the strong topology β(E, E 0 ) so {Pn x : n ∈ N, x ∈ B} is β(E, E 0 ) bounded. By Proposition 11.5 , E β ⊂ E 0 so {Pn x : n ∈ N, x ∈ B} is β(E, E β ) bounded. The result now follows from Theorem 11.2 since the strong topology is a Hellinger-Toeplitz topology. Recall the condition that E β ⊂ E 0 is important even in the scalar case (Example 2.9). From Corollary C.7 in Appendix C if X is a Frechet space, then l∞ (X) is a Frechet space with property AB so we have P Corollary 11.7. Let j Tj be bounded multiplier convergent. If X is a P∞ Frechet space and B ⊂ l ∞ (X) is bounded, then { j=1 Tj xj : x ∈ B} is β(Y, Y 0 ) bounded. We have the analogue of Theorem 2.11. Recall that E has 0-GHP if whenever xj → 0 in E and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that the coordinatewise sum of the series P∞ nj ∈ E (Appendix C). j=1 χInj x P Theorem 11.8. Let E be a K-space with 0-GHP. If j Tj is E multiplier convergent, then the summing operator S : E → Y is sequentially continuous and, therefore, bounded.

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We next consider the uniform convergence of operator valued series over bounded subsets of the multiplier space E. Recall that E has the signedSGHP if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, there exist a sequence of signs {sj } and a subsequence P∞ {nj } such that the coordinatewise sum of the series j=1 sj χInj xnj ∈ E (Appendix C). We have the analogue of Theorem 2.16. P Theorem 11.9. Let E have signed-SGHP. If j Tj is E multiplier converP∞ gent, then the series j=1 Tj xj converge uniformly for x = {xj } belonging to bounded subsets of E. Without the signed-SGHP assumption, the conclusion of Theorem 11.9 may fail. Example 11.10. Let 1 ≤ p < ∞ and define Qk : lp → lp by Qk t = tk ek . P P P For t ∈ lp and σ ⊂ N, k∈σ Qk t = k∈σ tk ek converges in l p so k Qk is m0 (lp ) multiplier convergent since m0 (lp ) = span{χσ t : σ ⊂ N, t ∈ P lp }. However, the series k Qk tk do not converge uniformly for t = {tk } belonging to bounded subsets of m0 (lp ). For let tk be the constant sequence P P∞ k k k in m0 (lp ) with ek in each coordinate. Then ∞ j=n Qj tj = j=n Qj e = e if k ≥ n. As a corollary of Theorem 11.9 we have an important property of bounded multiplier convergent series which was established for Banach spaces by Batt ([Bt]). P Corollary 11.11. Let j Tj be bounded multiplier convergent. Then the P∞ series j=1 Tj xj converge uniformly for x = {xj } belonging to bounded subsets of l∞ (X). Remark 11.12. We can give another interesting proof of Corollary 11.11 above by employing the lemma of Li (Lemma 3.29). Suppose that B ⊂ l∞ (X) is bounded and let p be a continuous semi-norm on X. There exists M > 0 such that sup{p(xj ) : x = {xj } ∈ B} ≤ M . Put Ej = {x ∈ X : P p(x) ≤ M } and define fj : Ej → X by fj (x) = Tj x. Then ∞ j=1 fj (xj ) = P∞ ∞ j xj converges for every x = {xj } ∈ Πj=1 Ej . By Lemma 3.29, the j=1 T P ∞ series j=1 Tj xj converge uniformly for x = {xj } ∈ Π∞ j=1 Ej so the series converge uniformly for {xj } ∈ B. The analogue of Theorem 2.22 also holds.

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P Theorem 11.13. Let E be a K-space with 0-GHP. If j Tj is E multiplier P ∞ convergent and xk → 0 in E, then the series j=1 Tj xkj converge uniformly for k ∈ N. We next consider uniform convergence results for families of E multiplier convergent series. The β-dual of E with respect to Y is E βY = {{Tj } : Tj ∈ L(X, Y ),

∞ X j=1

Tj xj converges for every {xj } ∈ E}.

The topology w(E βY , E) on E βY is defined to be the weakest topology such P βY that the mappings T = {Tj } → T · x = ∞ into Y are j=1 Tj xj from E continuous for every x ∈ E. If X is the scalar field, these notations agree with those employed in Chapter 2. The analogue of Theorem 2.26 holds. Recall E has signed-WGHP if for every x ∈ E and every increasing sequence {Ij }, there exist a sequence of signs {sj } and a subsequence {nj } such that P∞ the coordinatewise sum of the series j=1 sj χInj x ∈ E (Appendix C).

Theorem 11.14. Assume that E has signed-WGHP. If {T k } ⊂ E βY is such that limk T k · x exists for every x ∈ E, then for every x ∈ E the series P∞ k j=1 Tj xj converge uniformly for k ∈ N. We next consider the analogue of Stuart’s weak completeness theorem (Corollary 2.28). This is the point where the significant differences between the scalar and vector cases appear.

Definition 11.15. The pair (X, Y ) has the Banach-Steinhaus property if whenever {Tj } ⊂ L(X, Y ) is pointwise convergent, lim Tj x = T x exists for every x ∈ X, then T ∈ L(X, Y ). For example, if X is barrelled, then (X, Y ) has the Banach-Steinhaus property for every LCTVS Y ([Sw2] 24.12, [Wi] 9.3.7). Lemma 11.16. Let {T k } ⊂ E βY be such that there for every j, there exists Tj ∈ L(X, Y ) with limk Tjk x = Tj x for every x ∈ X. If for every x ∈ E, P k limk T k · x exists and the series ∞ j=1 Tj xj converge uniformly for k ∈ N, βY k then T = {Tj } ∈ E and T → T in w(E βY , E). Proof: Let x ∈ E and set u = limk T k · x. It suffices to show that P∞ u = j=1 Tj xj . Let U be a balanced neighborhood of 0 in Y and pick a balanced neighborhood of 0, V , such that V + V + V ⊂ U . There exists p

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P∞ such that j=n Tjk xj ∈ V for n ≥ p, k ∈ N. Fix n ≥ p. There exists k = kn P∞ Pn such that j=1 Tjk xj − u ∈ V and j=1 (Tjk − Tj )xj ∈ V . Then n X j=1

Tj xj −u = (

∞ X j=1

Tjk xj −u)−

and the result follows.

n ∞ X X (Tjk −Tj )xj − Tjk xj ∈ V +V +V ⊂ U j=1

j=n+1

Lemma 11.17. Let {T k } ⊂ E βY and let (X, Y ) have the BanachSteinhaus property. If for every x ∈ E, limk T k · x exists and the series P∞ k βY such j=1 Tj xj converge uniformly for k ∈ N, then there exists T ∈ E k βY that T → T in w(E , E). Proof: For each j define a linear map Tj : X → Y by Tj z = lim T k · (ej ⊗ z) = lim Tjk z. k

k

By the Banach-Steinhaus assumption, Tj ∈ L(X, Y ). The result now follows from Lemma 11.16. From Theorem 11.14 and Lemma 11.17, we can now obtain Stuart’s completeness result for vector valued sequence spaces. Corollary 11.18. (Stuart) Let E have signed-WGHP, Y be sequentially complete and (X, Y ) have the Banach-Steinhaus property. If {T k } is Cauchy in w(E βY , E), then there exists T ∈ E βY such that T k → T in w(E βY , E). That is, w(E βY , E) is sequentially complete. Proof: For each z ∈ X and j, the sequence {Tjk z}k = {T k · (ej ⊗ z)}k is Cauchy in Y . By the sequential completeness assumption, limk Tjk z = Tj z exists. The result now follows from Theorem 11.14 and Lemma 11.17. The assumption that the pair (X, Y ) has the Banach-Steinhaus property is necessary for the conclusion of Corollary 11.18 to hold. For suppose that Tk ∈ L(X, Y ) and lim Tk x = T x exists for every x ∈ X. Define T k ∈ E βY by T k = (Tk , 0, 0, ...). Then lim T k · x = lim Tk x1 exists for every {xj } ∈ E so {T k } is w(E βY , E) Cauchy. If w(E βY , E) is sequentially complete and T 0 = w(E βY , E) − lim T k , then T10 = T ∈ L(X, Y ) so (X, Y ) has the Banach-Steinhaus property. We state another corollary of Theorem 11.14. A subset F of E βY is conditionally w(E βY , E) sequentially compact if every sequence {T k } ⊂ F has a subsequence {T nk } which is such that lim T nk · x exists for every x ∈ E. From Theorem 11.14, we have

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Corollary 11.19. Let E have signed-WGHP and (X, Y ) have the BanachSteinhaus property. If F ⊂ E βY is conditionally w(E βY , E) sequentially P∞ compact and x ∈ E, then the series j=1 Tj xj converge uniformly for T ∈ F. We next consider a uniform convergence result for the strong topology of Y . This requires the ∞-GHP assumption. The space E has the ∞-GHP if whenever x ∈ E and {Ij } is an increasing sequence of intervals there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that the coordinatewise sum of P the series ∞ j=1 apj χIpj x ∈ E (Appendix C).

Theorem 11.20. Assume that E has ∞-GHP. If F ⊂ E βY is pointwise bounded on E with respect to β(Y, Y 0 ) , then for every x ∈ E the series P∞ 0 j=1 Tj xj converge uniformly in β(Y, Y ) for T ∈ F .

Proof: If the conclusion fails, there exist > 0, {T k } ⊂ F, {yk0 } ⊂ Y 0 σ(Y 0 , Y ) bounded and an increasing sequence of intervals {Ik } such that * + 0 X k Tl xl > for all k. (∗) yk , l∈Ik

By ∞-GHP, there exist {nk }, apk > 0, apk → 0 such that every subsequence of {nk } has a further subsequence {pk } such that the coordinatewise P∞ sum of the series j=1 apj χIpj x ∈ E. Define a matrix * + X M = [mij ] = yn0 i /ani , Tlni (apj xl ) . l∈Ij

We claim that M is a K-matrix (Appendix D.2). First, since F is pointwise β(Y 0 , Y ) bounded on E , {yi0 } is σ(Y 0 , Y ) bounded and 1/api → 0, the columns of M converge to 0. Next, given any subsequence there is a further P∞ subsequence {pj } such that u = j=1 apj χIpj x ∈ E. Therefore, * + ∞ ∞ X X pi 0 T uj → 0 mipj = yni /ani , j=1

j=1

by the same argument as above. Hence, M is a K-matrix and the diagonal of M converges to 0 by the Antosik-Mikusinski Matrix Theorem (Appendix D.2). But, this contradicts (∗). From Theorem 11.20 and Lemma 11.17, we have the following weak sequential completeness result.

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Corollary 11.21. Let E have ∞-GHP, Y be a sequentially complete, barrelled space and (X, Y ) have the Banach-Steinhaus property. Then w(E βY , E) is sequentially complete. Proof: Since Y is barrelled the original topology of Y is just β(Y, Y 0 ). Thus, the result follows from Theorem 11.20 and Lemma 11.17 since any w(E βY , E) Cauchy sequence {T k } is pointwise bounded on E and lim T k · x exists for every x ∈ E by the sequential completeness of Y . As noted in Remark 2.34, the ∞-GHP and signed-WGHP are independent so the results in Corollaries 11.18 and 11.21 cover different spaces. We have the vector analogues of Theorems 2.35 and 2.39. Theorem 11.22. Assume that E has signed-SGHP. If {T k } ⊂ E βY is such that lim T k · x exists for every x ∈ E and B ⊂ E is bounded, then the P k series ∞ j=1 Tj xj converge uniformly for k ∈ N, x ∈ B.

Theorem 11.23. Assume that E has 0-GHP. If {T k } ⊂ E βY is such that lim T k · x exists for every x ∈ E and xk → 0 in E, then the series P∞ k l j=1 Tj xj converge uniformly for k, l ∈ N.

Examples 2.30 and 2.40 show the gliding hump assumptions above are important. We next discuss the relationships between absolute convergence, bounded multiplier convergence and subseries convergence for operator valued series. For the topologies on the space L(X, Y ) which are employed, see Appendix C. P Theorem 11.24. Let Y be sequentially complete. If the series j Tj is absolutely convergent in Lb (X, Y ), then the series is bounded multiplier convergent. Proof: Let {xj } ⊂ X be bounded and set A = {xj : j ∈ N}. Let p be a continuous semi-norm on Y . If n > m, then n n n X X X p( T j xj ) ≤ p(Tj xj ) ≤ pA (Tj ) j=m

j=m

j=m

so the result follows from the completeness of Y .

The converse of the result above does not hold. Example 11.25. Define Tj : R → c0 by Tj t = (t/j)ej . Then Tj is continuous, linear with kTj k = 1/j. If {tj } is bounded in R, then the series

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P∞

P P j = ∞ j=1 (tj /j)e converges in c0 so the series j Tj is bounded multiplier convergent but not absolutely convergent. P Theorem 11.26. If the series j Tj is bounded multiplier convergent, then the series is subseries convergent in Lb (X, Y ). P Proof: We first claim that the series j Tj is convergent in Lb (X, Y ). If this fails to hold, there exist a continuous semi-norm p on Y , a bounded subset A ⊂ X, an increasing sequence of intervals {Ij } and > 0 such that P P pA ( i∈Ij Ti ) > for all j. Pick xj ∈ A such that p(( i∈Ij Ti )xj ) > for all j. Define z ∈ l∞ (X) by zi = xj if i ∈ Ij and zi = 0 otherwise. Then the P∞ series j=1 Tj zj is not convergent since the series fails the Cauchy criterion. Since the same argument can be applied to any subseries of the series P j Tj , the result follows. j=1 tj Tj

The converse of the result in Theorem 11.26 fails to hold.

Example 11.27. Let X = Y = l 1 and define Tj : l1 → l1 by Tj t = P

P

(tj /j)ej . If σ ⊂ N, then j∈σ Tj ≤ supj∈σ |1/j| so the series j Tj is subseries convergent in the uniform operator topology. However, the series P∞ P T is not bounded multiplier convergent since the series j=1 Tj ej = Pj∞ j j 1 j=1 (1/j)e is not convergent in l .

Obviously, a series which is subseries convergent in Lb (X, Y ) is subseries convergent in the strong operator topology, Ls (X, Y ), but the converse is false. Example 11.28. Let X = Y = l 1 and define Tj : l1 → l1 by Tj t = tj ej so Tj is continuous, linear with kTj k = 1. If σ ⊂ N and t ∈ l 1 , then

X

X

|tj | T t j j =

j∈σ

j∈σ P so the series j Tj is subseries convergent in the strong operator topology. However, since kTj k = 1 the series is not subseries convergent in the uniform operator topology. We establish partial converses to the statements above. Proposition 11.29. Let X be a normed space and Y a Banach space. Every l∞ (X) multiplier convergent series is absolutely convergent iff Y is finite dimensional.

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Proof: Suppose that Y is infinite dimensional. By the DvoretskyRogers Theorem ([Sw2] 30.1.1, [Day]), there is a subseries convergent series P Pick x0 ∈ X 0 , x0 6= 0. Define j yj in Y which is not absolutely convergent. P Tj ∈ L(X, Y ) by Tj x = hx0 , xi yj . Then j Tj is l∞ (X) multiplier conP vergent since the series j yj is also bounded multiplier convergent (2.54). P However, j Tj is not absolutely convergent since kTj k = kx0 k kyj k. P Suppose Y = Rn and j Tj is l∞ (X) multiplier convergent in L(X, Y ). P Since l∞ (X) is monotone, the series ∞ j=1 Tj xj is subseries convergent in P ∞ For each j, pick Y for every {xj } ∈ l (X). Therefore, ∞ j=1 kTj xj k < ∞.P ∞ xj ∈ X, kxj k ≤ 1, such that kTj k ≤ kTj xj k + 1/2j . Then j=1 kTj k < ∞. Proposition 11.30. If X = Rn and Y is a Banach space, then every P series j Tj which is subseries convergent in the strong operator topology P of L(X, Y ) is l∞ (X) multiplier convergent. In particular, the series j Tj is subseries convergent in Lb (X, Y ).

Proof: Let {ei }ni=1 be the canonical base in Rn . For i = 1, ..., n and P i i x ∈ l∞ (X), the series ∞ j=1 (e · xj )Tj e is bounded multiplier convergent since Y is complete (2.54). Therefore, the series n X ∞ X i=1 j=1

i

i

(e · xj )Tj e =

∞ X n X j=1 i=1

i

i

(e · xj )Tj e =

∞ X

T j xj

j=1

converges in Y . Proposition 11.31. Suppose X is a normed space and Y is a Banach space. Then every series in L(X, Y ) which is subseries convergent in the uniform operator topology is l ∞ (X) multiplier convergent iff X is finite dimensional. Proof: Suppose X is infinite dimensional. Since X 0 is infinite dimenP sional, there exists a series j x0j in X 0 which is subseries convergent in X 0 but ∞ X

0

x = ∞ j j=1

(Dvoretsky-Rogers Theorem ([Day], Pick y ∈ Y, y 6= 0. [Sw2] 30.1.1)). P Define Tj : X → Y by Tj x = x0j , x y. Then j Tj is subseries convergent in the uniform operator topology l ∞(X) multiplier convergent (pick

0 but not 0

xj ∈ X, kxj k ≤ 1, such that xj ≤ xj , xj + 1/2j ). The converse follows from Theorem 11.30.

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We next establish several results relating vector and scalar multiplier convergent series. For the definition of the spaces ν{X}, see Appendix C. P Proposition 11.32. If the series j Tj is ν{X} multiplier convergent, then the series is ν multiplier convergent in the strong operator topology. Proof: Let t ∈ ν and x ∈ X so tx ∈ ν{X} and P∞ j=1 tj Tj x converges.

P∞

j=1

Tj (tj x) =

The converse of the result above fails to hold.

P Example 11.33. Define Tj ∈ L(l2 , l2 ) by Tj t = tj ej . The series j Tj is bounded multiplier convergent in the strong operator topology since if P∞ P j 2 = ∞ s ∈ l∞ and t ∈ l2 , the series j=1 sj tj e converges in l . j=1 sj Tj tP P ∞ ∞ j ∞ 2 j j However, {e } ∈ l (l ) and j=1 Tj e = j=1 e does not converge in l 2 P so the series j Tj is not l∞ (l2 ) multiplier convergent. We do have a partial converse to Proposition 11.32.

