Vector Measures, Integration and Related Topics (Operator Theory: Advances and Applications)

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Author: Guillermo P. Curbera | Gerd Mockenhaupt | Werner J. Ricker

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Operator Theory: Advances and Applications Vol. 201

Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv Israel

Editorial Board: D. Alpay (Beer Sheva, Israel) J. Arazy (Haifa, Israel) A. Atzmon (Tel Aviv, Israel) J.A. Ball (Blacksburg, VA, USA) H. Bart (Rotterdam, The Netherlands) A. Ben-Artzi (Tel Aviv, Israel) H. Bercovici (Bloomington, IN, USA) A. Böttcher (Chemnitz, Germany) K. Clancey (Athens, GA, USA) R. Curto (Iowa, IA, USA) K. R. Davidson (Waterloo, ON, Canada) M. Demuth (Clausthal-Zellerfeld, Germany) A. Dijksma (Groningen, The Netherlands) R. G. Douglas (College Station, TX, USA) R. Duduchava (Tbilisi, Georgia) A. Ferreira dos Santos (Lisboa, Portugal) A.E. Frazho (West Lafayette, IN, USA) P.A. Fuhrmann (Beer Sheva, Israel) B. Gramsch (Mainz, Germany) H.G. Kaper (Argonne, IL, USA) S.T. Kuroda (Tokyo, Japan) L.E. Lerer (Haifa, Israel) B. Mityagin (Columbus, OH, USA)

V. Olshevski (Storrs, CT, USA) M. Putinar (Santa Barbara, CA, USA) A.C.M. Ran (Amsterdam, The Netherlands) L. Rodman (Williamsburg, VA, USA) J. Rovnyak (Charlottesville, VA, USA) B.-W. Schulze (Potsdam, Germany) F. Speck (Lisboa, Portugal) I.M. Spitkovsky (Williamsburg, VA, USA) S. Treil (Providence, RI, USA) C. Tretter (Bern, Switzerland) H. Upmeier (Marburg, Germany) N. Vasilevski (Mexico, D.F., Mexico) S. Verduyn Lunel (Leiden, The Netherlands) D. Voiculescu (Berkeley, CA, USA) D. Xia (Nashville, TN, USA) D. Yafaev (Rennes, France)

Honorary and Advisory Editorial Board: L.A. Coburn (Buffalo, NY, USA) H. Dym (Rehovot, Israel) C. Foias (College Station, TX, USA) J.W. Helton (San Diego, CA, USA) T. Kailath (Stanford, CA, USA) M.A. Kaashoek (Amsterdam, The Netherlands) P. Lancaster (Calgary, AB, Canada) H. Langer (Vienna, Austria) P.D. Lax (New York, NY, USA) D. Sarason (Berkeley, CA, USA) B. Silbermann (Chemnitz, Germany) H. Widom (Santa Cruz, CA, USA)

Vector Measures, Integration and Related Topics

Guillermo P. Curbera Gerd Mockenhaupt Werner J. Ricker Editors

Birkhäuser Basel · Boston · Berlin

Editors: Guillermo P. Curbera Facultad de Matemáticas Universidad de Sevilla 41080 Sevilla Spain e-mail: [email protected]

Gerd Mockenhaupt Fachbereich 6: Mathematik Universität Siegen 57068 Siegen Germany e-mail: [email protected]

Werner J. Ricker Mathematisch-Geographische Fakultät Katholische Universität Eichstätt-Ingolstadt 85072 Eichstätt Germany e-mail: [email protected]

2000 Mathematics Subject Classification: 26-06, 28-06, 42-06, 43-06, 46-06, 47-06

Library of Congress Control Number: 2009938995

Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-0346-0210-5 Birkhäuser Verlag AG, Basel – Boston – Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2010 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF∞ Printed in Germany

ISBN 978-3-0346-0210-5

e-ISBN 978-3-0346-0211-2

987654321

www.birkhauser.ch

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

List of Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

A.A. Albanese, J. Bonet and W.J. Ricker On Mean Ergodic Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

W. Arendt and S. Bu Fourier Series in Banach spaces and Maximal Regularity . . . . . . . . . . . . .

21

D. Atanasiu Spectral Measures on Compacts of Characters of a Semigroup . . . . . . . .

41

D. Barcenas and C.E. Finol On Vector Measures, Uniform Integrability and Orlicz Spaces . . . . . . . .

51

O. Blasco The Bohr Radius of a Banach Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

O. Blasco and J. van Neerven Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology . . . . . . . . . . . . . . . . . . . . . . .