P Proposition 11.34. Let X = Rn be finite dimensional. If the series j Tj is ν multiplier convergent in the strong operator topology of L(X, Y ), then the series is ν{X} multiplier convergent. Proof: Let {ei }ni=1 be the canonical basis for R n . If x ∈ ν{X}, then {kxj k} ∈ ν and since ν is normal and kxj k ≥ ei · xj for every i, {ei ·xj }j ∈ ν. Therefore, lim m

m X

Tj xj = lim m

j=1

= lim m

so the series

P∞

j=1

m X j=1

Tj (

n X

n X m X i=1 j=1

i=1

(ei · xj )ei )

(ei · xj )Tj ei =

n X ∞ X i=1 j=1

(ei · xj )Tj ei

Tj xj converges.

The results in Propositions 11.32 and 11.34 hold if ν = c0 or ν = lp (0 < p ≤ ∞). We now consider some applications to operator valued set functions. For this we assume that X and Y are normed spaces. Let A be an algebra of subsets of a set S and µ : A → L(X, Y ) be finitely additive. The (operator) semi-variation of µ is defined to be

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) ( n

X

µ(Ai )xi : {Ai } ⊂ A a partition of A, kxi k ≤ 1 . µ ˆ (A) = sup

i=1

It is clear that the semi-variation is subadditive and monotone. The operator semi-variation is employed when integrating X valued measurable functions with respect to operator valued measures with values in L(X, Y ) ([Bar]). Since µ ˆ (A) ≥ kµk(A) for any A ∈ A (as computed with respect to the norm in L(X, Y )), if µ ˆ (S) < ∞, then µ has bounded semi-variation and is, therefore, bounded (Proposition 3.45). However, µ can be bounded and have infinite operator semi-variation. P Example 11.35. Define µ : 2N → L(l2 , R) = l2 by µ(A) = j∈A Tj where Tj = (1/j)ej ∈ L(l2 , R) = l2 . Then µ is countably additive and P n Pn bounded. However, Tj ej = (1/j)ej · ej = 1/j so j=1 Tj ej = j=1 1/j Pn and µ ˆ ({1, ...n}) = j=1 1/j → ∞ so the operator semi-variation of µ is infinite. We have a multiplier convergent characterization of operator valued set functions with finite semi-variation. Theorem 11.36. The following conditions are equivalent: (i) µ has finite (operator) semi-variation. P (ii) For every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) is c0 (X) multiplier Cauchy. Proof: Assume (i). Let {Aj } ⊂ A be pairwise

let x ∈ c0 (X).

P disjoint and

ˆ (S). Hence, Put zj = xj / kxj k. For any finite σ ⊂ N, j∈σ µ(Aj )zj ≤ µ P j µ(Aj )zj is c0 multiplier Cauchy (Proposition 3.8). Therefore, X j

kxj k µ(Aj )zj =

X

µ(Aj )xj

j

is Cauchy so (ii) holds. Assume (ii). Suppose (i) fails so µ ˆ (S) = ∞. First, note that µ is bounded. For, if tj → 0 and {xj } ⊂ X is bounded, then {tj xj } ∈ c0 (X) so P j µ(Aj )tj xj is Cauchy and µ(Aj )tj xj → 0. Since {tj } ∈ c0 is arbitrary, {µ(Aj )xj } is bounded for any bounded sequence {xj }. Hence, {kµ(Aj )k} is bounded, and µ is bounded by Theorem 3.28. Put M = sup{kµ(A)k

: A ∈ A}. There exist a partition {A11 , ..., A1n , A1n+1 } and x11 , ..., x1n+1 , x1j ≤ 1,

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P

n+1 with j=1 µ(A1j )x1j > 1 + M , where some {A1j }, say A1n+1 , satisfies

µ ˆ (A1n+1 ) = ∞. Then

X

n 1 1

µ(Aj )xj ≥ 1 + M − µ(A1n+1 )x1n+1 ≥ 1.

j=1

Now treat A1n+1 as S above to obtain a partition {A21 , ..., A2m , A2m+1 }

P

2 2 of A1n+1 and x21 , ..., x2m+1 , x2j ≤ 1, with m j=1 µ(Aj )xj ≥ 2 and

µ ˆ (A2m+1 ) = ∞. Continuing this construction produces a pairwise disjoint sequence {Bj } = {A11 , ..., A1n , A21 , ..., A2m , ...} and a null sequence

{zj } = {x11 , ..., x1n , x21 /2, ..., x2m /2, ...} P such that the series j µ(Bj ) corresponding to the {Bj } is not c0 (X) multiplier Cauchy. For sequentially complete spaces, we can use Li’s Lemma 3.29 to give an improvement to Theorem 11.36. Corollary 11.37. Let Y be sequentially complete. The following are equivalent: (i) µ has finite (operator) semi-variation. P (ii) For every pairwise disjoint sequence {Aj } ⊂ A, the series j µ(Aj ) is c0 (X) multiplier convergent. (iii) For every pairwise disjoint sequence {Aj } ⊂ A and x = {xj } ∈ c0 (X), P the series j µ(Bj )zj converge uniformly for Bj ∈ A, Bj ⊂ Aj , kzj k ≤ kxj k. Proof: (i) and (ii) are equivalent by Theorem 11.36. Clearly, (iii) implies (ii). Assume (ii). We use Li’s Lemma 3.29. For this, set Ej = {(B, z) : B ∈ A, B ⊂ Aj , z ∈ X, kzk ≤ kxj k} and define fj : Ej → Y by fj (B, z) = µ(B)z. The condition (iii) now follows from Li’s Lemma.

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Another important property of operator valued set functions is that the semi-variation is continuous from above in the sense that if {Aj } ⊂ A and Aj ↓ ∅, then µ ˆ (Aj ) → 0. For example, this property was utilized by Bartle in developing properties of bilinear vector integrals ([Bar]). Note that if µ ˆ is continuous from above, then µ is countably additive (kµ(E)k ≤ µ ˆ (E) for E ∈ Σ). However, µ may be countably additive and µ ˆ may fail to be continuous from above. P Example 11.38. Let X be a Banach space and j Tj a series in L(X) which is subseries convergent in L(X) with respect to the operator norm but not bounded multiplier convergent (Example 11.27). Define µ : 2N → P L(X) by µ(E) = countably additive with respect j∈E Tj . Then µ is P to the operator norm. Since the series j Tj is not bounded multiplier convergent, there exist > 0, an increasing

sequence of intervals {Ij } and

P xj ∈ X, kxj k ≤ 1 such that k∈Ij Tk xk > . Put Ej = ∪k≥j Ik . Then Ej ↓ ∅ and µ ˆ (Ej ) ≥ µ ˆ (Ij ) ≥ so µ ˆ is not continuous from above. We give a series characterization for countably additive measures whose operator semi-variation is continuous from above. Theorem 11.39. Let Σ be a σ-algebra of subsets of a set S and µ : Σ → L(X, Y ) be countably additive in Ls (X, Y ). The following are equivalent: (i) µ ˆ is continuous from above. P (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series j µ(Aj ) is l∞ (X) multiplier Cauchy. Proof: Assume (i). Let {Aj } ⊂ Σ be pairwise disjoint and kxj k ≤ 1. Then

n+p

X n+p

ˆ (∪j=n Aj ) ≤ µ ˆ (∪∞ µ(Aj )xj j=n Aj ) → 0

≤µ

j=n

P since ∪∞ j=n Aj ↓ ∅. Therefore, j µ(Aj )xj is Cauchy and (ii) holds. Assume (ii). If (i) fails, there exist > 0, Aj ↓ ∅, Aj ∈ Σ, such that µ ˆ (Aj ) > . There exist n1 , {Ej : j = 1, ..., n1 } ⊂ Σ pairwise

disjoint and

P n 1 {xj : j = 1, ..., n1 }, kxj k ≤ 1, such that j=1 µ(A1 ∩ Ej )xj > . Since µ is countably additive in Ls (X, Y ), limi µ(Ai ∩ Ej )xj = 0 for j = 1, ..., n1 .

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Put k1 = 1. There exists k2 > k1 such that

n1

n1

X

X

(µ(Ak1 ∩ Ej )xj − µ(Ak2 ∩ Ej )xj ) =

µ((A A ) ∩ E )x k1 k2 j j > .

j=1

j=1

Treat Ak2 as A1 above to obtain a partition {Ej : j = n1 + 1, ..., n2 } of Ak2 ,

P 2 kxj k ≤ 1 for j = n1 + 1, ..., n2 , such that nj=n µ(Ak2 ∩ Ej )xj > 1 +1

P

n2

µ((Ak2 Ak3 ) ∩ Ej )xj > . Continuing and k3 > k2 such that j=n 1 +1 this construction produces a disjoint sequence {Ej } ⊂ Σ, kxj k ≤ 1, and increasing sequences {nj }, {kj } with

nj+1

X

µ((A A ) ∩ E )x kj+1 kj i i >

i=nj +1

for every j. The series

j+1 ∞ n X X

j=1 i=nj +1

µ((Akj+1 Akj ) ∩ Ei )xi

is not Cauchy so (ii) fails. Again if Y is sequentially complete, we can obtain an improvement of Theorem 11.39. Corollary 11.40. Let Y be sequentially complete and µ as in the theorem above. The following are equivalent: (i) µ ˆ is continuous from above. P (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series j µ(Aj ) is l∞ (X) multiplier convergent. (iii) For every pairwise disjoint sequence {Aj } ⊂ Σ, the series P∞ j=1 µ(Bj )xj converge uniformly for Bj ⊂ Aj , Bj ∈ Σ and xj ∈ X, kxj k ≤ 1. Proof: (i) and (ii) are equivalent by Theorem 11.39. Clearly, (iii) implies (ii). Assume (ii). We use Li’s Lemma 3.29. Set Ej = {(B, x) : B ∈ Σ, B ⊂ Aj , x ∈ X, kxk ≤ 1} and define fj : Ej → Y by fj (B, x) = µ(B)x. Then (iii) follows from Li’s Lemma.

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A sequence of finitely additive set functions µi : Σ → L(X, Y ) is uniformly continuous from above if Ej ↓ ∅ implies that limj µ ˆ i (Ej ) = 0 uniformly for i ∈ N. We have a characterization of uniform continuity from above in terms of multiplier convergent series. Theorem 11.41. Let µi : Σ → L(X, Y ) be countably additive in Ls (X, Y ) for every i ∈ N. The following are equivalent: (i) {µi } is uniformly continuous from above. (ii) For every pairwise disjoint sequence {Aj } ⊂ Σ the series P∞ { j=1 µi (Aj ) : i ∈ N} are uniformly l ∞ (X) multiplier Cauchy.

Proof: That (i) implies (ii) follows as in Theorem 11.39. Suppose that (i) fails to hold. Then there exist > 0, Aj ↓ ∅ such that for every j there exist nj > j and mj with µ ˆ mj (Anj ) > . For j = 1 there exist n1 , m1 such that µ ˆ m1 (An1 ) > . There exist pairwise disjoint {El : 1 ≤ l ≤ N1 } and xl ∈ X, kxl k ≤ 1 such that

N

1

X

µm1 (An1 ∩ El )xl > .

l=1

Since each µi is countably additive in Ls (X, Y ), limj µm1 (Aj ∩ El )xl = 0 for 1 ≤ l ≤ N1 . There exist k1 > n1 such that

N 1

X

(µm1 (An1 ∩ El )xl − µm1 (Ak1 ∩ El )xl )

l=1

N 1

X

= µm1 ((An1 Ak1 ) ∩ El )xl > .

l=1

There exists n2 > k1 , m2 such that µ ˆ m2 (An2 ) > . By the construction above there exist N2 , pairwise disjoint {El : N1 + 1 ≤ l ≤ N2 } ⊂ Σ, xl ∈ X, kxl k ≤ 1,N1 + 1 ≤ l ≤ N2 and k2 > n2 such that

N 2

X

µm2 (An2 ∩ El )xl >

l=N1 +1

and

N 2

X

µm2 ((An2 Ak2 ) ∩ El )xl > .

l=N1 +1

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Continuing this construction produces a disjoint sequence {El }, xl ∈ X, kxl k ≤ 1, a sequence {mi } and increasing sequences {nj }, {kj }, kj > nj satisfying

NX

j+1

µmi ((Anj Akj ) ∩ El )xl

> .

l=Nj +1

The series

j+1 ∞ N X X

j=1 l=Nj +1

µmi ((Anj Akj ) ∩ El )xl : i ∈ N

are not uniformly l ∞ (X) multiplier Cauchy so (ii) fails to hold.

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Chapter 12

Orlicz-Pettis Theorems for Operator Valued Series

In this chapter we consider Orlicz-Pettis type theorems for operator valued series and vector valued multipliers. Throughout this chapter let X and Y be LCTVS and L(X, Y ) the space of all continuous linear operators from X into Y . Let E be a vector space of X valued sequences which contains P the subspace c00 (X) of all sequences which are eventually 0. If j Tj is a P∞ series in L(X, Y ) and x = {xj } is a multiplier, then the series j=1 Tj xj has values in Y so any Orlicz-Pettis Theorem must focus on the topology of Y . We first have a straightforward result. P Theorem 12.1. Assume that E is monotone. If the series j Tj is E multiplier convergent with respect to the weak topology σ(Y, Y 0 ) of Y , then P the series j Tj is E multiplier convergent with respect to the topologies γ(Y, Y 0 ) and λ(Y, Y 0 ) (see Appendix A for these topologies). In particular, P the series j Tj is E multiplier convergent in the original topology of Y .

P Proof: Let x ∈ E. Since the space E is monotone, the series ∞ j=1 Tj xj 0 is subseries convergent in the weak topology σ(Y, Y ). By the Orlicz-Pettis result in Corollary 4.11, the result follows immediately. Examples of vector valued sequence spaces which are monotone are given in Appendix C. We also have the vector analogue of Theorem 4.5. Recall the β-dual of E (with respect to the scalar field) is ∞ X

x0j , xj converges for every x = {xj } ∈ E E β = {x0j } ⊂ X 0 : j=1

187

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and that E, E β form a dual pair under the bilinear pairing 0

x ·x=

{x0j }

· {xj } =

∞ X

x0j , xj

j=1

(Appendix C). Also, a locally convex topology w defined for dual pairs X, X 0 is a Hellinger-Toeplitz topology if whenever X, X 0 and Y, Y 0 are dual pairs and T : (X, σ(X, X 0 )) → (Y, σ(Y, Y 0 )) is linear and continuous, then T : (X, w(X, X 0 )) → (Y, w(Y, Y 0 )) is continuous (Appendix A.1). Theorem 12.2. Let w be a Hellinger-Toeplitz topology for dual pairs. If P (E, w(E, E β )) is an AK-space and j Tj is E multiplier convergent with P respect to σ(Y, Y 0 ), then j Tj is E multiplier convergent with respect to w(Y, Y 0 ). Proof: Let S : E → Y be the summing operator with respect to the P P 0 series j Tj and the topology σ(Y, Y 0 ), Sx = ∞ j=1 Tj xj (σ(Y, Y ) sum; β 0 Chapter 11). By Theorem 11.2, S is w(E, E ) − w(Y, Y ) continuous. If P x ∈ E, x = w(E, E β ) − limn nj=1 ej ⊗ xj so Sx = w(Y, Y 0 ) − lim n

n X j=1

S(ej ⊗ xj ) = w(Y, Y 0 ) − lim n

n X j=1

T j xj =

∞ X

T j xj .

j=1

Examples of vector valued AK-spaces are given in Appendix C. We next consider the strong topology on Y . The following example shows that without some condition on the multiplier space E the series P j Tj xj will not, in general, converge in the strong topology of Y .

Example 12.3. Equip X = l ∞ with the weak topology σ(l ∞ , l1 ) and let E = l∞ (X). Define Qk : l∞ → l∞ by Qk t = tk ek . If x = {xk } ∈ E, then P∞ P∞ k k Q k xk = x e is σ(l∞ , l1 ) convergent, but if x = {ek } ∈ E, k=1P k=1 P∞k k ∞ k then k=1 Qk e = k=1 e is not β(l∞ , l1 ) = k·k∞ convergent.

We now establish the analogue of Theorem 5.7. The space E has ∞GHP if whenever x ∈ E and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every

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subsequence of {nj } has a further subsequence {pj } such that the coordiP∞ natewise sum of the series j=1 apj χIpj x ∈ E (Appendix C). P Theorem 12.4. Let E have ∞-GHP. If j Tj is E multiplier convergent P with respect to σ(Y, Y 0 ), then j Tj is E multiplier convergent with respect to β(Y, Y 0 ). Proof: If the conclusion fails to hold, there exist x ∈ E, {yk0 } σ(Y 0 , Y ) bounded, > 0 and an increasing sequence of intervals {Ik } such that X 0 (∗) hyk , Tj xj i > for all k. j∈Ik

Since E has ∞-GHP, there exist {pk }, apk > 0, apk → ∞ such that every subsequence of {pk } has a further subsequence {qk } such that the coordiP natewise sum of the series ∞ k=1 aqk χIqk x ∈ E. Define a matrix X

yp0 i /api , Tl (apj xl ) ]. M = [mij ] = [ l∈Ipj

We claim that M is a K-matrix (Appendix D.2). First, the columns of M converge to 0 since {yi0 } is σ(Y 0 , Y ) bounded and api → ∞. Next, given a subsequence there is a further subsequence {qj } such that y = P∞ j=1 aqj χIqj x ∈ E. Let z=

∞ X l=1

Tl y l =

∞ X X

Tl (aqj xl )

j=1 l∈Iqj

be the σ(Y, Y 0 ) sum of this series. Then ∞ X j=1

miqj = yp0 i /api , z → 0

so M is a K-matrix. By the Antosik-Mikusinski Matrix Theorem (Appendix D.2), the diagonal of M converges to 0. But, this contradicts (∗). Examples of vector valued sequence spaces with ∞-GHP are given in Appendix C.

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Chapter 13

Hahn-Schur Theorems for Operator Valued Series

In this chapter we consider versions of the Hahn-Schur Theorem for series with values in the space of continuous linear operators and with multipliers which are vector valued with values in the domain of the operators. Throughout this chapter let X and Y be LCTVS, L(X, Y ) the space of all continuous linear operators from X into Y , and E a vector space of X valued sequences which contains c00 (X), the space of all X valued sequences which are eventually 0. The analogue of the hypothesis (H) for the Hahn-Schur Theorem in Chapter 7 for E multiplier convergent series is straightforward. P (H) Let j TijPbe E multiplier convergent in L(X, Y ) for every i. ∞ Assume that limi j=1 Tij xj exists for every x = {xj } ∈ E and that Tj x = limi Tij x exists for every x ∈ X. We, of course, need to have that Tj ∈ L(X, Y ) for every j for reasonable conclusions to hold. We will either assume this or impose conditions which guarantee that this holds. Similarly, the analogues of the conclusions of the Hahn-Schur Theorems of Chapter 7 are also straightforward. P∞ P (C1) The series j Tj is E multiplier convergent and limi j=1 Tij xj = P∞ every {xj } ∈ E. j=1 Tj xj for P∞ P∞ (C2) limi j=1 Tij xj = j=1 Tj xj uniformly for x = {xj } belonging to bounded subsets of E. P∞ (C3) The series j=1 Tij xj converge uniformly for x = {xj } belonging to bounded subsets of E. From Theorem 11.14 and Lemma 11.16, we have the following connection between (H) and conclusion (C1). Theorem 13.1. Assume (H), that E has signed-WGHP, and that there 191

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exist Tj ∈ L(X, Y ) such that limi Tij x = Tj x for every x ∈ E. Then (C1) holds. Hypothesis (H) implies that limi Tij x = Tj x exists for every x ∈ X so if the pair (X, Y ) has the Banach-Steinhaus property, it follows that Tj ∈ L(X, Y ) for every j and the hypothesis in Theorem 13.1 is satisfied. The hypothesis in (H) implies in particular, that if Tj ∈ L(X, Y ), then (i) For each j ∈ N, limi Tij = Tj in the strong operator topology of L(X, Y ). To see this, take x = ej ⊗ z for j ∈ N and z ∈ X. The following example shows that even in the presence of the BanachSteinhaus property, property SGHP, and condition (i), the hypothesis (H) does not imply conclusion (C2). Example 13.2. Let X = l 1 and Y = R so L(X, Y ) = (l 1 )0 = l∞ . For i, j ∈ N, let Tij = ei /2j so Tij t = (ei /2j ) · t = ti /2j for t ∈ l1 . Thus, for each j, limi Tij = 0 in the strong operator topology of L(X, Y ) = l ∞ which in this case is just σ(l ∞ , l1 ). Note, however, that {Tij }i does not converge to 0 in the norm topology k·k∞ of L(X, Y ) = l∞ . Let E = l∞ (l1 ), where j l 1 has

the norm topology k·k1 and E has the sup norm. If {t } ∈ E with

tj ≤ 1 for every j, we have 1 ∞ X

Tij tj =

j=1

∞ X j=1

ei · tj /2j =

∞ X

tji /2j

j=1

P∞ and since tji ≤ 1, the series j=1 Tij tj converges (absolutely). That is, P for each i, the series j Tij is E multiplier convergent. Since ∞ ∞ ∞ ∞ X ∞ ∞ X ∞ X X j j X X j j X Tij tj = 1/2j < ∞, ti /2 ≤ ti /2 = i=1 j=1

P∞

j=1 i=1

i=1 j=1

j

j=1

limi j=1 Tij t = 0. Thus, conditions (H) and (i) hold. However, condition (C2) does not hold. Indeed, for any i, let ti ∈ E be the constant sequence ti = {ei }. Then ∞ ∞ ∞ X X X Tij ti = ei · ei /2j = 1/2j = 1 j=1

j=1

j=1

while

∞ X j=1

Tkj ti = 0

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P∞ for k 6= i. Thus, limi j=1 Tij tk = 0 does not hold uniformly for {tk } belonging to bounded subsets of E and conclusion (C2) fails. Note that E = l ∞ (l1 ) has SGHP so a straightforward analogue of the Hahn-Schur Theorem given in Theorem 7.10 does not hold. In order to obtain versions of the Hahn-Schur Theorem in which conclusions (C2) and (C3) hold, it is necessary to strengthen condition (i). In particular, we need to replace the assumption that the sequence {Tij }i converges to Tj in the strong operator topology with convergence in Lb (X, Y ) in order to obtain the conclusion (C2). Actually, we consider the more general situation where the sequences {Tij }i converge in LA (X, Y ) (see Appendix A). We first consider the conclusion (C3). P Theorem 13.3. Let in L(X, Y ) for j Tij be E multiplier convergent P∞ every i. Assume that E has signed-SGHP. If limi j=1 Tij xj exists for P∞ every x = {xj } ∈ E, then for every bounded set A the series j=1 Tij xj converge uniformly for x ∈ A, i ∈ N, i.e., conclusion (C3) holds. Proof: If the conclusion fails to hold there exist a closed neighborhood of 0,U , such that for every i there exist ki > i, a finite interval Ii with min Ii > i, xi ∈ A with X Tki k xik ∈ / U. k∈Ii

Put i1 = 1. By the condition above there exist k1 > 1, an interval I1 with min I1 > i1 , x1 ∈ A with X Tk1 k x1k ∈ / U. k∈I1

By Theorem 11.9 (this uses signed-SGHP), there exists j1 such that ∞ X k=j

Tik xk ∈ U

for every x ∈ A, j ≥ j1 , 1 ≤ i ≤ k1 . Set i2 = max[I1 + 1, j1 ]. By the condition above, there exist k2 > i2 , an interval I2 with min I2 > i2 , x2 ∈ P 2 / U . Note that k2 > k1 . Continuing this A such that k∈I2 Tk2 k xk ∈ construction produces an increasing sequence {ki }, an increasing sequence of intervals {Ii }, xi ∈ A such that X (∗) Tki xik ∈ / U. k∈Ii

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Define an infinite matrix X Tki k xjk ]. M = [mij ] = [ k∈Ij

We claim that M is a signed K-matrix (Appendix D). First the columns of M converge by hypothesis. Next, given any subsequence there exist a further subsequence {pj } and a sequence of signs {sj } such that x = P∞ pj ∈ E. Then j=1 sj χIpj x ∞ ∞ ∞ X X X X p T k i k xk Tki k xkj = sj mipj = sj j=1

j=1

P∞

k=1

k∈Ipj

so limi j=1 sj mipj exists by hypothesis. Hence, M is a signed K-matrix so the diagonal of M converges to 0 by the signed version of the AntosikMikusinski Matrix Theorem (Appendix D.3). But, this contradicts (∗).

We next consider the converse of the Hahn-Schur result above. In what follows A will denote a family of bounded subsets of X whose union is X(see Appendix A). P Theorem 13.4. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ) and for every A ∈ A the series j=1 Tij xj converge uniformly P∞ for x ∈ A, i ∈ N. Then the sequences { j=1 Tij xj }i satisfy a Cauchy condition uniformly for x ∈ A. Proof: Let U be a neighborhood of 0 in Y and pick a symmetric neighborhood of 0, V , such that V + V + V ⊂ U . There exists n such P that ∞ Tij xj ∈ V for every x ∈ A, i ∈ N. There exists m such that Pn−1 j=n (T − Tkj )xj ∈ V for x ∈ A, and i, k ≥ m. If x ∈ A and i, k ≥ m, ij j=1 then ∞ ∞ X X Tij xj − Tkj xj j=1

j=1

=

n−1 X j=1

(Tij − Tkj )xj +

and the conclusion holds.

∞ X

j=n

Tij xj −

∞ X

j=n

Tkj xj ∈ V + V + V ⊂ U

From Theorem 13.4, we have condition (H). P Corollary 13.5. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in

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P Ls (X, Y ). If for each x ∈ E the series ∞ j=1 Tij xj converge uniformly for P∞ i ∈ N, then { j=1 Tij xj }i is a Cauchy sequence. Hence, if Y is sequentially complete, hypothesis (H) is satisfied. We can also obtain a boundedness result from uniform convergence of series. P Proposition 13.6. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ) and for every A ∈ A the series j=1 Tij xj converge uniformly P∞ for x ∈ A, i ∈ N. Then B = { j=1 Tij xj : i ∈ N, x ∈ A} is bounded for every A ∈ A . Proof: Let U be a neighborhood of 0 in Y and pick a balanced neighborhood, V , such that V + V ⊂ U . There exists n such that P∞ j=n Tij xj ∈ V for i ∈ N, x ∈ A. For every j, {Tij xj : i ∈ N, xj for x = {xj } ∈ A} is bounded by the hypothesis so there exists t > 1 Pn−1 P∞ such that { j=1 Tij xj : i ∈ N, x ∈ A} ⊂ tV . Therefore, j=1 Tij xj = Pn−1 P∞ j=1 Tij xj + j=n Tij xj ∈ tV + V ⊂ tU for i ∈ N, x ∈ A.

Theorem 13.3 gives sufficient conditions for the uniform convergence hypothesis in Theorem 13.4, Corollary 13.5 and Proposition 13.6 to hold. We next consider the conclusions (C1) and (C2). P Theorem 13.7. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ LA (X, Y ). If limi j=1 Tij xj exists for every x = {xj } ∈ E and for every P∞ A ∈ A the series for x ∈ A, i ∈ N, then j=1 Tij xj converge uniformly P P∞ P∞ T is E multiplier convergent and lim T j i j j=1 ij xj = j=1 Tj xj uniformly for x ∈ A. That is, conclusions (C1) and (C2) hold for the family A. P Proof: That j Tj is E multiplier convergent and lim i

∞ X j=1

Tij xj =

∞ X

T j xj

j=1

for x ∈ E follows from Lemma 11.16. Let U be a neighborhood of 0 in Y and pick a closed, symmetric neighborhood of 0, V , such that V + V + V ⊂ U . Pn There exists N such that j=m Tij xj ∈ V for n > m ≥ N, i ∈ N and Pn x ∈ A. Hence, j=m Tj xj ∈ V for n > m ≥ N, i ∈ N and x ∈ A and,

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P therefore, ∞ j=m Tj xj ∈ V for m ≥ N, x ∈ A. There exists M such that PN −1 j=1 (Tij − Tj )xj ∈ V for i ≥ M, x ∈ A. If x ∈ A and i ≥ M , we have ∞ X j=1

Tij xj −

∞ X j=1

T j xj =

∞ X

j=N

and the result follows.

Tij xj −

∞ X

j=N

T j xj +

N −1 X j=1

(Tij −Tj )xj ∈ V +V +V ⊂ U

From Theorems 13.3 and 13.7, we have a Hahn-Schur Theorem for LA (X, Y ). P Corollary 13.8. Assume that E has signed-SGHP. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist P∞ Tj ∈ L(X, Y ) such that limi Tij = Tj in LA (X, Y ). If limi j=1 Tij xj P exists for every x = {xj } ∈ E, then j Tj is E multiplier converP P∞ gent, limi ∞ T x = T x uniformly for x ∈ A and the series j=1 ij j j=1 j j P∞ j=1 Tij xj converge uniformly for x ∈ A, i ∈ N. That is, conclusions (C2) and (C3) hold for the family A. A sufficient condition for there to exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every x ∈ X is that the pair (X, Y ) has the Banach-Steinhaus property. However, as Example 13.2 shows we must have convergence in LA (X, Y ) in order to obtain the conclusion in Theorem 13.7. Corollary 13.9. Assume that E has signed-SGHP and that Y is sequenP tially complete. Let j Tij be E multiplier convergent in L(X, Y ) for every i. Assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in LA (X, Y ). The following are equivalent: (i) (H). P∞ (ii) For each x ∈ E the series j=1 Tij xj converge uniformly for i ∈ N. P∞ (iii) For each A ∈ A the series j=1 Tij xj converge uniformly for i ∈ N, x ∈ A. P (iv) For each A ∈ A, the series j Tj is E multiplier convergent and P∞ P∞ limi j=1 Tij xj = j=1 Tj xj uniformly for x ∈ A. Proof: That (i) implies (iii) follows from Theorem 13.3; (iii) implies (iv) from Theorem 13.7; (ii) implies (i) from Corollary 13.5; clearly (iii) implies (ii) and (iv) implies (i).

We give a statement of the results above for the case of bounded (l ∞ (X)) multiplier convergent series. Note that l ∞ (X) has SGHP so these results apply.

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P Corollary 13.10. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj P∞ in Lb (X, Y ). If limi j=1 Tij xj exists for every x = {xj } ∈ l∞ (X), P∞ P then the series j Tj is bounded multiplier convergent, limi j=1 Tij xj = P∞ for x = {xj } belonging to bounded subsets A ⊂ l ∞ (X) j=1 Tj xj uniformly P∞ and the series j=1 Tij xj converge uniformly for x = {xj } belonging to bounded subsets A ⊂ l ∞ (X). P Corollary 13.11. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in Ls (X, Y ). If Y is sequentially complete and for every x = {xj } ∈ l∞ (X), P∞ P∞ the series j=1 Tij xj converge uniformly for i ∈ N, then limi j=1 Tij xj exists for every x = {xj } ∈ l∞ (X). From Proposition 13.6, we have a boundedness result. P Corollary 13.12. Let j Tij be bounded multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij = Tj in P∞ Lb (X, Y ). If for each bounded subset A of l ∞ (X) the series j=1 Tij xj P∞ converge uniformly for i ∈ N, x ∈ A, then B = { j=1 Tij xj : i ∈ N, x ∈ A} is bounded. Unlike the case of scalar multipliers, the space m0 (X) (or the subset {χσ x : σ ⊂ N, x ∈ X}) does not have SGHP so the results above do not apply to m0 (X) multiplier convergent series. However, m0 (X) does have WGHP so Theorem 13.1 applies and gives the following result. P Theorem 13.13. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) such that limi Tij x = Tj x for P∞ every x ∈ X. If limi j=1 Tij xj exists for every x = {xj } ∈ m0 (X), then P∞ P∞ P j=1 Tj xj j=1 Tij xj = j Tj is m0 (X) multiplier convergent and limi for every x = {xj } ∈ m0 (X). The following example shows that the uniform convergence conclusions in (C2) and (C3) do not hold for m0 (X) multiplier convergent series. Example 13.14. Let 1 ≤ p < ∞ and define Qj : lp → lp by Qj t = tj ej as P in Example 11.10. As noted in Example 11.10, the series j Qj is m0 (lp ) multiplier convergent. Now define Tij = Qj if j ≤ i and Tij = 0 if j > i. We have that ∞ ∞ X X Q j tj Tij tj = (∗) lim i

j=1

j=1

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for any t = {tj } ∈ m0 (lp ) and limi Tij = Qj in Lb (lp ) for every j. However, the limit in (∗) is not uniform for t belonging to bounded subsets of m0 (lp ) [take t to be the constant sequence {ek } in m0 (lp ) so P∞ Pi k k j=1 Tij tj = j=1 Qj e = e if i ≥ k]. Similarly, the uniform convergence condition in (C3) does not hold. From Theorems and 7.17 and Corollary 7.19, we do have the following results for m0 (X) multiplier convergent series. P Theorem 13.15. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every P x ∈ X. If limi ∞ j=1 Tij xj exists for every {xj } ∈ m0 (X), then

P∞ P Tj is m0 (X) multiplier convergent and limi j=1 Tij xj = j P∞ } ∈ m0 (X), j=1 Tj xj for every {x P Pj (ii) for every x ∈ X, limi j∈σ Tij x = j∈σ Tj x uniformly for σ ⊂ N, P (iii) for every x ∈ X, the series j∈σ Tij x converge uniformly for i ∈ N, σ ⊂ N. (i)

P Theorem 13.16. Let Y be sequentially complete. Let j Tij be m0 (X) multiplier convergent for every i and assume that there exist Tj ∈ L(X, Y ) with limi Tij x = Tj x for every x ∈ X. The following are equivalent: P∞ (1) limi j=1 Tij xj exists for every {xj } ∈ m0 (X). P (2) The series j Tj is m0 (X) multiplier convergent and for every x ∈ X, P P limi j∈σ Tij x = j∈σ Tj x uniformly for σ ⊂ N. P (3) For every x ∈ X the series j∈σ Tij x converge uniformly for i ∈ N, σ ⊂ N. P (4) For every x ∈ X and σ ⊂ N the series j∈σ Tij x converge uniformly for i ∈ N. P In the scalar case, if X is sequentially complete, j xij is m0 multiplier P convergent for every i, limi xij = xj exists for every j and limi j∈σ xij P exists for every σ ⊂ N, then the series j xj is l∞ multiplier convergent and P∞ P∞ limi j=1 tj xij = j=1 tj xj uniformly for k{tj }k1 ≤ 1 (Theorem 7.29). As the following example shows, the analogue of this statement does not hold for m0 (X) and l∞ (X) multiplier convergent series. P Example 13.17. Let X = l 1 and let j Tj be a series in L(X) which is subseries convergent but not l ∞ (X) multiplier convergent (Example 11.27). Set Tij = Tj if j ≤ i and Tij = 0 if j > i. If σ ⊂ N and x ∈ X,

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P P∞ P P then limi j∈σ Tij x = j∈σ Tj x so limi ∞ j=1 Tij xj = j=1 Tj xj for any {xj } ∈ m0 (X). However, if x = {xj } ∈ l∞ (X), k{xj }k ≤ 1, is such that P∞ P∞ j=1 Tj xj does not converge, then lim i j=1 Tij xj does not exist.