65

A. Boccuto and D. Candeloro Defining Limits by Means of Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

D. Bongiorno A First Return Examination of Vector-valued Integrals . . . . . . . . . . . . . .

89

G. Buskes, R. Page, Jr. and R. Yilmaz A Note on Bi-orthomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99

R. del Campo, A. Fern´ andez, I. Ferrando, F. Mayoral and F. Naranjo Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure . . . . . . . . . . . . . . 109 K. Cicho´ n and M. Cicho´ n Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 S.-O. Corduneanu Equations Involving the Mean of Almost Periodic Measures . . . . . . . . . . 125

vi

Contents

G.P. Curbera How Summable are Rademacher Series? . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135

O. Delgado Rearrangement Invariant Optimal Domain for Monotone Kernel Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149

L. Di Piazza and V. Marraffa The Fubini and Tonelli Theorems for Product Local Systems . . . . . . . . . 159 L. Di Piazza and K. Musial A Decomposition of Henstock-Kurzweil-Pettis Integrable Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 P.G. Dodds and B. de Pagter Non-commutative Yosida-Hewitt Theorems and Singular Functionals in Symmetric Spaces of τ -measurable Operators . . . . . . . . . . . . . . . . . . . . .

183

L. Drewnowski and I. Labuda Ideals of Subseries Convergence and Copies of c0 in Banach Spaces . . . 199 J.H. Fourie On Operator-valued Measurable Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 T.A. Gillespie Logarithms of Invertible Isometries, Spectral Decompositions and Ergodic Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 H. Hunziker and H. Jarchow Norms Related to Binomial Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

T.P. Hyt¨ onen Vector-valued Extension of Linear Operators, and T b Theorems . . . . . . 245 B. Jefferies Some Recent Applications of Bilinear Integration . . . . . . . . . . . . . . . . . . . .

255

G. Knowles A Complete Classification of Short Symmetric-antisymmetric Multiwavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 M. de Kock and D. Puglisi On the Range of a Vector Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 H. K¨ onig Measure and Integration: Characterization of the New Maximal Contents and Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 J. van Neerven and L. Weis Vector Measures of Bounded γ-variation and Stochastic Integrals . . . .

303

Contents S. Okada Does a Compact Operator Admit a Maximal Domain for its Compact Linear Extension? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

313

B. de Pagter and W.J. Ricker A Note on R-boundedness in Bidual Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Ch. Papadimitropoulos Salem Sets in the p-adics, the Fourier Restriction Phenomenon and Optimal Extension of the Hausdorff-Young Inequality . . . . . . . . . . .

327

H. Pfitzner L-embedded Banach Spaces and a Weak Version of Phillips Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 L. Rodr´ıguez-Piazza When is the Space of Compact Range Measures Complemented in the Space of All Vector-valued Measures? . . . . . . . . . . . . . . . . . . . . . . . . .

345

A.R. Schep When is the Optimal Domain of a Positive Linear Operator a Weighted L1 -space? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 R.G. Venter Liapounoff Convexity-type Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

371

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Preface Slowly, but surely, it is becoming a tradition that biannual international conferences on “Vector Measures and Integration” are taking place. The first meeting was held in Valencia (Spain) in 2004 with the respectable total of 35 participants and the second meeting in Sevilla (Spain) in 2006 with 50 participants. It became clear at the latter meeting that there was already a broader interest level in the area generally (as it should be). Related areas from operator theory, functional analysis, Banach (and Fréchet) spaces and lattices, non-commutative integration, function theory, classical and harmonic analysis, mathematical physics, and applied mathematics have traditionally used methods and techniques from vector measures and integration theory and, simultaneously, have themselves provided new problems, directions and impetus for the theory of vector measures and integration. So, for the third meeting, held in Eichst¨ att (Germany) in September 2008, a natural and deliberate step was taken to put a larger emphasis on applications and connections with other areas of mathematics. Accordingly, the conference title was modified to “Vector Measures, Integration and Applications”, which is also reflected in the title of this volume. Correspondingly, the attendance grew to 84 participants, which illustrates that the area is really thriving. Most importantly, there was also a healthy mixture of “oldies” and younger researchers in attendance, coming from 21 countries. In addition, there was a pleasant and interesting combination of abstract theory, concrete applications and open problems. Needless to say, the fourth meeting is already fixed; it will take place in late 2010 in Murcia (Spain). This volume consists of a selection of refereed papers based on talks presented at the conference. The included papers represent rather well most of the topics covered in the conference. The organising committee, consisting of O. Blasco, B. Cascales, S. Hilger, G. Mockenhaupt, P. Ressel and W.J. Ricker, wishes to take this opportunity to thank all of the participants and the local support staff from Eichst¨ att for providing a pleasant and stimulating meeting. We also wish to thank Birkh¨ auser Verlag (via T. Hempfling) and Springer Verlag (via C. Byrne) for presenting an excellent book display. For financial support we are indebted to the Maximilian Bickhoff Stiftung, the Alexander von Humboldt Foundation, the Katholische Universit¨ at Eichst¨ att-Ingolstadt, and Professors W. Bischoff, P. Ressel, W.J. Ricker, J. Rohlfs and M. Sommer from the Mathematics Department of the Katholische Universität