In Chapter 7 it was noted that a scalar matrix [aij ] which maps m0 into c also maps l∞ into c (condition (S) following Theorem 7.2). It was also shown that a vector valued matrix [xij ] which maps m0 into c(X), the space of convergent X valued sequences, also maps l ∞ into c(X) (see condition (S’) following Theorem 7.29). The example above shows that an operator valued matrix [Tij ] may map m0 (X) into c(Y ) but fail to map l ∞ (X) into c(Y ). Finally, we give an application of the operator version of the HahnSchur Theorem to obtain a version of the Nikodym Convergence Theorem for operator valued measures. Let Σ be a σ-algebra of subsets of a set S and let µi : Σ → LA (X, Y ) be countably additive. The Nikodym Convergence Theorem given in Theorem 7.47 implies that if limi µi (E) = µ(E) exists in LA (X, Y ) for every E ∈ Σ, then µ : Σ → LA (X, Y ) is countably additive and {µi } is uniformly countably additive in LA (X, Y ). We seek to obtain a version of the Nikodym Convergence Theorem for operator valued measures whose operator semi-variation is continuous from above (see the definition following Corollary 11.37). Let X, Y be normed spaces and µ : Σ → L(X, Y ) be finitely additive. Recall the operator semi-variation of µ, µ ˆ , is defined to be

n

X

: {Aj }n a partition of E, kxj k ≤ 1 µ ˆ (E) = sup µ(A )x j j j=1

j=1

(see the definition preceding Theorem 11.34). The semi-variation of µ, µ ˆ , is continuous from above if Ej ↓ ∅ implies that µ ˆ (Ej ) ↓ 0. Theorem 11.39 gives necessary and sufficient conditions for µ ˆ to be continuous from above. We derive a Nikodym Convergence Theorem for measures whose variation is continuous from above. First, the following example shows that a straightforward analogue of the Nikodym Convergence Theorem fails for such measures. P Example 13.18. Let X be a Banach space and j Tj be a series in Lb (X) which is subseries convergent in Lb (X) with respect to the operator norm but not bounded multiplier convergent (Example 11.27). Define µi : 2N → P Lb (X) by µi (E) = j∈E∩[1,i] Tj . Then each µi is countably additive and its semi-variation is continuous from above. However, limi µi (E) = µ(E) =

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P

j∈E Tj exists in Lb (X) but the semi-variation of µ is not continuous from above (Example 11.38).

We obtain a version of the Nikodym Convergence Theorem for measures whose semi-variation is continuous from above by strengthening the condition that limi µi (E) = µ(E) exists in Lb (X, Y ) for every E ∈ Σ to the condition that P∞ (#) limi j=1 µi (Ej )xj exists in Y for every pairwise disjoint sequence {Ej } ⊂ Σ and kxj k ≤ 1. Recall that a sequence of finitely additive set functions {µi } with finite semi-variation is uniformly continuous from above if Ej ↓ ∅ implies that limj µi (Ej ) = 0 uniformly for i ∈ N. Theorem 11.41 gives necessary and sufficient conditions for a sequence of set functions to be uniformly continuous from above. Theorem 13.19. Let Y be sequentially complete and let µi : Σ → Ls (X, Y ) be countably additive with µ ˆ i continuous from above. Assume that condition (#) above holds and for every E ∈ Σ there exists µ(E) ∈ L(X, Y ) with limi µi (E) = µ(E) in Lb (X, Y ). Then the semi-variation of µ, µ ˆ , is continuous from above and the semi-variations {ˆ µi } are uniformly continuous from above. Proof: Let {Ej } ⊂ Σ be pairwise disjoint and xj ∈ X, kxj k ≤ 1. P Then condition (#) and Corollary 13.10 imply that the series j µ(Ej ) P is bounded multiplier convergent and the series { j µi (Ej )}i∈N are uniformly bounded multiplier convergent. Theorems 11.39 and 11.41 now give the desired conclusion.

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Chapter 14

Automatic Continuity for Operator Valued Matrices

In this chapter we will establish several automatic continuity results for operator valued matrices analogous to those for scalar valued matrices established in Chapter 10. We first consider the relationship between an operator matrix and its transpose which require some technical assumptions. Let X, Y be TVS with L(X, Y ) the space of all continuous linear operators from X into Y . Let E [F ] be a vector space of X [Y ] valued sequences which contains c00 (X) [c00 (Y )], the space of X [Y ] valued sequences which are eventually 0 and let A = [Aij ] be an infinite matrix with Aij ∈ L(X, Y ). P∞ We say that A maps E into F if the series j=1 Aij xj converges for P∞ every i ∈ N and x = {xj } ∈ E and Ax = { j=1 Aij xj }i ∈ F for every x = {xj } ∈ E; we write A : E → F if A maps E into F . Note that if A : E → F , then the rows of A must be E multiplier convergent. We begin by considering the analogue of Theorem 10.3. Recall that the (scalar) β-dual of E is defined to be ∞ X E β = {yj } : yj ∈ X 0 , hyj , xj i = y · x converges for every x = {xj } ∈ E j=1

and E, E β form a dual pair under the pairing y · x (Appendix C).

Theorem 14.1. Let A : E → F and assume that σ(E β , E) is sequentially complete. Then A is σ(E, E β ) − σ(F, F β ) continuous and, therefore, w(E, E β ) − w(F, F β ) continuous for every Hellinger-Toeplitz topology w (Appendix A.1). 201

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Proof: Let y ∈ F β , x ∈ E. Then + + * * m ∞ ∞ ∞ X X X X yi , Aij xj y · Ax = yi , Aij xj = lim i=1

= lim m

j=1 m ∞ X X j=1 i=1

m

i=1

j=1

A0ij yi , xj = lim z m · x, m

Pm 0 m where zjm = = {zjm }j ∈ E β . Then {z m } is σ(E β , E) i=1 Aij yi , z Cauchy and, therefore, there exists z ∈ E β such that z m → z with respect to σ(E β , E) with y · Ax = z · x for every x ∈ E. This implies that A is σ(E, E β )−σ(F, F β ) continuous. The last statement follows from Appendix A.1. A sufficient condition for σ(E β , E) to be sequentially complete is that E have the signed-WGHP and X 0 be σ(X 0 , X) sequentially complete (Corollary 11.18). In the scalar case Theorem 14.1 also contains a statement concerning the transpose matrix of A; however, the operator case which we now discuss is more complicated. The transpose of the matrix A is defined to be AT = [A0ji ]. In order for the transpose AT to be defined on F β it is necessary P∞ that the series i=1 A0ij yi converge in X 0 for each y ∈ F β with respect to some locally convex topology. In order for this to be the case we henceforth assume (*) (X 0 , σ(X 0 , X)) is sequentially complete. Under the assumption in (*) the transpose matrix AT will map F β into s(X 0 ), the space of all X 0 valued sequences (Appendix C). Theorem 14.1 gives sufficient conditions for AT to map F β into E β . Corollary 14.2. Let A : E → F and assume that σ(E β , E) is sequentially complete. Under assumption (*), AT : F β → E β and y · Ax = AT y · x

for x ∈ E, y ∈ F β . Hence, AT is σ(F β , F ) − σ(E β , E) continuous and, therefore, w(F β , F )− w(E β , E) continuous with respect to any HellingerToeplitz topology w (Appendix A.1). Proof: Let the notation be as in the proof of Theorem 14.1. Then y · Ax = lim z m · x = z · x = AT y · x m P∞ 0 m since limm zj = zj = i=1 Aij yi for each j. The last statements are immediate from the equation y · Ax = AT y · x.

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Recall that even in the scalar case the matrix A may map E into F but the transpose matrix AT may fail to map F β into E β (Example 10.6). It can also be the case that A : E → F and AT : F β → E β but the condition y · Ax = AT y · x may fail to hold (Example 10.9). We next consider the problem of when AT : F β → E β will imply that A : E → F . For this we always assume that dual spaces carry the strong topology and we write E ββ = (E β )β and AT T = (AT )T . Then E ββ is a space of X 00 valued sequences and AT T consists of linear operators A00ij : X 00 → Y 00 whose restriction to X is just Aij . In order that the transpose matrix AT T map E ββ into s(Y 00 ), we assume (**) (Y 00 , σ(Y 00 , Y 0 )) is sequentially complete. From Theorem 14.1 and Corollary 14.2, we have Corollary 14.3. Assume that σ(F ββ , F β ) is sequentially complete. If AT : F β → E β , then AT T : E ββ → F ββ and z · AT y = AT T z · y for all z ∈ E ββ , y ∈ F β . Moreover, AT is σ(F β , F ββ ) − σ(E β , E ββ ) continuous and AT T is σ(E ββ , E β ) − σ(F ββ , F β ) continuous. Proof: Note that the assumption that σ(F ββ , F β ) is sequentially complete implies condition (**) so the transpose matrix AT T is defined on E ββ and the result follows from Theorem 14.1 and Corollary 14.2. To consider the problem of when AT : F β → E β will imply that A : E → F we use the second transpose AT T . For this we establish a lemma. Lemma 14.4. Let Y be semi-reflexive. If σ(F, F β ) is sequentially complete, then F = F ββ . Proof: Let z ∈ F ββ . Since Y is semi-reflexive and F ⊃ c00 (Y ), z n = (z1 , ...zn , 0, 0, ...) ∈ F , and since z ∈ F ββ , {z n } is σ(F, F β ) Cauchy. Since σ(F, F β ) is sequentially complete, {z n } converges to an element of F which must be z. Note that semi-reflexivity of Y is a necessary condition for the identity F = F ββ to hold. From Corollary 14.3 and Lemma 14.4, we obtain Corollary 14.5. Assume that σ(F, F β ) is sequentially complete and Y is semi-reflexive. If AT : F β → E β , then AT T : E ββ → F and z · AT y = AT T z · y for z ∈ E ββ , y ∈ F β so AT T is σ(E ββ , E β ) − σ(F, F β ) continuous and AT is σ(F β , F ) − σ(E β , E) continuous. In particular, A : E → F and y · Ax = AT y · x holds for x ∈ E, y ∈ F β and A is σ(E, E β ) − σ(F, F β )

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continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Corollary 14.6. Assume that Y is semi-reflexive and that σ(F, F β ) and σ(E β , E) are sequentially complete. The following are equivalent: (i) A : E → F (ii) AT : F β → E β (iii) AT T : E ββ → F. We now give applications of the automatic continuity results to concrete sequences spaces. Recall that l 1 (X) is the space of all absolutely convergent X valued series and if X is quasi-barrelled, the β-dual of l 1 (X) is l∞ (Xb0 ), the space of all strongly bounded sequences with values in X 0 (Appendix C.25). Theorem 14.7. Let X be barrelled and Y be quasi-barrelled. If A : l1 (X) → l1 (Y ), then AT : l∞ (Yb0 ) → l∞ (Xb0 ) and A is σ(l1 (X), l∞ (Xb0 )) − σ(l1 (Y ), l∞ (Yb0 )) continuous and AT is ∞ 0 1 ∞ 0 1 σ(l (Yb ), l (Y )) − σ(l (Xb ), l (X)) continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Proof: Since X is barrelled, condition (*) is satisfied so the transpose map AT is defined on l∞ (Yb0 ). The space l1 (X) is monotone and (X 0 , σ(X 0 , X)) is sequentially complete so Corollary 11.18 implies that (l∞ (Xb0 ), σ(l∞ (Xb0 ), l1 (X))) is sequentially complete. Theorem 14.1 and Corollary 14.2 give the result. We next consider matrices acting between l ∞ spaces. The β-dual of l∞ (X) is l1 (Xb0 ) (Appendix C.23). Theorem 14.8. Let X be barrelled. If A : l ∞ (X) → l∞ (Y ), then AT : l1 (Yb0 ) → l1 (Xb0 ) and A is σ(l∞ (X), l1 (Xb0 )) − σ(l∞ (Y ), l1 (Yb0 )) continuous and AT is σ(l1 (Yb0 ), l∞ (Y )) − σ(l1 (Xb0 ), l∞ (X)) continuous. Similar continuity statements hold for any Hellinger-Toeplitz topology w. Proof: The space l ∞ (X) is monotone and (X 0 , σ(X 0 , X)) is sequentially complete so Corollary 11.18 implies that (l 1 (Xb0 ), σ(l1 (Xb0 ), l∞ (X))) is sequentially complete. Theorem 14.1 and Corollary 14.2 give the result. We give an additional automatic continuity result. The space E has the property I if the injections x → ej ⊗ x from X into E are continuous for every j.

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Theorem 14.9. Assume that every T ∈ E βY induces a continuous linear operator belonging to L(E, Y ), (E, F ) has the Banach-Steinhaus property and F is an AK-space with property I. If A : E → F , then A is continuous. Proof: Let Ri be the ith row of A so that Ri ∈ L(E, Y ). By hypothesis x → Rj · x → (Rj · x)ej is continuous from E into F so the operator P An : E → F defined by An x = nj=1 (Rj · x)ej is continuous. By the AK assumption, An → A pointwise so A is continuous by the Banach-Steinhaus assumption. Sufficient conditions for each T ∈ E βY to induce a continuous linear operator are given in Theorem 11.2. The analogue of Theorem 10.10 also holds with essentially the same proof. Theorem 14.10. Let (E, τ ) be a K-space with 0-GHP and assume that condition (*) is satisfied. If A : E → F , then A is τ − σ(F, F β ) continuous.

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Appendix A Topological Vector Spaces

In this appendix we will record some of the results pertaining to topological vector spaces (TVS) which will be used throughout the text. For convenience we assume that all vector spaces are real. A topological vector space (TVS) is a vector space X supplied with a topology τ such that the operations of addition and scalar multiplication are continuous with respect to τ . A subset U of a TVS X is symmetric (balanced) if x ∈ U implies −x ∈ U (x ∈ U implies tx ∈ U for |t| ≤ 1). Any TVS has a neighborhood base at 0 which consists of symmetric (balanced, closed) sets. See [Sch], [Sw2] or [Wi] for discussions of TVS. One other result pertaining to TVS will be used. A quasi-norm on a vector space X is a map |·| : X → [0, ∞) satisfying |0| = 0, |x + y| ≤ |x|+|y| and |x| = |−x| for x, y ∈ X, and if tk → t in R and xk , x ∈ X with |xk − x| → 0, then |tk xk − tx| → 0. If the quasi-norm satisfies |x| = 0 iff x = 0, then the quasi-norm is said to be total. If |·| is a quasi-norm on X, then d (x, y) = |x − y| defines a semi-metric on X which is a metric iff |·| is total. The semi-metric d is translation invariant in the sense that d(x + z, y + z) = d(x, y) for x, y, z ∈ X. The space X is a TVS under the semi-metric d. A useful fact which we will use is that the topology of any TVS is generated by a family of quasi-norms ([BM ]). That is, if τ is the vector topology of X, then there exists a family δ of quasi-norms {|·|a : a ∈ A} which generate δ τ in the sense that a net {x } in X converges to 0 with respect to τ iff x − x a → 0 for every a ∈ A. A TVS X is locally convex (LCT V S) if X has a neighborhood base at 0 consisting of convex sets. Any LCTVS also has a base at 0 consisting of closed, absolutely convex sets. The topology τ of any LCTVS is generated by a family of semi-norms {pa : a ∈ A} as above. See [Sch], [Sw2] or [Wi] 207

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for the basic properties of LCTVS. We now give a description of polar topologies which will play an important role when we discuss Orlicz-Pettis Theorems. A pair of vector spaces X, X 0 are said to be in duality if there is a bilinear map h·, ·i : X 0 × X → R such that (i) {h·, xi : x ∈ X, x 6= 0} separates the points of X and (ii) {hx0 , ·i : x0 ∈ X 0 , x0 6= 0} separates the points of X 0 . If X, X 0 are in duality, the weak topology of X (X 0 ), σ(X, X 0 ) (σ(X 0 X)), is the locally convex vector topology generated by the semi-norms p(x) = |< x0 , x >| , x0 ∈ X 0 (p (x0 ) = |< x0 , x >|) , x ∈ X). A subset A ⊂ X is σ(X, X 0 ) bounded iff sup {|< x0 , x >| : x ∈ A} < ∞ for every x0 ∈ X 0 . Let A be a family of σ(X 0 X) bounded subsets of X 0 . For A ∈ A, set pA (x) = sup{|hx, x0 i| : x0 ∈ A}. The semi-norms {pA : A ∈ A} generate a locally convex topology τA on X called the polar topology of uniform convergence on A (for the reason the topology is called a polar topology, see [Sw2] 17). Thus, a net {xδ } converges to 0 in τA iff x0 , xδ → 0 uniformly for x0 ∈ A for every A ∈ A. We will use the following polar topologies in the text. (1) The weak topology σ(X, X 0 ) is generated by the family A of all finite subsets of X 0 . (2) The strong topology of X, denoted by β(X, X 0 ), is generated by the family of all σ(X 0 , X) bounded subsets of X 0 . (3) The Mackey topology, denoted by τ (X, X 0 ), is generated by the family of all absolutely convex, σ(X 0 , X) compact subsets of X 0 . (4) The polar topology generated by the family of all σ(X 0 , X) compact subsets of X 0 is denoted by λ(X, X 0 ). (5) A subset A ⊂ X 0 is said to be conditionally σ(X 0 , n X) sequentially o compact if every sequence x0j ⊂ A has a subsequence x0nj which is E D σ(X 0 , X) Cauchy, i.e., lim x0nj , x exists for every x ∈ X. The polar topology generated by the family of conditionally σ(X 0 , X) sequentially compact sets is denoted by γ(X, X 0 ). The topology λ(X, X 0 ) was introduced by G. Bennett and Kalton ([BK]) and is obviously stronger than the Mackey topology τ (X, X 0 ); it can be strictly stronger ([K1] 21.4).

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Topological Vector Spaces

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Let w(X, X 0 ) be a polar topology defined for all dual pairs X, X 0 . We have the following useful notion introduced by Wilansky ([Wi]). Definition A.1. The topology w(·, ·) is a Hellinger-Toeplitz topology if whenever T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then T : (X, w (X, X 0 )) → (Y, w (Y, Y 0 )) is continuous. Wilansky has given a very useful criterion for Hellinger-Toeplitz topologies ([Wi] 11.2.2). If T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then the adjoint (transpose) operator of T is the linear operator T 0 : Y 0 → X 0 defined by hT 0 y 0 , xi = hy 0 , T xi for x ∈ X, y 0 ∈ Y 0 . The adjoint T 0 is σ (Y 0 , Y ) − σ(X 0 , X) continuous. Let A(X 0 , X) be a family of σ(X 0 , X) bounded subsets which is defined for all dual pairs X, X 0 . Let w(X, X 0 ) be the polar topology generated by the elements of A(X 0 , X). We have Theorem A.2. The topology w(X, X 0 ) is a Hellinger-Toeplitz topology if whenever T : (X, σ (X, X 0 )) → (Y, σ (Y, Y 0 )) is linear and continuous, then T 0 A (Y 0 , Y ) ⊂ A(X 0 , X). 0 Proof: Let {xδ } be a net in X which converges to 0 in

0w(X,δ X ). 0 0 0 0 0 = Let , X) so y , T x

0 0 Aδ ∈ A(Y , Y ). Then {T0 y : y ∈ A} ∈ A(X T y , x → 0 uniformly for y ∈ A. That is, T xδ → 0 in w(Y, Y 0 ).

Theorem A.2 clearly implies that the polar topologies given in (1) - (5) are all Hellinger-Toeplitz topologies. We next consider another notion due to Wilansky which is useful in treating Orlicz-Pettis results.

Definition A.3. Let X be a vector space and σ and τ two vector topologies on X. We say that τ is linked to σ if τ has a neighborhood base at 0 consisting of σ closed sets. [The terminology is that of Wilansky ([Wi] 6.1.9).]

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For example, the polar topologies β(X, X 0 ), τ (X, X 0 ), γ(X, X 0 ) and λ(X, X 0 ) are linked to the weak topology σ(X, X 0 ). Lemma A.4. Let X be a vector space and σ and τ two vector topologies on X such that τ is linked to σ. (i) If {xj } ⊂ X is τ Cauchy and if σ-lim xj = x, then τ -lim xj = x. (ii) If (X, σ) is sequentially complete and σ ⊂ τ, then (X, τ ) is sequentially complete. Proof: (i): Let U be a τ neighborhood of 0 which is σ closed. There exists N such that j, k ≥ N implies xj − xk ∈ U . Since U is τ closed, xj − x ∈ U for j ≥ N . (ii) follows from (i). Remark A.5. It is important that the topologies σ and τ are linked in the Lemma. For example, consider the space c with its weak topology P σ(c, l1 ) and the topology of pointwise convergence p. The series j ej is p convergent, the partial sums of the series are σ(c, l 1 ) Cauchy, but the series is not σ(c, l1 ) convergent [here we are using the pairing between c and l 1 where l1 is the topological dual of c ([Sw2] 5.12)]. We now establish a basic lemma. Lemma A.6. Let X be a vector space and σ and τ two vector topologies P on X such that τ is linked to σ. If every series j xj which is σ subseries convergent satisfies τ − lim xj = 0, then every series in X which is σ subseries convergent is τ subseries convergent. Proof: By the previous lemma it suffices to show that every σ subseries P Pn convergent series j xj is such that its partial sums sn = j=1 xj form P a τ Cauchy sequence. If j xj is σ subseries convergent but {sn } is not τ Cauchy, there exists a τ neighborhood of 0, U , and a pairwise disjoint sequence of finite subsets, {Ik }, of N such that max Ik < min Ik+1 and P P ∈ / U . The series k zk is σ subseries convergent, being zk = j∈Ik xj P a subseries of / U and j xj , so τ − lim zk = 0. This contradicts zk ∈ establishes the result. We next consider topologies for spaces of continuous linear operators. Let X, Y be LCTVS and L (X, Y ) the space of all continuous linear operators T : X → Y. Let A be a family of bounded (σ (X, X 0 )) subsets of X

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and let Q be the family of all continuous semi-norms on Y. Then A and Q generate a locally convex topology on L (X, Y ) defined by (∗) pqA (T ) = sup {q(T x) : x ∈ A} , q ∈ Q, A ∈ A. We denote by LA (X, Y ), L(X, Y ) with the locally convex topology generated by the semi-norms in (∗). A net {T δ } in L (X, Y ) converges to 0 in LA (X, Y ) iff T δ x → 0 uniformly for x ∈ A for every A ∈ A and for this reason, the topology is called the topology of uniform convergence on A or A -uniform convergence. We have the following examples which will be considered. (i) If A is the family of all finite subsets of X, we denote the topology generated by A by Ls (X, Y ). This is just the topology of pointwise convergence on X and is called the strong operator topology. Thus, a net {T δ } converges to 0 in Ls (X, Y ) iff T δ x → 0 in Y for every x ∈ X. (ii) Let A again be the family of all finite subsets of X but equip Y with the weak topology σ(Y, Y 0 ). This topology is called the weak operator topology. Thus, a net {T δ } in L(X, Y ) converges to 0 in the weak operator iff for every x ∈ X , T δ x → 0 in σ(Y, Y 0 ) or, equivalently,

0 topology δ y , T x → 0 for every x ∈ X, y 0 ∈ Y 0 . (iii) Let A be the family of all bounded subsets of X. Then the topology generated by A is denoted by Lb (X, Y ). If X and Y are normed spaces, the topology Lb (X, Y ) is generated by the operator norm kT k = sup{kT xk : kxk ≤ 1} and is called the uniform operator topology. (iv) Let A be the family of all precompact subsets of X. Then the topology generated by A is denoted by Lpc (X, Y ). (v) Let A be the family of all compact subsets of X. Then the topology generated by A is denoted by Lc (X, Y ). (vi) Let A be the family of all sequences {xj } ⊂ X which converge to 0. Then the topology generated by A is denoted by L→0 (X, Y ).

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Appendix B Scalar Sequence Spaces

In this appendix we will list the sequence spaces and their properties which will be used in the text. If λ is a vector space of (real) sequences containing c00 , the space of all sequences which are eventually 0, the β-dual of λ is defined to be λβ ( =

s = {sj } :

∞ X

)

sj tj = {sj } · {tj } = s · t converges for every t = {tj } ∈ λ .

j=1

Since λ ⊃ c00 , the pair λ, λβ are in duality with respect to the pairing s · t = {sj } · {tj } for s ∈ λβ , t ∈ λ. We now list some of the scalar valued sequence spaces which will be encountered in the text. • • • • • •

c00 = {{tj } : tj = 0 eventually} c0 = {{tj } : lim tj = 0} cc = {{tj } : tj is eventually constant} c= {{tj } : lim tj exists} m0 = {{tj } : the range of {tj } is finite} = span{χσ : σ ⊂ N} l∞ = {{tj } : supj {|tj |} = k{tj }k∞ < ∞}

All of the sequence spaces above are usually equipped with the supnorm, k·k∞ , defined above. For 0 < p < 1, • lp = {{tj } :

P∞

j=1

p

|tj | = |{tj }|p < ∞} 213

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The space lp (0 < p < 1) is usually equipped with the quasi-norm |·|p which generates the metric dp (s, t) = |s − t|p under which it is complete. For 1 ≤ p < ∞, • lp = {{tj } : (

P∞

j=1

p

|tj | )1/p = k{tj }kp < ∞}

The space lp is usually equipped with the norm k·kp under which it is a Banach space. P n • bs= {{tj } : supn { j=1 tj = ktkbs < ∞}

The space bs is called the space of bounded series and is usually equipped with the norm k·kbs under which it is a Banach space. • cs= {{tj } :

P∞

converges}

P∞

|tj+1 − tj | < ∞}

j=1 tj

The space cs is a subspace of bs and is called the space of convergent series; cs is a closed subspace of bs under the norm k·kbs . • bv= {{tj } :

j=1

The space bv is called the space of sequences of bounded variation and P∞ is a Banach space under the norm k{tj }kbv = j=1 |tj+1 − tj | + |t1 | . • bv0 = bv ∩ c0

The space bv0 is a closed subspace of bv. • s= the space of all real valued sequences. The space s is a Frechet space under the metric d(s, t) =

∞ X j=1

|sj − tj | /2j (1 + |sj − tj |)

of coordinatewise convergence. We give a list of the β-duals and topological duals of the spaces above. For these, see [HK] and [Bo].

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space c00 c0 cc c m0 l∞ lp (0 < p < 1) lp (1 ≤ p < ∞) bs cs bv0 s

β-dual s l1 cs l1 l1 l1 l∞ lq ( p1 + bv0 bv bs c00

1 q

= 1)

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topological dual l1 l1 l1 l1 ba ba l∞ lq ( p1 + q1 = 1) bv bv bs c00

We now list some of the properties of sequence spaces which will be encountered in the sequel. Throughout the remainder of this appendix λ will denote a sequence space containing c00 . Suppose that λ is equipped with a Hausdorff vector topology. Definition B.1. The space λ is a K-space if the coordinate functionals t = {tj } → tj are continuous from λ into R for every j. If the K-space λ is a Banach (Frechet) space, λ is called a BK-space (FK-space). All of the spaces listed above are K-spaces under their natural topologies. Let ej be the sequence with a 1 in the j th coordinate and 0 in the other coordinates. Definition B.2. The K-space λ is an AK-space if the {ej } form a Schauder Pn basis for λ, i.e., if t = {tj } ∈ λ, then t = limn j=1 tj ej , where the convergence is in λ. The spaces c00 , c0 , lp (0 < p < ∞), cs, bv and bv0 are AK-spaces. The spaces m0 and l∞ are not AK-spaces. For each n let Pn : λ → λ be the sectional projection (operator) defined Pn by Pn t = j=1 tj ej = (t1 , ..., tn , 0, 0, ...).

Definition B.3. The K-space λ is an AB-space if {Pn t : n ∈ N} is bounded for each t ∈ λ, i.e., if the {Pn } are pointwise bounded on λ.

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Definition B.4. The K-space λ has the sections uniformly bounded property (SUB) if {Pn t : n ∈ N, t ∈ B} is bounded for every bounded subset B of λ, i.e., if the {Pn } are uniformly bounded on bounded subsets of λ. Definition B.5. The K-space λ has the property SE (sections equicontinuous) if the sectional operators {Pn } from λ into λ are equicontinuous. Obviously, property SUB implies that λ is an AB-space and property SE implies property SUB. If λ is a barrelled AB-space, then λ has property SE and, therefore, SUB ([Sw2], [Wi]). If λ is a metric linear space whose topology is generated by the quasi-norm |·| which satisfies |Pn t| ≤ M |t| for some M and all t ∈ λ, then λ has property SE; e.g., s, l ∞ and its subspaces, lp (0 ≤ p < ∞), and bs and its subspace cs. Throughout this text numerous gliding hump properties are employed. We now list these gliding hump properties and give examples of sequence spaces which satisfy the various gliding hump properties. If σ ⊂ N, χσ will denote the characteristic function of σ and if t = {tj } is any sequence (scalar or vector), χσ t will denote the coordinatewise product of χσ and t. A sequence space λ is monotone if χσ t ∈ λ for every σ ⊂ N and t ∈ λ. A sequence space λ is normal (solid ) if t ∈ λ and |sj | ≤ |tj | implies that s = {sj } ∈ λ. Obviously, a normal space is monotone; the space m0 is monotone but not normal. The spaces c00 , c0 , lp (0 < p ≤ ∞) and s are normal whereas cc , c, bs, cs, bv and bv0 are not monotone. An interval in N is a subset of the form [m, n] = {j ∈ N : m ≤ j ≤ n}, where m, n ∈ N with m ≤ n. A sequence of intervals {Ij } is increasing if max Ij < min Ij+1 for every j. A sequence of signs is a sequence {sj } with sj = ±1 for every j. We begin with 2 gliding hump properties which are algebraic and require no topology on the sequence space λ. Definition B.6. Let Λ ⊂ λ. Then Λ has the signed weak gliding hump property (signed-WGHP) if for every t ∈ Λ and every increasing sequence of intervals {Ij }, there is a subsequence {nj } and a sequence of signs {sj } P∞ such that the coordinatewise sum of the series j=1 sj χInj t belongs to Λ. If the signs sj can all be chosen to be equal to 1 for every t ∈ Λ, then Λ has the weak gliding hump property (WGHP). The weak gliding hump property was introduced by Noll ([No]) and the signed weak gliding hump property was introduced by Stuart ([St1],

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[St2]). We now give some examples of sequence spaces with WGHP and signed-WGHP. Example B.7. Any monotone space has WGHP. We now show that the space cs of convergent series is not monotone but has WGHP. Example B.8. cs has WGHP but is not monotone. Let t ∈ cs and {Ij } P be an increasing sequence of intervals. Since the series j tj converges, X tk → 0 χIj ∩J · t = k∈Ij ∩J

for any interval J. Pick a subsequence {nj } such that χInj ∩J · t < 1/2j P∞ for every interval J. Then j=1 χInj t ∈ cs since this series satisfies the Cauchy criterion.

If t = {(−1)j /j}, then t ∈ cs, but if σ = {1, 3, 5, ...}, then χσ t ∈ / cs so cs is not monotone. Example B.9. The space c of convergent sequences does not have WGHP. P∞ For example, t = {1, 1, ...} ∈ c and if Ij = {2j}, then j=1 χInj t ∈ / c for any subsequence {nj }. We show later in Example B.26 that the space bs of bounded series has signed-WGHP but not WGHP so the inclusion of signs in Definition B.6 is important. We next establish a general criterion for a space to have WGHP. A TVS X is a K-space if whenever {xj } is a null sequence in X and {nj } is a subsequence, then there is a further subsequence {mj } of {nj } such that P∞ the series j=1 xmj converges in X. For example, any complete metric linear space is a K-space ([Sw1] 3.2.3, there are further examples in this text). Theorem B.10. If λ is a K-space and an AK-space, then λ has WGHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Since P P∞ t = j=1 tj ej converges in λ, limk j∈Ik tj ej = 0. Since λ is a K-space, P∞ P j there exists {nk } such that k=1 j∈Ink tj e converges to some s ∈ λ. P ∞ P Since λ is a K-space, the series k=1 j∈In tj ej also converges to s pointk wise. Theorem B.10 applies to spaces such as c0 , lp (0 < p < ∞) and cs.

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We next consider results related to ideas due to Garling ([Ga]). For this we need the following observation. Example B.11. Let B0 be the closed unit ball of bv0 . We show that B0 has WGHP. Let t ∈ B0 and {Ik } be an increasing sequence of intervals. P∞ Then k=1 |tk+1 − tk | ≤ 1 and lim tk = 0. Choose {nk } such that ∞ X X

k=1 j∈Ink

|tj+1 − tj | < 1/2

and ∞ X k=1

2 max{ tmin Ink , tmax Ink } < 1/2

(this is possible since t ∈ c0 , so we can extract a subsequence which belongs to l1 ). Then χ∪∞ I t has total variation less than or equal to k=1 nk ∞ X X

k=1 j∈Ink

so χ

∪∞ k=1 Ink

|tj+1 − tj | +

t ∈ B0 .

∞ X k=1

2 max{ tmin Ink , tmax Ink } < 1

Definition B.12. (Garling) Let S be any subset of sequences. The space λ is S invariant if Sλ = λ. For example, λ is monotone iff λ is m0 invariant. Proposition B.13. If S has signed-WGHP (WGHP) and λ is S invariant, then λ has signed-WGHP (WGHP). Proof: For t ∈ λ there exists u = {uk } ∈ S,v ∈ λ such that t = uv. Let {Ik } be an increasing sequence of intervals. By hypothesis there exist a subsequence {Ink } and a sequence of signs {sk } such that u0 = P∞ sk χInk u ∈ S (coordinate sum). Then {u0k vk } ∈ λ and since {u0k vk } = Pk=1 ∞ k=1 sk χInk t, λ has signed-WGHP. From Example B.11 and Proposition B.14, we have

Corollary B.14. If λ is B0 invariant, then λ has WGHP. We next consider a notion introduced by Noll ([No]). Definition B.15. The multiplier space of λ, M (λ), is defined to be {s : st ∈ λ for all t ∈ λ}.

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Proposition B.16. If M (λ) has signed-WGHP (WGHP), then λ has signed-WGHP (WGHP). Proof: Consider the case of WGHP. The constant sequence e with 1 in each coordinate belongs to M (λ). Let t ∈ λ and {Ik } be an increasing P sequence of intervals. There exist {nk } such that ∞ k=1 χInk e ∈ M (λ) so P∞ P∞ ( k=1 χInk e)t = k=1 χInk t ∈ λ.

The results above in Propositions B.14 and B.16 are used in Chapter 2 to establish weak sequential completeness for β-duals. We next consider gliding hump properties which depend on the topology of the sequence space λ. In what follows we assume that λ is a K-space.

Definition B.17. Let Λ ⊂ λ. Then Λ has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {tj } ⊂ Λ and every increasing sequence of intervals {Ij }, there is a subsequence {nj } and a sequence of signs {sj } such that the coordinatewise sum of the series P∞ nj ∈ Λ. If the signs sj can be chosen equal to 1 for every j=1 sj χInj t t ∈ Λ, then Λ is said to have the strong gliding hump property (SGHP). The strong gliding hump property was introduced by Noll ([No]) and the signed strong gliding hump property was introduced in [Sw4]. Example B.18. The space l ∞ has SGHP. The spaces l p (0 < p < ∞) and c0 do not have SGHP (consider {ej } and Ij = {j}). Example B.19. The subset M0 = {χσ : σ ⊂ N} ⊂ m0 has SGHP while the space m0 = spanM0 does not have SGHP. We consider smaller subsets of 2N whose characteristic functions have SGHP. Definition B.20. A family F of subsets of N is an FQσ family if F contains the finite sets and if whenever {Ij } is a pairwise disjoint sequence of finite subsets, there is a subsequence {Inj } such that ∪∞ j=1 Inj ∈ F. This notion is due to Sember and Samaratanga ([SaSe]). We give an example of an FQσ family which is a proper subset of 2N . Example B.21. Haydon has given an example of an algebra H of subsets of N such that H is an FQσ family but for no infinite A ⊂ N do we have 2A = {A ∩ B : B ∈ H} ([Hay]).

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Example B.22. If F is an FQσ family and Λ = {χσ : σ ∈ F} ⊂m0 , then Λ has SGHP (if {χσj } ⊂ Λ and {Ij } is an increasing sequence of intervals, then there exists a subsequence {nj } such that I = ∪∞ j=1 σnj ∩ Inj ∈ F so P∞ χI = j=1 χInj χσnj ∈ Λ). Definition B.23. A family of subsets F of N is an IQσ family if F contains the finite sets and if whenever {Ij } is an increasing sequence of intervals, there is a subsequence {Inj } such that ∪∞ j=1 Inj ∈ F.