x

Preface

Eichst¨ att-Ingolstadt, all of whom assisted our cause. And, of course, special thanks go to the referees and the publisher, without whose support this volume would not have been possible. The Editors, July, 2009 G. Curbera, G. Mockenhaupt, W.J. Ricker Sevilla Siegen Eichst¨ att

List of Talks Plenary Talks W. Arendt The vector-valued Marcinkiewicz Fourier multiplier theorem and applications to evolution equations J. Bonet∗∗ Mean ergodic operators on Fréchet spaces G.P. Curbera∗ How summable are Rademacher series? J. Diestel∗ The Radon-Nikodym property: some old and some not so old questions A. Fern´ andez∗ Lattice isomorphisms between spaces of integrable functions (with respect to vector measures) T.A. Gillespie∗ Logarithms of invertible isometries, spectral decompositions and ergodic multipliers M. Girardi∗ Operator-valued martingale transforms and R-boundedness B. Jefferies Some recent applications of bilinear integration B. de Pagter∗ Double operator integrals with applications to perturbations, differentiation and commutator estimates L. Rodr´ıguez-Piazza∗ When is the subspace of compact range measures complemented in the space of X-valued measures? L. Weis All kinds of integrals. . . and Gauss function spaces Shorter Communications E. Albrecht Unbounded extensions and operator moment problems D. Atanasiu∗ Moment problems and semigroups of operators D. Barcenas On vector integrals, uniform integrability and Orlicz spaces

xii

List of Talks

R. Becker Conical measures and ordered Banach spaces M. Biegert Lattice homomorphisms between Sobolev spaces O. Blasco∗ On the Bohr radius of a Banach space A. Boccuto Ideal-convergence and integrals in Riesz spaces D. Bongiorno A first return examination of the Pettis integral J.M. Calabuig Vector measure representations of Banach function spaces containing Lp -spaces R. del Campo The role of Fatou and Lebesgue properties in the relationship between L1 and L1w for a Fréchet-space valued measure B. Cascales∗ Measurability and selections of multi-functions in Banach spaces M. Cicho´ n & K. Cicho´ n Some applications of non-absolute integrals in the theory of differential equations in Banach spaces S.-O. Corduneanu Equations involving the mean of almost periodic measures A. Croitoru On a generalized Gould type set-valued integral O. Delgado Rearrangement invariant optimal domain for monotone kernel operators L. Di Piazza Approximation of Banach space-valued Riemann type integrable functions by step functions J. Diestel Sequences in the range of a vector measure P.G. Dodds∗ A Yosida-Hewitt decomposition and applications to K¨ othe duality in non-commutative spaces I.R. Doust An integration theory for AC(σ)-operators I. Ferrando Trace duality and summing operators on Lp (m) of a vector measure m L. Florescu Compactness via tightness J.H. Fourie∗ On operator-valued measurable functions L.M. Garc´ıa Raffi Bilinear integration and applications to quantum scattering

List of Talks

xiii

J. Haluska Toeplitz operators with distribution symbols as operator-valued integrals T.P. Hyt¨ onen∗ A new vector-valued maximal operator and its applications H. Jarchow Some relatives of harmonic series M.A. Juan Representation theorems of p-convex Banach lattices J. Kawabe The continuity of Riesz space-valued non-additive measures: the Alexandroff theorem G. Knowles A complete classification of three and four tap symmetric-antisymmetric multiwavelets with approximation order two Heinz K¨ onig New foundations for measure and integration M. Kunze Integrable semigroups on norming dual pairs I. Labuda Ideals of unconditional convergence and copies of c0 in Banach spaces M. Leinert Two problems in non-commutative integration Z. Lipecki Characteristic properties of the semivariation of a vector measure W. Lusky On L1 -subspaces consisting of holomorphic functions P. Macansantos Differential inclusions in separable Hilbert spaces using the generalized Riemann integral S.M. Maepa Classification of the Grothendieck natural tensor norms according to the RadonNikodym property and the Lewis-Radon-Nikodym property V. Marraffa Product local system and Fubini and Tonelli theorems F. Mayoral The two complex interpolation methods applied to spaces of integrable functions with respect to a vector measure P.J. Miana Hilbert, Dirichlet and Fejér families of operators arising from C0 -groups and cosine functions K. Musial∗ Gauge integrals and the Radon-Nikodym-property

xiv

List of Talks

F. Naranjo Continuity and compactness of multiplication operators on spaces of integrable functions with respect to a vector measure J. van Neerven Lie-Trotter approximations for Ornstein-Uhlenbeck processes S. Okada∗ Does a compact operator admit a maximal domain for its compact linear extension? C. Papadimitropoulos∗ p-adic Salem sets, Fourier restriction and optimal extension of the Hausdorff-Young inequality H. Pfitzner∗ Phillips’ lemma for L-embedded Banach-spaces E. Popa A weak integral arising in potential theory B. Satco Decomposability in the space of l.c.s.-valued Pettis integrable functions A.R. Schep∗∗ When are optimal domains of positive linear operators weighted L1 -spaces? M. Amin Sofi Hilbert space-valued absolutely summing maps and their relatives on Banach spaces P. Tradecete Operators preserving c0 or 1 between vector-valued Banach lattices A. Uglanov∗ Vector integrals and applications M. V¨ ath Compactness of integral operators of vector functions R. Venter Some vector measures on fields and Liapounoff convexity-type theorems M. Veraar R-boundedness of smooth operator-valued functions W. Wnuk Properties of Banach lattices E related to properties of E-valued measures D. Yost Decomposability of polyhedra

∗

∗∗

Partially supported by the Maximilian Bickhoff Stiftung of the Katholische Universit¨ at Eichst¨ att-Ingolstadt Partially supported by the Alexander von Humboldt Foundation.

Operator Theory: Advances and Applications, Vol. 201, 1–20 c 2009 Birkh¨ auser Verlag Basel/Switzerland

On Mean Ergodic Operators Angela A. Albanese, José Bonet and Werner J. Ricker Abstract. Aspects of the theory of mean ergodic operators and bases in Fréchet spaces were recently developed in [1]. This investigation is extended here to the class of barrelled locally convex spaces. Duality theory, also for operators, plays a prominent role. Mathematics Subject Classification (2000). Primary 46A04, 46A08, 46A35, 47A35; Secondary 46G10. Keywords. Mean ergodic operator, power bounded, barrelled locally convex space, basis, Schauder decomposition.

1. Introduction Certain aspects of the theory of mean ergodic operators in Banach spaces (see, e.g., [14] and the references therein) are related to the theory of bases. This is well documented in [10] (see also the references) where it is shown, amongst other results, that a Banach space with a basis is reflexive if and only if every power bounded operator is mean ergodic. The proof is based on classical results of A.A. Pelczynski and of M. Zippin, connecting bases with reflexivity. In order to extend the results of [10] to the Fréchet space setting, it is necessary to have available the corresponding results of Pelczynski and of Zippin. These were established, and then applied to mean ergodic operators, in the recent article [1]. Since much of modern analysis also occurs in locally convex Hausdorff spaces (briefly, lcHs) which are not metrizable, there is some interest in extending the recent results of [1] beyond the Fréchet space setting. This is our aim here.

The first author gratefully acknowledges the support of the Alexander von Humboldt Foundation. The second author gratefully acknowledges the support of MEC and FEDER project MTM 2007–626443 and GV project Prometeo/2008/101 (Spain) and of the Alexander von Humboldt Foundation.