This notion is also due to Sember and Samaratanga ([SaSe]). We give an example of an IQσ family containing N. P∞ Example B.24. Let j=1 tj be a conditionally convergent scalar series. P Put F = {σ : j∈σ tj converges}. Then F is an IQσ family containing N. For suppose that {Ij } is an increasing sequence of intervals. There exists P a subsequence {Inj } such that i∈In ∩J ti < 1/2j for every j and every j P the interval J. Then I = ∪∞ j=1 Inj ∈ F since the series j∈I tj satisfies P N Cauchy condition. Note that N ∈ F but F 6=2 since the series j tj is conditionally convergent. As in Example B.22, we have Example B.25. Let F be an IQσ family and Λ = {χσ : σ ∈ F} ⊂m0 . Then Λ has SGHP. We next show that the space bs of bounded series has signed-SGHP but not SGHP. Example B.26. The space bs has signed-SGHP but not SGHP. Let {tj } ⊂ bs be bounded and {Ij } be an increasing sequence of intervals. Put ( ) X j ti : j ∈ N, I an interval in N < ∞. M = sup i∈I

Define signs inductively by setting s1 = signχI1 · t1 and sn+1 = −[sign

n X

k=1

sk χIk · tk ][signχIn+1 · tn+1 ].

P∞ k Put y = k=1 Psk χIk t . We show kykbs ≤ 2M . We first Pshow by in max In max I duction that j=1 yj ≤ M for every n. For n = 1, j=1 1 yj =

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P j∈I1 s1 t1j ≤ M . Suppose the inequality holds for n. Then max max In+1 X XIn X yj + yj ≤ M yj = j=1 j=1 j∈In+1 P P P max I since j∈In+1 yj = j∈In+1 sn+1 yj ≤ M and j=1 n yj ≤ M and both of these terms have opposite signs. Now for arbitrary n, let k = kn be the largest integer such that max Ik ≤ n. Then n max Ik n n XIk X X max X X k+1 yj = yj + yj ≤ yj + sk+1 tj ≤ 2M j=1 j=1 j=1 j=min Ik+1 j=min Ik+1

so kykbs ≤ 2M as desired.

Note that bs does not have WGHP (consider t = {1, −1, 1, −1, ...} and Ij = {2j − 1}). The proof above is essentially that of Stuart who showed that bs has signed-WGHP but not WGHP ([St1], [St2]). Further examples of spaces with SGHP (WGHP) and signed-SGHP (signed-WGHP) are constructed later in this appendix. Note that in the proof of Example B.26 it was not necessary to pass to a subsequence in the definition of signed-SGHP. Definition B.27. The K-space λ has the zero gliding hump property (0GHP) if whenever tj → 0 in λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that the coordinate sum of the series P∞ nj belongs to λ. j=1 χInj t

The 0-GHP was essentially introduced by Lee Peng Yee ([LPY]); see also [LPYS]. We give some examples of spaces with 0-GHP. Recall the section operPn ators Pn : λ → λ are defined by Pn t = j=1 tj ej .

Proposition B.28. Let λ be a K-space with property SE. Then λ has 0GHP.

Proof: Let tj → 0 in λ and let {Ij } be an increasing sequence of intervals. Then χIj tj → 0 in λ by property SE. Since λ is a K-space, there is P a subsequence {nj } such that the subseries j χInj tnj converges to some P t ∈ λ. Since λ is a K-space, the series j χInj tnj converges coordinatewise to t.

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From Proposition B.28, it follows that the spaces l p (0 < p ≤ ∞), s, c, cs, bs and c0 have 0-GHP. The space c00 does not have 0-GHP. Further examples are given later in this appendix and can be found in [Sw1] 12.5. The gliding hump properties WGHP and 0-GHP are independent; the space c has 0-GHP but not WGHP while the space c00 has WGHP but not 0-GHP. We give a simple proposition which relates the two conditions. Proposition B.29. If λ is an AK-space with 0-GHP, then λ has WGHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Since λ has AK, χIj t → 0. By 0-GHP, there is a subsequence {nj } such that P∞ j=1 χInj t ∈ λ, where the series is coordinatewise convergent since λ is a K-space. We next define a gliding hump property which is used to establish uniform boundedness principles. Let µ be sequence space containing c00 . Definition B.30. The K-space λ has the strong µ gliding hump property (strong µ-GHP) if whenever {tj } is a bounded sequence in λ and {I } is an increasing sequence of intervals, the coordinate sum of the series Pj∞ j j=1 uj χIj t belongs to λ for every u = {uj } ∈ µ.

Definition B.31. The K-space λ has the weak µ gliding hump property (weak µ-GHP) if whenever {tj } is a bounded sequence in λ and {Ij } is an increasing sequence of intervals, there is a subsequence {nj } such that P∞ nj belongs to λ for every the coordinate sum of the series j=1 uj χInj t u = {uj } ∈ µ. Of course, the difference in the strong µ-GHP and the weak µ-GHP is the necessity to pass to a subsequence in Definition B.31. We refer to the elements u = {uj } ∈ µ as multipliers since the coordinates of u multiply the blocks or ”humps”, {χIj tj }, determined by the {tj } and {Ij }. This is analogous to the situation in the signed-WGHP or signed-SGHP where the humps are multiplied by {±1}. We give examples of spaces with µ-GHP. Proposition B.32. If λ is a locally complete K-space with property SUB, then λ has strong l 1 -GHP. Proof: Let {tj } ⊂ λ be bounded and {Ij } be an increasing sequence of intervals. By SUB, {χIj tj } is bounded in λ so if u ∈ l1 , the series P∞ j is absolutely convergent in λ and, therefore, converges to j=1 uj χIj t

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an element t ∈ λ by local completeness. Since λ is a K-space, the series P∞ j is coordinatewise convergent to t. j=1 uj χIj t

Example B.33. From Proposition B.32, it follows that the spaces l p (0 < p ≤ ∞), s, c, cs, bs and c0 have strong l 1 -GHP. We also have Example B.34. The spaces l ∞ and c0 have strong c0 -GHP; lp (0 < p ≤ ∞) has strong l∞ -GHP. We next give examples of non-complete spaces with weak l p -GHP. This example requires some properties of integration with respect to finitely additive set functions. We refer the reader to [RR] for a discussion of the integrals. Example B.35. Let 1 ≤ p < ∞. Let P be the power set of N and let α : P → [0, ∞) be a finitely additive set function with α({j}) > 0 for every j. Let lp (α) = Lp (α) beR the space of all pth power α-integrable functions p with the norm kf kp = ( N |f | dα)1/p [see [RR] for details; the assumption p that α({j}) > 0 makes l (α) a K-space]. We show that l p (α) has weak lp -GHP. Let {fj } ⊂ lp (α) be bounded with kfj kp ≤ 1 and {Ij } be an increasing sequence of intervals. By Drewnowski’s Lemma (Appendix E.2), there is a subsequence {nj } such that α is countably additive on the σP∞ algebra generated by {Inj }. Suppose that t ∈ l p . Put f = j=1 tj χInj fnj P∞ [coordinate sum]. We claim that f ∈ l p (α) and the series j=1 tj χInj fnj p converges to f in l (α) by using Theorem 4.6.10 of [RR]; this will establish Pn the result. Put sn = j=1 tj χInj fnj and note that sn → f α-hazily [i.e., in α measure] since if > 0, α({j : |sn (j) − f (j)| ≥ }) ≤ α(∪∞ j=n+1 Inj ) =

∞ X

j=n+1

α(Inj ) → 0

by the countable additivity. Next, {sn } is Cauchy in lp (α) since

p

n n X X

p p

|tj | → 0. ksn − sm kp = t j χ In j f n j ≤

j=m+1 j=m+1 p

R

p

It follows that { |sn | dα} is uniformly α-continuous and using [RR] this justifies the claim.

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We next consider a gliding hump property called the infinite gliding hump property which has been proven to be useful when considering strong convergence. This property is algebraic and requires no topology on λ. Definition B.36. The sequence space λ has the infinite gliding hump property (∞-GHP) if whenever t ∈ λ and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that every subsequence of {nj } has a further subsequence {pj } such that P the coordinate sum of the series ∞ j=1 apj χIpj t ∈ λ. The term ”infinite” gliding hump property is used to suggest that the humps {χIj t} are multiplied by a sequence of scalars which tend to infinity. We now give examples of sequence spaces with ∞-GHP. For this, we introduce another property for a sequence space.

Definition B.37. The space λ is c0 -factorable (called c0 -invariant by Garling ([Ga])) if t ∈ λ implies that there exist s ∈ c0 , u ∈ λ with t = su [coordinate product]. Proposition B.38. If λ is normal and c0 -factorable, then λ has ∞-GHP. Proof: Let t ∈ λ with t = su, s ∈ c0 , u ∈ λ and let {Ij } be an increasing sequence of intervals. Pick an increasing sequence {nj } such that sup{|si | : i ∈ Inj } = bnj > 0 (if this choice is not possible there is nothing to do). Note that bnj → 0 so anj = 1/bnj → ∞. Define vj = sj ank if j ∈ Ink and vj = 0 otherwise; then v ∈ l ∞ so vu ∈ λ since λ is normal. We P∞ P j have ∞ k=1 ank χInk t ∈ λ. Since the same argument can be j=1 (vu)j e = applied to any subsequence, λ has ∞-GHP. We now give some examples of spaces which satisfy the conditions of Proposition B.38. Example B.39. The space c0 is normal and c0 -factorable so has ∞-GHP. Example B.40. Let 0 < p < ∞. Let t ∈ l p . Pick an increasing sequence Pnj+1 p {nj } with n0 = 0 such that i=n |tj | < 1/2j(p+1) and set Ij = [nj + Pj +1 P∞ −j ∞ j p 1, nj+1 ], s = j=1 2 χIj t. Then t = su so l is c0 j=1 2 χIj , u = factorable and is obviously normal so Proposition B.38 applies. Example B.41. As in Example B.40, it can be shown that the spaces d = {t : sup{|tj |1/j < ∞} and δ = {t : lim |tj |1/j = 0} are c0 -factorable and are both clearly normal so Proposition B.38 applies [see [KG] for these spaces].

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We give examples of non-normal spaces with ∞-GHP. Proposition B.42. If λ is a Banach AK-space, then λ has ∞-GHP. Proof: Let t ∈ λ and {Ij } be an increasing sequence of intervals. Choose

P

a subsequence {nj } such that i∈In ∩J tj ej < 1/j2j for any interval J. j P Consider s = ∞ jχInj t. If J is any interval contained in the interval j=1

P P

P

P

∞ k j [min Inj , ∞), then j∈J sj ej = ∞ i=j k∈In ∩J itk e ≤ i=j 1/2 = j

2−j+1 so the partial sums of the series generated by s are Cauchy and, therefore, convergent. Hence, s ∈ λ and λ has ∞-GHP since λ is a K-space.

Example B.43. For example, it follows from Proposition B.42 that the non-normal space cs of convergent series has ∞-GHP. Likewise, bv0 has ∞-GHP. Example B.44. The spaces l ∞ , m0 , bs and bv do not have ∞-GHP. To this point the only example of a space with signed-SGHP (signedWGHP) is bs. We will now describe a method which can be used to construct more sequence spaces with these and other gliding hump properties ([BSS]). Let A = [aij ] be an infinite matrix with scalar entries. We use A and λ to generate a further sequence space. The matrix domain of A and λ is defined to be λA = {t = {tj } : At ∈ λ}.

Thus, A is a linear map from λA → λ. Some of the familiar sequence spaces can be generated by this procedure. In particular, cA and c0A = (c0 )A are the spaces of all sequences which are A-summable and A-summable to 0, respectively ([Bo]). Example B.45. Let B = [bij ] be the matrix with bij = 1 for j ≤ i and ∞ bij = 0 otherwise. Then lB = bv, the space of sequences with bounded variation, and cB = cs, the space of convergent series. Example B.46. Let B1 = [bij ] be the matrix with bii = 1, bi+1,i = −1 and 1 = bv. bij = 0 otherwise. Then lB 1 Example B.47. Let C = [cij ] be the Cesaro matrix , cij = 1/i for 1 ≤ j ≤ i ∞ and cij = 0 otherwise. Then lC is the space of sequences {tj } with bounded P j averages, supj i=1 ti /j < ∞.

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Let {dj } be a scalar sequence and let D be the diagonal matrix with the {dj } as the entries down the diagonal. Theorem B.48. If λ has WGHP (signed-WGHP), then λD has WGHP (signed-WGHP). Proof: Let t ∈ λD and {Ij } be an increasing sequence of intervals. Since Dt ∈ λ, there is a subsequence {nj } such that ∞ X v= χInj Dt ∈ λ. j=1

Then

u=

∞ X j=1

χ In j t ∈ λ D

since Du = v. The proof for the signed-WGHP case is similar. Corollary B.49. bsD has signed-WGHP for any diagonal matrix D. If each dj 6= 0, the spaces bs and bsD are algebraically isomorphic, but, in general, the spaces may have very different topological properties depending on the growth of the sequence {dj }. For example, if dj → ∞ and t ∈ bsD , then t ∈ c0 so bsD ⊂ c0 in this case. On the other hand, if dj → 0 and dj 6= 0 for all j, then u = (1/d1 , −1/d2 , 1/d3 , ...) ∈ bsD

and lim uj = ∞ so u ∈ / l∞ in this case. Also, if dj = 0 for some j, then th the j coordinate of any element in bsD can be arbitrarily large so bsD will be very different from bs. Thus, the spaces bsD furnish a large class of sequence spaces with signed-WGHP. We next consider the construction above when the sequence space λ is a K-space. Assume that λ is a K-space whose locally convex topology is generated by a family of semi-norms P. We give the space λA the locally convex topology generated by the semi-norms pA (t) = p(At) f or p ∈ P and pk (t) = |tk | , k ∈ N.

Since only triangular matrices A will be considered below, this agrees with the usual topology defined on λA ([Wi2] 4.3.12). Note that λA is a K-space and A : λA → λ is a linear, continuous operator. Theorem B.50. If λ has SGHP (signed-SGHP), then λD has SGHP (signed-SGHP) for any diagonal matrix D.

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Proof: Let {tj } be bounded in λD and {Ij } be an increasing sequence of intervals. Since D : λD → λ is continuous, {Dtj } is bounded in λ. There exists a subsequence {nj } such that the coordinate sum of the series P∞ nj v = ∈ λ. Then the coordinate sum of the series u = j=1 χInj Dt P∞ nj χ t ∈ λ since Du = v. D j=1 Inj The other case is treated similarly. ∞ Corollary B.51. lD has SGHP and bsD has signed-SGHP for any diagonal matrix D. ∞ As noted earlier the spaces lD furnish examples of spaces of a different nature with SGHP. The only spaces other than l ∞ with SGHP seem to have been constructed by Noll ([No]).

Theorem B.52. If λ has 0-GHP, then λD has 0-GHP. Proof: Let tj → 0 in λD and {Ij } be an increasing sequence of intervals. Since D : λD → λ is continuous, Dtj → 0 in λ. There exists a subsequence P∞ {nj } such that the coordinate sum of the series v = j=1 χInj Dtnj ∈ λ. P∞ Then the coordinate sum of the series u = j=1 χInj tnj ∈ λD since Du = v. Again Theorem B.52 furnishes a large number of examples of spaces with 0-GHP. Let µ be a sequence space containing c00 . As in Theorems B.48, B.50 and B.52, we have Theorem B.53. If λ has strong µ-GHP (weak µ-GHP), then λD has strong µ-GHP (weak µ-GHP). Further examples of sequence spaces λA with various gliding hump properties for matrices which are not diagonal matrices are given in [BSS].

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Appendix C Vector Valued Sequence Spaces

In this appendix we give a list of vector valued sequence spaces, review the gliding hump properties of sequence spaces and give examples of the sequence spaces satisfying various gliding hump properties. Let X be a TVS. We give a list of X valued sequence spaces and their natural topologies. • • • • • •

c00 (X) : all X valued sequences which are eventually 0. c0 (X) : all X valued sequences which converge to 0. cc (X) : all X valued sequences which are eventually constant. c(X) : all X valued sequences which are convergent. m0 (X) : all X valued sequences with finite range. l∞ (X) : all X valued sequences which are bounded.

Let X be an LCTVS whose topology is generated by the family of seminorms X . The natural topology of all the spaces above is generated by the semi-norms q∞ ({xj }) = sup q(xj ), q ∈ X . j

There is a significant difference between the scalar and vector case for spaces of bounded sequences. In the scalar case m0 is dense in l∞ with respect to k·k∞ . However, when X is an infinite dimensional Banach space, m0 (X) is not dense in l∞ (X). For suppose that X is an infinite dimensional Banach space. By Riesz’s Lemma ([Sw2] 7.6), There exists a sequence {xj } such that kxi − xj k ≥ 1 for i 6= j with kxj k = 1 for all j. Then x = {xj } ∈ l∞ (X). However, if m0 (X) is dense in l∞ (X), then {xj : j ∈ N} has a finite -net for every > 0 and is, therefore, precompact which is clearly not the case. Let 0 < p < ∞. Then 229

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• lp (X) : all X valued sequences such that

P∞

j=1

q(xj )p < ∞, q ∈ X .

If 1 ≤ p < ∞, the topology of l p (X) is generated by the semi-norms 1/p ∞ X qp ({xj }) = q(xj )p , q ∈ X . j=1

If 0 < p < 1, the topology of l p (X) is generated by the quasi-norms qp ({xj }) =

∞ X j=1

q(xj )p , q ∈ X .

Pn • BS(X) : all X valued sequences {xj } satisfying {{xj } : { j=1 xj }n is bounded}. P • CS(X) : all X valued sequences {xj } satisfying {{xj } : j xj is Cauchy}. If X is the scalar field, then bs = BS(X) and cs = CS(X). We define a locally convex topology on BS(X) ⊃ CS(X) by the semi-norms X q 0 ({xj }) = sup q xj : I a finite interval , q ∈ X . j∈I

• BV (X) : all X valued sequences satisfying X. • BV0 (X) = BV (X) ∩ c0 (X).