2

A.A. Albanese, J. Bonet and W.J. Ricker

A continuous linear operator T in a lcHs X (the space of all such operators is denoted by L(X)) is called mean ergodic if the limits n 1 m T x, n→∞ n m=1

P x := lim

x ∈ X,

(1.1)

exist in X. An operator T ∈ L(X) is said to be power bounded if {T m}∞ m=1 is an equicontinuous subset of L(X). Of course, for X a Banach space, this means that supm≥0 T m < ∞. A power bounded operator T is mean ergodic precisely when X = Ker(I − T ) ⊕ Im(I − T ),

(1.2)

where I is the identity operator, Im(I − T) denotes the range of (I − T ) and the bar denotes the “closure in X”. In general, n the right-hand side of (1.2) is the set of all x ∈ X for which the sequence { n1 m=1 T m x}∞ m=1 converges to 0 in X. Let us indicate some of our main results. Technical terms concerning lcHs’ X and certain aspects of L(X) will be defined in later sections. Let us recall at this stage that if ΓX is a system of continuous seminorms determining the topology of X, then the strong operator topology τs in L(X) is determined by the family of seminorms qx (S) := q(Sx),

S ∈ L(X),

for each x ∈ X and q ∈ ΓX (in which case we write Ls (X)). Denote by B(X) the collection of all bounded subsets of X. The topology τb of uniform convergence on bounded sets is defined in L(X) via the seminorms qB (S) := sup q(Sx), x∈B

S ∈ L(X),

for each B ∈ B(X) and q ∈ ΓX (in which case we write Lb (X)). For X a Banach space, τb is the operator norm topology in L(X). If ΓX is countable and X is complete, then X is called a Fréchet space. Given T ∈ L(X), let T[n] :=

n 1 m T , n m=1

n ∈ N,

(1.3)

denote the Cesàro means of T (see also (1.1)). Then T is mean ergodic precisely ∞ when {T[n]}∞ n=1 is a convergent sequence in Ls (X). If {T[n] }n=1 happens to be convergent in Lb (X), then T is called uniformly mean ergodic. The space X itself is called mean ergodic (resp. uniformly mean ergodic) if every power bounded operator on X is mean ergodic (resp. uniformly mean ergodic). The natural setting for mean ergodic operators seems to be the class of barrelled lcHs’. We show in Section 3 that most of the results on mean ergodicity that were established in [1] for operators on Fréchet spaces carry over to barrelled spaces; see also [21, Ch. VIII, §3]. The same is also true of Propositions 2.3 and 2.4 below. Since the strong dual of a distinguished Fréchet space is barrelled, the current results can be combined with those of [1] via duality theory.

On Mean Ergodic Operators

3

An important class of Fréchet spaces consists of the Köthe echelon spaces λp (A), whose mean ergodicity properties were thoroughly investigated in [1]. Just as important is the class of K¨ othe co-echelon spaces kp (V ), for p ∈ {0} ∪ [1, ∞], all barrelled, but, typically not Fréchet spaces except for very special cases. In Section 4 the general results of Sections 2 and 3 are applied to give a complete description of the ergodicity properties of these co-echelon spaces. For instance, if 1 < p < ∞, then the reflexive space kp (V ), necessarily mean ergodic, is uniformly mean ergodic iff it is Montel. The non-reflexive co-echelon spaces k1 (V ) and k∞ (V ) are mean ergodic iff they are uniformly mean ergodic iff they are Montel. Provided it is complete, k0 (V ) is mean ergodic iff it is uniformly mean ergodic iff it is a Schwartz space.

2. Preliminary results Given a lcHs X and T ∈ L(X) we have (I − T )T[n] = T[n] (I − T ) =

1 (T − T n+1 ), n

n ∈ N,

(2.1)

and also, with T[0] := I, that (n − 1) 1 n T = T[n] − T[n−1] , n n If T ∈ L(X) is power bounded, then

n ∈ N.

(2.2)

Im(I − T ) = {x ∈ X : lim T[n] x = 0}

(2.3)

n→∞

and hence, in particular, Im(I − T ) ∩ Ker(I − T ) = {0}, [21, Ch. VIII, §3]. Moreover, such a T clearly satisfies 1 lim T n = 0, in Ls (X). n→∞ n The following fact, without a proof, occurs in [1, Proposition 2.3].

(2.4)

(2.5)

Proposition 2.1. Let X be a barrelled lcHs. If T ∈ L(X) satisfies (2.5) and {T[n] x}∞ n=1 is bounded in X, for each x ∈ X,

(2.6)

then T satisfies both (2.3) and (2.4). Proof. It follows from (2.1) and (2.5) that limn→∞ T[n] w = 0 for w ∈ Im(I − T). Since X is barrelled, condition (2.6) implies that {T[n] }∞ n=1 is an equicontinuous subset of L(X), [13, p. 137]. So, given any q ∈ ΓX there exists p ∈ ΓX such that q(T[n] x) ≤ p(x),

x ∈ X, n ∈ N.

Fix z ∈ Im(I − T ). Given ε > 0 there exists wε ∈ Im(I−T) satisfying p(z−wε ) < ε. Then we have q(T[n] z) ≤ ε + q(T[n] wε ), n ∈ N,

4


which implies that lim supn q(T[n] z) ≤ ε. Since ε > 0 is arbitrary, we can conclude that T[n] z → 0 in X. This establishes one containment in (2.3). Conversely, let x ∈ X satisfy limn→∞ T[n] x = 0. It follows from (1.3) that x − T[n] x = (I − T )

n 1 (I + T + · · · + T m−1 )x ∈ Im(I − T ), n m=1

for each n ∈ N. Combined with T[n] x → 0 in X, it is immediate that x ∈ Im(I − T ). This establishes equality in (2.3). Finally, observe if x ∈ Im(I − T ) ∩ Ker(I − T ), then x = T x and hence, via (1.3), we have x = T[n] x for all n ∈ N. It then follows from (2.3) that x = 0. So, (2.4) is also valid. Given T ∈ L(X), its dual operator T t : X → X , where X is the continuous dual space of X, is defined by T x, x = x, T t x for all x ∈ X, x ∈ X . By Xσ we denote X equipped with its weak topology σ(X, X ). A subset A ⊆ X is called relatively sequentially σ(X, X )-compact if every sequence in A contains a subsequence which is convergent in Xσ . Such sets belong to B(X), [12, §24;(1)], after recalling that every sequentially compact set in any lcHs is also relatively countably compact, [12, p. 310]. The following version of the Mean Ergodic Theorem for Banach spaces occurs in [8, Ch. VIII, 5.1–5.3], [16, p. 214], and for lcHs’ in [1, Theorem 2.4]. Proposition 2.2. Let X be a barrelled lcHs and T ∈ L(X). Then T is mean ergodic if and only if it satisfies (2.5) and {T[n] x}∞ n=1 is relatively sequentially σ(X, X )-compact, ∀x ∈ X.

(2.7)

Setting P := τs –limn→∞ T[n] , the operator P is a projection which commutes with T and satisfies Im(P ) = Ker(I − T ) and Ker(P ) = Im(I − T ). Moreover, X has a direct sum decomposition as given by (1.2). Many lcHs’ X have the property that all relatively σ(X, X )-compact sets are also relatively sequentially σ(X, X )-compact. This includes all Fréchet spaces (actually, all (LF)-spaces), all (DF)-spaces, and many more, [6, Theorem 11, Examples 1.2]. The following fact is an extension of [1, Corollary 2.7]. Proposition 2.3. Let X be a reflexive lcHs in which every relatively σ(X, X )compact set is relatively sequentially σ(X, X )-compact. Then X is mean ergodic. Proof. Let T ∈ L(X) be power bounded. Then clearly (2.5) holds as does (2.6); see [1, Remark 2.6(i)]. By definition, all reflexive spaces are barrelled and all bounded sets in a reflexive lcHs X are relatively σ(X, X )-compact, [12, p. 299]. By the hypotheses on X, all bounded sets are then relatively sequentially σ(X, X )-compact. This, together with (2.6), implies that (2.7) holds. The mean ergodicity of T then follows from Proposition 2.2. A special case of the following fact occurs in [1, Proposition 2.8].