P∞

j=1

q(xj+1 −xj ) < ∞, q ∈

If X is the scalar field, then bv = BV (X) and bv0 = BV0 (X). These spaces are topologized by the semi-norms ∞ X qˆ({xj }) = q(x1 ) + q(xj+1 − xj ), q ∈ X . j=1

• s(X) : all X valued sequences. We now describe a method of constructing vector valued sequence spaces from scalar valued sequence spaces. Let ν be a scalar valued sequence space which is normal and a K-space under a locally convex topology generated by the semi-norms N . If t = {tj } ∈ ν , we write |t| = {|tj |}; note |t| ∈ ν since ν is normal. We will also consider a monotone property for ν. The sequence space ν satisfies condition

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(M ) if there is a family of semi-norms N generating the topology of ν which is such that q(s) ≤ q(t) whenever s, t ∈ ν and |s| ≤ |t| and q ∈ N . Define ν{X} = {{xj } : xj ∈ X, {q(xj )} ∈ ν for all q ∈ X }. Since ν is normal, ν{X} is a vector space. We supply ν{X} with the locally convex topology generated by the semi-norms (#) πq,p (x) = q({p(xj }), p ∈ X , q ∈ N . Thus, we have c00 (X) = c00 {X}, c0 (X) = c0 {X} and lp (X) = lp {X} for 0 < p ≤ ∞. We extend some of the topological properties of scalar sequence spaces to vector valued sequence spaces. Let E be a vector space of X valued sequences equipped with a locally convex Hausdorff topology. Definition C.1. The space E is a K-space if the maps x = {xj } → xj from E into X are continuous for every j ∈ N. Since ν is a K-space, then ν{X} is a K-space. If z ∈ X, then ej ⊗z will denote the sequence with z in the j th coordinate and 0 in the other coordinates. For every n ∈ N, the sectional operator (projection) is defined to be the map Pn : E → E defined by Pn (x) =

n X j=1

ej ⊗ xj = (x1 , ..., xn , 0, ...).

Definition C.2. The K-space E is an AK-space if for every x ∈ E, we P∞ have x = j=1 ej ⊗ xj , with convergence in E.

Definition C.3. The K-space E is an AB-space if {Pn x} is bounded for every x ∈ E.

Definition C.4. The K-space E has the sections uniformly bounded property (SUB) if {Pn x : n ∈ N, x ∈ B} is bounded for every bounded set B ⊂ E. Definition C.5. The K-space E has the property SE (sections equicontinuous) if the sectional operators {Pn } are equicontinuous.

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The spaces ν{X} supply a large number of spaces with the properties defined above. Proposition C.6. Consider the following properties for ν{X}: (i) (ii) (iii) (iv) (v)

If ν is an AK-space, then ν{X} is an AK-space. If ν is an AB-space, then ν{X} is an AB-space. If ν has property SUB, then ν{X} has property SUB. If ν has property SE, then ν{X} has property SE. If X is sequentially complete, ν is sequentially complete, has property (M ) and the topology of ν is linked to the topology of coordinatewise convergence, then ν{X} is sequentially complete.

Proof: We prove (i); the other proofs of (ii)-(iv) are similar. Let x ∈ E and let πq,p be a basic semi-norm for ν{X}. Then ∞ ∞ X X p(xj )ej ) → 0 ej ⊗ xj ) = q( πq,p ( j=n

j=n

since {p(xj )} ∈ ν, so the result follows. (v): Suppose {xk } is Cauchy in ν{X}. Since X is sequentially complete, limk xkj = xj exists for every j. Set x = {xj }. Let p ∈ X and q ∈ N . By (M ), q({p(xkj )}j − {p(xlj )}j ) ≤ q({p(xkj − xlj )}j ) so {p(xkj )}j is a Cauchy sequence in ν. Since ν is sequentially complete, there exists u ∈ ν such that {p(xkj )} → u as k → ∞. Also, limk p(xkj ) = p(xj ) for each j so u = {p(xj )} ∈ ν and, therefore, x = {xj } ∈ ν{X}. Since {xk } is Cauchy in ν and xk → x ∈ ν{X} in the topology of coordinatewise convergence, xk → x in ν since the topology of ν is linked to the topology of coordinatewise convergence (Appendix A.4). From (v) above, we have Corollary C.7. If X is sequentially complete, then l p (X) and c0 (X) are sequentially complete for 1 ≤ p ≤ ∞. We now give statements of various gliding hump properties for vector valued sequence spaces. These statements are straightforward generalizations of the corresponding properties for scalar sequence spaces (see

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Appendix B). If σ ⊂ N and x = {xj } is an X valued sequence, the coordinatewise product of χσ and x is denoted by χσ x. Definition C.8. The space E has the signed weak gliding hump property (signed-WGHP) if for every x ∈ E and every increasing sequence of intervals {Ij }, there exist a subsequence {nj } and a sequence of signs {sj } such that P∞ the coordinatewise sum of the series j=1 sj χInj x ∈ E. If the signs above can be chosen to be equal to 1 for every x ∈ E, then E has the weak gliding hump property (WGHP). The space E is monotone if χσ x ∈ E for every σ ⊂ N and x ∈ E. Any monotone space has WGHP. The spaces ν{X} are all monotone since ν is assumed to be normal so ν{X} always has WGHP. The proof in Example 8 of Appendix B shows that the non-monotone space CS(X) has WGHP (use the semi-norms |·| which generates the topology of X in place of absolute value). Definition C.9. The K-space E has the signed strong gliding hump property (signed-SGHP) if for every bounded sequence {xj } from E and every increasing sequence of intervals {Ij }, there exist a subsequence {nj } and a sequence of signs {sj } such that the coordinatewise sum of the series P∞ nj ∈ E. If the signs above can be chosen equal to 1 for every j=1 sj χInj x x ∈ E, then E has the strong gliding hump property (SGHP). The space l∞ (X) has SGHP. Proposition C.10. If ν has SGHP and if X is normed, then ν{X} has SGHP. Proof: Let {xj } be bounded in ν{X} and {Ij } be an increasing sequence

j of intervals. Then {{ xi }i : j ∈ N} is bounded in ν so there exists a

nj P

}i ∈ ν so subsequence {nj } such that the coordinate sum ∞ j=1 χInj { xi P∞ nj ∈ ν{X}. j=1 χInj x

Definition C.11. The K-space E has the zero gliding hump property (0GHP) if for every null sequence {xj } in E and every increasing sequence of intervals {Ij }, there is a subsequence {nj } such that the coordinatewise P∞ sum of the series j=1 χInj xnj ∈ E. As in Proposition C.10, we have

Proposition C.12. If ν has 0-GHP and if X is normed, then ν{X} has 0-GHP.

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Let µ be a scalar sequence space containing c00 . Definition C.13. The K-space E has the strong µ gliding hump property (strong µ-GHP) if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, then for every u = {uj } ∈ µ the P∞ coordinatewise sum of the series j=1 uj χIj xj ∈ E.

Definition C.14. The K-space E has the weak µ gliding hump property (weak µ-GHP) if whenever {xj } is a bounded sequence in E and {Ij } is an increasing sequence of intervals, then there is a subsequence {nj } such that for every u = {uj } ∈ µ the coordinatewise sum of the series P∞ nj ∈ E. j=1 uj χInj x Proposition C.15. If ν has strong µ-GHP, then ν{X} has strong µ-GHP.

Proof: Let {xj } be bounded in ν{X} and let {Ij } be an increasing P j sequence of intervals. Let u = {uj } ∈ µ and set x = ∞ j=1 uj χIj x (coordiP∞ j nate sum). Let p ∈ X and note that p(x(·)) = j=1 |uj | χIj p(x (·)), where x(·) is the function j → xj . Now {{p(xji )}i : j ∈ N} is bounded in ν by definition. By strong µ-GHP, {p(xj )} ∈ ν so x ∈ ν{X}. As in Propositions C.10 and C.12, we have Proposition C.16. If ν has weak µ-GHP and X is normed, then ν{X} has weak µ-GHP. We give an example of a non-monotone space with strong l 1 -GHP. Example C.17. CS(X) has strong l 1 -GHP. Suppose that {xj } is bounded in CS(X) and {Ij } is an increasing sequence of intervals. Let u = P∞ j and set {uj } ∈ l1 . Put x = j=1 uj χIj x . Let > 0 and p ∈ X P j M = sup{p( i∈I xi ) : I a finite interval, j ∈ N}. Pick N such that P∞ Suppose I is a finite interval such that min I > N . Then j=N |uj | < . P P∞ p( j∈I xj ) ≤ j=N |uj | M < M so x ∈ CS(X). As in the example above, BS(X) also has strong l 1 -GHP. We next extend the ∞-GHP to vector valued sequence spaces (Appendix B.36).

Definition C.18. The space E has the infinite gliding hump property (∞-GHP) if whenever x ∈ E and {Ij } is an increasing sequence of intervals, there exist a subsequence {nj } and anj > 0, anj → ∞ such that

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every subsequence of {nj } has a further subsequence {pj } such that the P∞ coordinatewise sum of the series j=1 apj χIpj ∈ E. As in Propositions C.10 and C.12, we have

Proposition C.19. If X is normed and ν has ∞-GHP, then ν{X} has ∞-GHP. In particular, if X is normed, then the spaces c0 (X) and lp (X) (0 < p < ∞) have ∞-GHP (Examples B.39 and B.40 of Appendix B). As in Proposition B.42 of Appendix B, we also have Proposition C.20. If E is a Banach AK-space, then E has ∞-GHP. Let Y be a TVS. The β-dual of E with respect to Y is defined to be ∞ X E βY = {Tj } : Tj ∈ L(X, Y ), Tj xj converges for every {xj } ∈ E . j=1

Thus, E βY consists of all operator valued series in L(X, Y ) which are E multiplier convergent. If Y is the scalar field, we write E βR = E β ; in this case, E, E β form a dual pair under the pairing ∞ X

0 xj , xj , x = {xj } ∈ E, x0 = {x0j } ∈ E β . x0 · x = j=1

We give several examples of β-duals.

Example C.21. Let X be a normed space and assume that the dual space X 0 is equipped with the dual norm. Then ν{X}β = ν β {X 0 }. {x {kxj k} ∈ ν First, let {x0j } ∈ ν β {X 0 } and suppose j } ∈ ν{X}. Then P∞ 0 P∞ 0 β 0

≤ and { xj } ∈ ν . Hence, j=1 xj , xj j=1 xj kxj k < ∞ so 0 β {x0j } ∈ ν{X}β and ν β {X } ⊂ ν{X} . Next, let {x0j } ∈ ν{X}β . If

0 0 β β 0

{xj } ∈ / ν {X }, then { xj } ∈ / ν . Since ν is normal, there exists P∞ t ∈ ν such that j=1 |tj | x 0j = ∞. For each j pick xj ∈ X such j 0

0 and that kxj k = 1 and xj < xj , xj0 + 1/2 . β Then {tj xβj } ∈ βν{X} P∞ 0 0 / ν{X} so ν{X} ⊂ ν {X } and = ∞. Thus, {xj } ∈ j=1 |tj | xj , xj equality follows. Let Xb0 be the dual space of X equipped with the strong topology β(X 0 , X). Proposition C.22. BS(X)β = BV0 (Xb0 ).

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Proof: Let y ∈ BS(X)β . To show that yj → 0 strongly, it suffices to show that hyj , xj i → 0 for every bounded sequence {xj } ⊂ X. If x0 = 0, P∞ then {xj − xj−1 } ∈ BS(X) so j=1 hyj , xj − xj−1 i converges, and we have that limj hyj , xj − xj−1 i = 0 for every bounded sequence {xj }. This implies that lim hyj , xj i = 0 for every bounded sequence {xj } [ define a bounded sequence {zj } by 0, x1 , 0, x3 , 0, ...; then the sequence {hyj , zj+1 − zj i} contains the sequence {hy2j+1, x2j+1 i} as a subsequence so lim hy2j+1, x2j+1 i = 0 and similarly, lim hy2j , x2j i = 0 so lim hyj , xj i = 0 ]. Thus, y ∈ c0 (X). Let {xj } be bounded in X and set zj = xj+1 − xj . Then {zj } ∈ BS(X) P∞ and j=1 hyj , zj i converges. Now (∗)

n X j=1

hyj , zj i =

n X j=1

hyj , xj+1 − xj i =

n X j=1

hyj − yj+1 , xj i − hyn , xn i .

P∞ By what was established above, hyn , xn i → 0 so j=1 hyj − yj+1 , xj i converges absolutely for every bounded sequence {xj } by (∗). This implies P∞ that j=1 (yj − yj+1 ) is absolutely convergent in Xb0 [ if B ⊂ X is bounded, for every j pick xj ∈ B such that pB (yj − yj+1 ) = sup{|hyj − yj+1 , xi| : x ∈ B} < |hyj − yj+1 , xj i| + 1/2j P∞ so j=1 pB (yj − yj+1 ) < ∞ ]. Thus, y ∈ BV0 (Xb0 ). Pn Next, let y ∈ BV0 (Xb0 ) and x ∈ BS(X). Then {sn = j=1 xj } is P∞ bounded so j=1 hyj − yj+1 , sj i converges absolutely. Now (∗∗)

n X j=1

hyj , xj i =

n−1 X j=1

hyj − yj+1 , sj i + hyn , sn i .

Since yn → 0 strongly, hyn , sn i → 0 so (∗∗) implies that converges. That is, y ∈ BS(X)β so BV0 (Xb0 ) ⊂ BS(X)β .

P∞

j=1

hyj , xj i

Proposition C.23. l∞ (X)β = l1 (Xb0 ). Proof: Let y ∈ l∞ (X)β . Suppose that {xj } ⊂ X is bounded and t ∈ P∞ P∞ l . Then {tj xj } ∈ l∞ (X) so j=1 |hyj , tj xj i| = j=1 |tj | |hyj , xj i| < ∞. P ∞ Thus, {hyj , xj i} ∈ l1 or j=1 |hyj , xj i| < ∞. Hence, y ∈ l 1 (Xb0 ) [see the argument in Proposition C.22] and l ∞ (X)β ⊂ l1 (Xb0 ). Let y ∈ l1 (Xb0 ). Suppose that {xj } ∈ l∞ (X) and set B = {xj : j ∈ N}, P∞ P∞ pB (x0 ) = sup{|hx0 , xi| :x ∈ B}. Then j=1 |hyj , xj i| ≤ j=1 pB (yj ) < ∞ so y ∈ l∞ (X)β and l1 (Xb0 ) ⊂ l∞ (X)β . Similarly, we have ∞

Proposition C.24. c0 (X)β = l1 (Xb0 ).

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Let le∞ (X 0 ) be the sequences in l ∞ (Xb0 ) which are equicontinuous (such sequences are always strongly bounded ([Sw2] 19.5)). Proposition C.25. We have the following relationships: (i) le∞ (X 0 ) ⊂ l1 (X)β . (ii) l1 (X)β ⊂ l∞ (Xb0 ). (iii) If X is quasi-barrelled, then l 1 (X)β = l∞ (Xb0 ). Proof: (i): Let y = {yj } ∈ le∞ (X 0 ). Then there exists a continuous semi-norm p on X such that |hyj , xi| ≤ p(x) for all x ∈ X, j ∈ N. Let P∞ P∞ {xj } ∈ l1 (X). Since j=1 |hyj , xj i| ≤ j=1 p(xj ) < ∞, y ∈ l1 (X)β and (i) holds. (ii): Let y ∈ l1 (X)β and let {xj } be bounded. If t ∈ l1 , then {tj xj } ∈ P∞ P∞ 1 l (X) so j=1 |hyj , tj xj i| = j=1 |tj | |hyj , xj i| < ∞. Hence, {hyj , xj i} ∈ l∞ which implies that {yj } is strongly bounded [if B ⊂ X is bounded, then for every j there exists xj ∈ B such that pB (yj ) = sup{|hyj , xi| : x ∈ B} ≤ |hyj , xj i| + 1 so {pB (yj )} is bounded]. Therefore, (ii) holds. (iii) follows from the definition of quasi-barrelled and (i) and (ii) ([Sw2] 19.13, [Wi] 10.1.11). Finally, we compute one example of an operator valued β-dual. Below when X and Y are normed spaces, the dual spaces are always assumed to be equipped with the dual norm. Proposition C.26. Let X and Y be normed spaces with Y complete. Let 1 < p < ∞ and p1 + 1q = 1. Then {Tj } ∈ lp (X)β iff {{Tj0y 0 } : y 0 ∈ Y 0 , ky 0 k ≤ 1} is bounded in l q (X 0 ). Proof: Suppose {{Tj0y 0 } : y 0 ∈ Y 0 , ky 0 k ≤ 1} is bounded in l q (X 0 ) and M is a bound for the norms of the elements in this set. Then

n

n n X

0 0

X 0

X

Tj y kxj k : ky 0 k ≤ 1} ≤ sup

= sup{

hy , T x i T x j j j j

ky 0 k≤1 j=m

j=m

j=m ≤(

n X

j=m

kxj kp )1/p sup {(

P

ky 0 k≤1

n n X X

0 0 q 1/q

Tj y ) ≤ M( kxj kp )1/p

j=m

j=m

which implies that j Tj xj is Cauchy in Y and, therefore, converges. Suppose {Tj } ∈ lp (X)β . First, we claim that {Tj0 y 0 } ∈ lq (X 0 ) for every

P∞

T 0 y 0 q = ∞ y 0 ∈ Y 0 . Suppose that this condition fails to hold. Then j=1 j

P ∞ for some y 0 ∈ Y 0 . Then there exists t ∈ l p such that j=1 Tj0 y 0 |tj | = ∞.

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For every j there exists xj ∈ X, kxj k ≤ 1 with hy 0 , Tj xj i ≥ Tj0 y 0 /2. Then P∞ 0 p / lp (X)β . j=1 |tj | |hy , Tj xj i| = ∞. Since {tj xj } ∈ l (X), {Tj } ∈ Now lq (X 0 ) is complete (Corollary C.7) and the linear map y 0 → {Tj0 y 0 } from Y 0 into lq (X 0 ) has a closed graph. Hence, from the Closed Graph Theorem the map is continuous and the result follows. Other examples of β-duals for vector valued sequence spaces can be found in [FP2], [Fo], [GKR] and [Ros].

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Appendix D The Antosik-Mikusinski Matrix Theorems

In this appendix we will present two versions of the Antosik-Mikusinski Matrix Theorems. These matrix theorems have proven to be very useful in treating applications in functional analysis and measure theory where gliding hump techniques are employed (see [Sw1] for more versions of the matrix theorem and applications). These theorems are used at various points in the text in gliding hump proofs. Let X be a (Hausdorff) TVS. We begin with a simple lemma. Lemma D.1. Let xij ∈ X for i, j ∈ N. If limi xij = 0 for every j and limj xij = 0 for every i and if {Uk } is a sequence of neighborhoods of 0 in X, then there exists an increasing sequence {pi } such that xpi pj ∈ Uj for j˙ > i. Proof: Set p1 = 1. There exists p2 > p1 such that xip1 ∈ U2, xp1 j ∈ U2 for i, j ≥ p2 . Then there exists p3 > p2 such that xip1 , xip2 , xp1 j , xp2 j ∈ U for i, j ≥ p3 . Now just continue the construction. We now establish our version of the Antosik-Mikusinski Matrix Theorem. Theorem D.2. (Antosik-Mikusinski) Let xij ∈ X for i, j ∈ N. Suppose (I) limi xij = xj exists for each j and (II) for each increasing sequence of positive integers {mj } there is a subseP∞ quence {nj } of {mj } such that { j=1 xinj }i is Cauchy. Then limi xij = xj uniformly for j ∈ N. In particular, lim lim xij = lim lim xij = 0 and lim xii = 0. i

j

j

i

i

239

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Proof: If the conclusion fails, there is a closed, symmetric neighborhood U0 of 0 and increasing sequences of positive integers {mk } and {nk } such that xmk nk −xnk ∈ / U0 for all k. Pick a closed, symmetric neighborhood U1 of 0 such that U1 + U1 ⊆ U0 and set i1 = m1 , j1 = n1 . Since xi1 j1 − xj1 = / U1 for (xi1 j1 − xij1 ) + (xij1 − xj1 ), there exists i0 such that xi1 j1 − xij1 ∈ i ≥ i0. Choose k0 such that mk0 > max{i1 , i0 }, nk0 > j1 and set i2 = mk0 , j2 = nk0 . Then xi1 j1 − xi2 j1 ∈ / U1 and xi2 j2 − xj2 ∈ / U0. Proceeding in this manner produces increasing sequences {ik }, {jk } such that xik jk − xjk ∈ / U0 and xik jk − xik+1 jk ∈ / U1 . For convenience, set zk,l = xik jl − xik+1 jl so zk,k ∈ / U1 . Choose a sequence of closed, symmetric neighborhoods of 0, {Un }, such that Un + Un ⊆ Un−1 for n ≥ 1. Note that m X U3 + U 4 + · · · + U m = Uj ⊆ U 2 j=3

for each m ≥ 3. By (I) and (II), limk zkl = 0 for each l and liml zkl = 0 for each k so by Lemma D.1 there is an increasing sequence of positive integers {pk } such that zpk pl , zpl pk ∈ Uk+2 for k > l. By (II), {pk } has a P ∞ subsequence {qk } such that { ∞ k=1 xiqk }i=1 is Cauchy so lim k

Thus, there exists k0 such that m X

z qk 0 ql =

l=1,l6=k0

kX 0 −1

z qk 0 ql +

m X

Ul+2 ⊆

l=k0 +1

P∞

l=1,l6=k0

zqk ql = 0.

l=1 P∞ l=1 zqk0 ql

zqk0 ql ∈ U2. Thus,

z qk 0 qk 0 =

∞ X l=1

∈ U2 . Then for m > k0 ,

m X

l=k0 +1

l=1

+ so zk0 =

∞ X

m+2 X l=3

z qk 0 ql ∈

kX 0 −1

Uk0 +2

l=1

Ul ⊆ U 2

z qk 0 ql − z k0 ∈ U 2 + U 2 ⊆ U 1

This is a contradiction and establishes the result. A matrix [xij ] satisfying conditions (I) and (II) of Theorem D.2 is called a K-matrix . [The appellation ”K” here refers to the Katowice branch of

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the Mathematical Institute of the Polish Academy of Science where the matrix theorems and applications were developed by Antosik, Mikusinski and other members of the institute.] At other points in the text we will also require another version of the matrix theorem which was developed by Stuart ([St1], [St2]) to treat weak sequential completeness of β-duals. Theorem D.3. (Stuart) Let xij ∈ X for i, j ∈ N. Suppose (I) limi xij = xj exists for all j and (II) for each increasing sequence of positive integers {mj } there is a subsequence {nj } and a choice of signs sj ∈ {−1, 1} such that P∞ { j=1 sj xinj }∞ i=1 is Cauchy. Then limi xij = xj uniformly for j ∈ N. In particular,

lim lim xij = lim lim xij = 0 and lim xii = 0. i

j

j

i

i

Proof: If the conclusion fails, there is a closed, symmetric neighborhood of 0,U0 , and increasing sequences of positive integers {mk } and {nk } such / U0 for all k. Pick a closed, symmetric neighborhood of that xmk nk − xnk ∈ 0,U1 , such that U1 + U1 ⊂ U0 and set i1 = m1 , j1 = n1 . Since xi1 j1 − xj1 = (xi1 j1 − xij1 ) + (xij1 − xj1 ), there exists i0 such that xi1 j1 − xij1 ∈ / U1 for i ≥ i0 . Choose k0 such that mk0 > max{i1 , i0 }, nk0 > j1 and set i2 = mk0 , j2 = nk0 . Then / U0 . Proceeding in this manner produces / U1 and xi2 j2 − xj2 ∈ xi 1 j 1 − x i 2 j 1 ∈ / U0 and xik jk − increasing sequences {ik } and {jk } such that xik jk − xjk ∈ / U1 . / U1 . For convenience, set zkl = xik jl − xik+1 jl so zkk ∈ xik+1 jk ∈ Choose a sequence of closed, symmetric neighborhoods of 0, {Un }, such that Un + Un ⊂ Un−1 for n ≥ 1. Note that U3 + U4 + ... + Um =

m X j=3

Uj ⊂ U2 f or each m ≥ 3.

By (I), limk zkl = 0 for each l and by (II), liml zkl = 0 for each k so by Lemma D.1 there is an increasing sequence of positive integers {pk } such that zpk pl , zpl pk ∈ Uk+2 for k > l. By (II) there is a subsequence {qk } of {pk } and a choice of signs sk such that {

∞ X k=1

sk xiqk }∞ i=1

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is Cauchy so ∞ X

lim k

sl zqk ql = 0.

l=1

Thus, there exists k0 such that ∞ X l=1

s l z qk 0 ql ∈ U 2 .

Then for m > k0 , m X

l=1,l6=k0

so

s l z qk 0 ql =

kX 0 −1

s l z qk 0 ql +

l=1

m X

l=k0 +1 ∞ X

z k0 =

l=1,l6=k0

Thus, s k0 z qk 0 qk 0 =

∞ X l=1

since U1 is symmetric

s l z qk 0 ql ∈

kX 0 −1

Uk0 +2 +

l=1

m X

l=k0 +1

Ul ⊂ U 2

s l z qk 0 ql ∈ U 2 .

s l z qk 0 ql − z k0 ∈ U 2 + U 2 ⊂ U 1 z qk 0 qk 0 ∈ U 1

as well. This is a contradiction.

A matrix which satisfies conditions (I) and (II) of Theorem D.3 will be called a signed K-matrix and Theorem D.3 will be referred to as the signed version of the Antosik-Mikusinski Matrix Theorem. We give an example of a matrix which is a signed K-matrix but is not a K-matrix. Example D.4. Let t = {tj } ∈ bs, the space of bounded series, and let X be bs equipped with the topology of coordinatewise convergence, σ(bs, c00 ) [Appendix A]. Define a matrix M = [mij ] with entries from X by mij = ej . Then no row of M has a subseries which converges in X so M is not a P∞ K-matrix. However, given any subsequence {nj } the series j=1 (−1)j enj converges in X so M is a signed K-matrix. Other refinements and comments on the matrix theorems can be found in [Sw1] 2.2. The text [Sw1] contains numerous applications of the matrix theorems to topics in topological vector spaces, measure and integration theory and sequence spaces.

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Appendix E Drewnowski’s Lemma

In this appendix we establish a remarkable result of Drewnowski which asserts that a strongly bounded, finitely additive set function defined on a σ-algebra is in some sense not ”too far” from being countably additive ([Dr]). This result is very useful in treating finitely additive set functions. Let Σ be a σ-algebra of subsets of a set S, X be a TVS whose topology is generated by the quasi-norm |·| and let µ : Σ → X be finitely additive and strongly bounded. (Recall µ is strongly bounded if µ(Ej ) → 0 whenever {Ej } is a pairwise disjoint sequence from Σ.) For E ∈ Σ, set µ0 (E) = sup{|µ(A)| : A ⊂ E, A ∈ Σ};

µ0 is called the submeasure majorant of µ and µ0 is also strongly bounded in the sense that µ0 (Ej ) → 0 whenever {Ej } is a pairwise disjoint sequence from Σ. Lemma E.1. (Drewnowski) If µ : Σ → X is finitely additive and strongly bounded and {Ej } is a pairwise disjoint sequence from Σ, then {Ej } has a subsequence {Enj } such that µ is countably additive on the σ-algebra generated by {Enj }. Proof: Partition N into a pairwise disjoint sequence of infinite sets 0 0 {Kj1 }∞ j=1 . By the strong additivity of µ , µ (∪j∈Ki1 Ej ) → 0 as i → ∞ so 0 there exists i such that µ (∪j∈Ki1 Ej ) < 1/2. Set N1 = Ki1 and n1 = inf N1 . Now partition N1 \ {n1 } into a pairwise disjoint sequence of infinite sub0 2 sets {Kj2 }∞ j=1 . As above there exists i such that µ (∪j∈Ki2 Ej ) < 1/2 . Let 2 N2 = Ki and n2 = inf N2 . Note N2 ⊂ N1 and n2 > n1 . Continuing this construction produces a subsequence nj ↑ ∞ and a sequence of infinite subsets of N, {Nj }, such that Nj+1 ⊂ Nj and µ0 (∪i∈Nj Ei ) < 1/2j . If Σ0 is the σ-algebra generated by {Enj }, then µ is countably additive on Σ0 . 243

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We also have a version of Drewnowski’s Lemma for a sequence of finitely additive set functions. Corollary E.2. Let µi : Σ → X be finitely additive and strongly bounded for each i ∈ N. If {Ej } is a pairwise disjoint sequence from Σ, then there is a subsequence {Enj } such that each µi is countably additive on the σ-algebra generated by {Enj }. 0

Proof: Define a quasi-norm |·| on X N by 0

0

|x| = |{xi }| = N

∞ X i=1

|xi | /((1 + |xi |)2i ).

Define µ : Σ → X by µ(E) = {µi (E)}. Then µ is finitely additive and 0 strongly bounded with respect to |·| so by Lemma 1 there is a subsequence {Enj } such that µ is countably additive on the σ-algebra Σ0 generated by {Enj }. Thus, each µi is countably additive on Σ0 .

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References

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[St3] C. Stuart, Interchanging the limit in a double series, Southeast Asia Bull. Math., 18(1994), 81 - 84. [SS] C. Stuart and C. Swartz, A Projection Property and Weak Sequential Completeness of α-duals, Collect. Math.l, 43(1992), 177 - 185. [Sw1] C. Swartz, Infinite Matrices and the Gliding Hump, World Sci. Publ., Singapore, 1996. [Sw2] C. Swartz, An Introduction to Functional Analysis, Marcel Dekker, N.Y., 1992. [Sw3] C. Swartz, Measure Integration and Function Spaces, World Sci., Pub., Singapore, 1994. [Sw4] C. Swartz, Orlicz-Pettis Theorems for Multiplier Convergent Operator Valued Series, Proy. J. Math., 23(2004), 61-72. [Sw5] C. Swartz, Subseries Convergence in Spaces with Schander Basis, Proc. Amer. Math. Soc.,129(1995), 455-457. [Th] G.E.F. Thomas, L’integration par rapport a une mesure de Radon vectorielle, Ann. Inst. Fourier, 20(1970), 55 - 191. [Thr] B.L. Thorp, Sequential Evaluation Convergence, J. London. Math. Soc., 44(1969), 201-209. [Tw] I. Tweddle, Unconditional Convergence and Vector-valued Measures, J. London Math. Soc., 2(1970), 603 - 610. [Wi] A. Wilansky, Modern Methods in Topological Vector Spaces, McGraw - Hill, N.Y., 1978. [Wi2] A. Wilansky, Summability through Functional Analysis, North Holland, Amsterdam, 1984. [Wu] Wu, Junde, The compact sets in the infinite matrix topological algebras, Acta Math. Sinica, to appear. [WL] Wu Junde and Lu Shijie, A Summation Theorem and its Applications, J. Math. Anal. Appl., 257(2001), 29 - 38. [Y] K. Yosida, Functional Analysis, Springer - Verlag, N.Y., 1966.

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World Scientific Book - 9in x 6in

Index

A : E → F , 201 δ1 (X, X 0 ), 57 δ2 (X, X 0 ), 62 Dλ (X, X 0 ), 63 FQσ, 219 IQσ, 220 0 τ (X, X P ), 208 M ∞ ( xj ), 140 t(X, X 0 ), 60 tλ (X, X 0 ), 68 s, 214 β-dual, 5 X b , 34 B(S, Σ), 60 Lb (X, Y ), 211 l∞ , 213 bs, 214 k·kbs , 214 bv, 214 k{tj }kbv , 214 M0 , 9, 219 χσ , 216 K(X, Y ), 94 Ac , 37 γ(X, X 0 ), 208 CX (S), 154 cs, 214 c, 213 ca(Σ), 60, 72 Lc (X, Y ), 211 λ(X, X 0 ), 208 c0 -factorable, 224

c0 -invariant, 224 δt , 60 cc , 213 c00 , 213 m0 , 9, 213 ∞-GHP, 15, 85, 224, 234 [m, n], 216 λβ , 5 A : λ → µ, 157 M (λ), 218 bv0 , 214 L→0 (X, Y ), 92, 211 c0 , 213 E βY , 173 µ ˆ (A), 180 LA (X, Y ), 211 LC (X, Y ), 91 Ls (X, Y ), 211 Lpc (X, Y ), 92, 211 χσ t, 216 k·kp , 214 lp , 213, 214 |·|p , 214 s · t, 5 E ββ , 203 AT T , 203 Pn , 231 X s , 34 β(X, X 0 ), 208 Kb (X, Y ), 97 k·k∞ , 213 t · x, 12 251

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World Scientific Book - 9in x 6in

Multiplier Convergent Series

ej ⊗ z, 170 AT , 158, 202 E β , 170, 187 l∞ (X), 229 BS(X), 230 BV (X), 230 c(X), 229 CS(X), 230 c0 (X), 229 c00 (X), 229 cc (X), 229 lp (X), 230 m0 (X), 229 BV0 (X), 230 w(E βY , E), 173 σ(X, X 0 ), 208 ω(λβX , λ), 12 0-GHP, 8, 221, 233 AB-space, 215, 231 absolutely convergent, 23, 45 AK-space, 215, 231 Antosik, 146 Antosik Interchange Theorem, 145 Antosik-Mikusinski, 239 Banach Mackey pair, 138 Banach-Mackey space, 34 Banach-Steinhaus property, 173 BK-space, 215 bounded multiplier convergent, 5, 169 Cesaro matrix, 225 conditionally sequentially compact, 14 continuous from above, 182 coordinate functionals, 36 countably additive, 36

finite/co-finite, 37 finitely additive, 36 FK-space, 215 Hahn-Schur Theorem, 193, 196 Hellinger-Toeplitz topology, 6, 51, 170, 188, 209 increasing, 8, 216 infinite gliding hump property, 15, 85, 224, 234 interval, 8, 216 invariant, 218 K-matrix, 240 K-space, 215, 231 LCTVS, 207 linked, 50, 209 locally complete, 27 Mackey topology, 208 matrix domain, 225 monotone, 216, 233 multiplier Cauchy, 5, 169 multiplier convergent, 5, 169 multiplier space, 218 multipliers, 5, 222 Nikodym Convergence Theorem, 200 normal, 10, 216 Orlicz-Pettis Theorem, 49, 53, 85, 90, 152, 187 polar topology, 208 quasi-norm, 207 rearrangement convergent, 22

DF property, 95 Dierolf topology, 57, 62, 63 double series, 145 Drewnowski, 243 exhaustive, 41

Schauder basis, 36 Schauder decomposition, 153 SE, 216, 231 sectional operator, 231 sectional projection, 215

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Index

sections equicontinuous, 216, 231 sections uniformly bounded, 133, 216 semi-variation, 43, 179 SGHP, 9, 219, 233 sign, 9 signed K-matrix, 242 signed strong gliding hump property, 9, 219, 233 signed weak gliding hump property, 12, 216, 233 signed-SGHP, 9, 219, 233 signed-WGHP, 12, 216, 233 signs, 216 solid, 10, 216 Stiles, 151 strong gliding hump property, 9, 219, 233 strong µ-GHP, 138 strong µ gliding hump strong property, 138 strong operator topology, 211 strong topology, 208 strongly additive, 41 strongly bounded, 41, 243 Stuart, 174 SUB, 133, 216 submeasure majorant, 243 subseries convergent, 5 summing operator, 6, 169

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TVS, 207 Tweddle topology, 60, 68 unconditional, 36 unconditionally Cauchy, 22 unconditionally convergent, 22 unconditionally converging, 32, 71 uniform operator topology, 211 uniform tails, 63 unordered convergent, 22 variation, 45 weak gliding hump property, 12, 216, 233 weak µ-GHP, 139 weak µ gliding hump property, 139 weak operator topology, 211 weak topology, 208 weakly compact, 71 weakly unconditionally Cauchy, 26 WGHP, 12, 216, 233 wuc, 26 zero gliding hump property, 8, 221, 233

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