5

Proposition 2.4. Let X be a Montel space in which every relatively σ(X, X )compact set is relatively sequentially σ(X, X )-compact. Then X is uniformly mean ergodic. Proof. Let T ∈ L(X) be power bounded. Since X is reflexive, [12, p. 369], it follows from Proposition 2.3 that T is mean ergodic. The proof can now be completed as in that of Proposition 2.8 of [1]. For Banach spaces the following result is due to M. Lin [15] and for general Fréchet spaces it occurs in [1, Proposition 2.16]. An examination of the proof given in [1] shows that the Fréchet space condition is only used to conclude that the inverse of a certain linear bijection is again continuous. So, we can replace this requirement with the property that every continuous linear surjection is an open map (i.e., the open mapping theorem is valid). Of course, (ii) ⇒ (i) is immediate from the identities n 1 m−1 I − T[n] = (I + T + · · · + T ) (I − T ), n ∈ N. n m=1 So, we have the following result. Proposition 2.5. Let X be a lcHs with the property that every continuous linear surjection from X onto itself is an open map. Let T ∈ L(X) satisfy Ker(I − T ) = {0} and n1 T n → 0 in Lb (X) as n → ∞. Consider the following statements. (i) I − T[n] is surjective for some n ∈ N. (ii) I − T is surjective. (iii) T[n] → 0 in Lb (X) as n → ∞. Then (i) ⇔ (ii) ⇒ (iii). If, in addition, X is a Banach space, then also (iii) ⇒ (i). The class of all lcHs’ which satisfy the hypothesis of Proposition 2.5 includes all ultrabornological spaces which possess a web, [17, Theorem 24.30], and in particular, includes all (LF)-spaces, the space of distributions D , and many more. It is shown in Example 2.17 of [1], that the implication (iii)⇒(i) of Theorem 2.5 fails for Fréchet spaces in general. It might be hoped that (iii)⇒(i) holds at least for (LB)-spaces. We will see in Section 4 that this is not the case. The strong topology in a lcHs X (resp. in X ) is denoted by β(X, X ) (resp. β(X , X)) and we write Xβ (resp. Xβ ); see [12, §21.2] for the definition. The final three results are concerned with duality. The first one occurs in [2, Lemma 2.1]. Lemma 2.6. Let X, Y be lcHs’ with Y quasi-barrelled. Then the linear map Φ : Lb (X, Y ) → Lb (Yβ , Xβ ) defined by Φ(T ) := T t , for T ∈ L(X, Y ), is continuous. In particular, if X is quasi-barrelled and {Tn }∞ n=1 ⊆ L(X) is a sequence which satisfies τb –limn→∞ Tn = T in Lb (X), then also τb –limn→∞ Tnt = T t in Lb (Xβ ). A useful consequence is the following observation.

6


Corollary 2.7. Let X be a lcHs and T ∈ L(X). (i) If T is uniformly mean ergodic, then T t ∈ L(Xβ ) is mean ergodic. (ii) Suppose that X is quasi-barrelled. If T is uniformly mean ergodic, then T t ∈ L(Xβ ) is uniformly mean ergodic. (iii) Suppose that X is sequentially complete and that both X and Xβ are quasibarrelled. If T t ∈ L(Xβ ) is uniformly mean ergodic, then T itself is uniformly mean ergodic. Proof. (i) By assumption there is P ∈ L(X) such that limn→∞ T[n] = P in Lb (X). Fix x ∈ X and B ∈ B(X). Since WB := {S ∈ L(X) : | Sx, x | ≤ 1 for all x ∈ B} is a 0-neighbourhood in Lb (X) there is n(0) ∈ N such that (T[n] − P ) ∈ WB for all n ≥ n(0), that is, t | x, (T[n] − P t )x | = | (T[n] − P )x, x | ≤ 1,

x ∈ B.

t x −P t x ) ∈ B ◦ (the polar of B). Since B ∈ B(X) is arbitrary, we Equivalently, (T[n] t t conclude that limn→∞ T[n] x = P t x in Xβ for each x ∈ X , i.e., limn→∞ T[n] = Pt in Ls (Xβ ). t (ii) Since (T t )[n] = T[n] for all n ∈ N (see (1.3)), it follows from Lemma 2.6 t that T is uniformly mean ergodic in Xβ . t (iii) Let T[n] → Q in Lb (Xβ ). By Lemma 2.6 applied to T t in Xβ we have that tt t (T )[n] → Q in Lb (Xβ ) as n → ∞. Observe that the restriction (T tt )[n] |X = T[n] , for n ∈ N. Interpreting any given element x ∈ X (and then also T[n] x) as an element of Xβ we have that Qt x = limn→∞ T[n] x in Xβ . Since X is quasi-barrelled, Xβ induces the original topology on X, [12, p. 301], that is, {T[n] x}∞ n=1 is Cauchy in X and hence, by sequential completeness, converges in X. It follows that the limit must be Qt x, that is, Qt x ∈ X. Hence, Qt (X) ⊆ X and, since the topology of X is that induced by Xβ , it follows that P := Qt |X belongs to L(X). Moreover, (T tt )[n] → Qt in Lb (Xβ ) implies that T[n] → P in Lb (X) as n → ∞.

Remark 2.8. Every reflexive lcHs X satisfies the hypotheses of (iii) in Corollary 2.7. So does every distinguished Fréchet space (such spaces are not necessarily reflexive). Also, every sequentially complete, quasi-barrelled (DF)-space X has the required properties (as Xβ , being a Fréchet space, is surely quasi-barrelled). Of course, every sequentially complete, quasi-barrelled space is actually barrelled, [12, p. 368]. If X is a quasi-barrelled lcHs, then the general theory of such spaces ensures that T ∈ L(X) is power bounded if and only if T t ∈ L(Xβ ) is power bounded, [13, (6), p. 138]. Combining this with Corollary 2.7(iii) gives the following result; see also Proposition 2.4.


7

Corollary 2.9. Let X be a sequentially complete, barrelled lcHs with Xβ quasibarrelled. Then X is uniformly mean ergodic if and only if Xβ is uniformly mean ergodic. We mention that if T ∈ L(X) is mean ergodic, then so is T t ∈ L(Xσ ), where denotes X equipped with its weak-star topology σ(X , X). Actually, it suffices for T to be mean ergodic in Xσ . As an application of Corollary 2.9 we have some examples.

Xσ

Example. (i) The separable (LB)-spaces Lp+ := ∪r>p Lr ([0, 1]), for 1 < p < ∞, are all reflexive. The corresponding strong duals (Lp+ )β = ∩1≤r 12 for all j ∈ N.

Proof. Adapt the proof of Lemma 4.4 in [1].

Lemma 3.2. Let X be a barrelled lcHs which admits a Schauder decomposition without property (M ). Then there exists a Schauder decomposition {Pj }∞ j=1 ⊆ L((X) of X, a seminorm q ∈ ΓX and a bounded sequence {zj }∞ j=1 ⊂ X with zj ∈ (Pj+1 − Pj )(X) such that q(zj ) > 12 for all j ∈ N.

Proof. The proof of Lemma 4.5 in [1] also applies here.

Remark 3.3. Let {Pj }∞ j=1 be any Schauder decomposition in the complete barrelled lcHs X with ΓX a system of continuous seminorms generating the topology of X. Then {Pj }∞ j=1 is an equicontinuous sequence. Hence, for every p ∈ ΓX there exist q ∈ ΓX and Mp > 0 such that p(Pj x) ≤ Mp q(x),

x ∈ X,

for all j ∈ N. By setting r˜(x) := supj∈N r(Pj x), for every r ∈ ΓX , we obtain p(x) ≤ p˜(x) ≤ Mp q(x) ≤ Mp q˜(x),

x ∈ X.

˜ X := {˜ p : p ∈ ΓX } is also a system of continuous seminorms Accordingly, Γ generating the topology of X and satisfies p˜(Pj x) ≤ p˜(x),

x ∈ X, j ∈ N.

(3.1)


9

The proof of the next result and Theorems 3.6 and 3.8 below follow those given in [1] for the corresponding result in Fréchet spaces. We include the essential parts of these proofs to illustrate certain differences in the current setting and for the sake of self containment. Theorem 3.4. Let X be a complete barrelled lcHs which admits a non-shrinking Schauder decomposition. Then there exists a power bounded operator on X which is not mean ergodic. Proof. Let (Pj )j ⊂ L(X) denote a Schauder decomposition as given by Lemma 3.1 and define projections Qj := Pj − Pj−1 (P0 := 0) and closed subspaces Xj := Qj (X), j ∈ N. By Lemma 3.1 there exist a bounded sequence {zj }∞ j=1 ⊂ X with zj ∈ Xj+1 , and ξ ∈ X such that | zj , ξ| > 12 for all j ∈ N. Set ej := zj / zj , ξ ∈ Xj+1 . Then {ej }∞ j=1 is a bounded sequence of X and ej , ξ = 1 for all j ∈ N. By Remark 3.3 there exists a system ΓX of continuous seminorms generating the topology of X such that p(Pj x) ≤ p(x),

x ∈ X,

(3.2)

for all p ∈ ΓX and j ∈ N. Moreover, since ξ ∈ X , there exists p0 ∈ ΓX such that | x, ξ| ≤ p0 (x) for all x ∈ X. ∞ As in [10, p. 150], take a sequence a = {aj }∞ ⊆ R with j=1 j=1 aj = 1, n aj > 0, and set An := j=1 aj . For x ∈ X and integers m > n ≥ 2 we have m

⎛ Ak Qk x = ⎝

n−1

⎞ aj ⎠

j=1

k=n

m k=n

Qk x

+

m

⎛ aj ⎝

j=n

m

⎞ Qk x⎠ .

k=j

m ∞ ∞ Since k=1 Qk x sums to x in X, we see that { k=1 Ak Qk x}m=1 is a Cauchy sequence and hence, converges in X. Moreover, for each p ∈ ΓX , by (3.2) we have p

m

Ak Qk x

=

⎛ ⎞ m p⎝ aj (Pm − Pj−1 )x⎠ j=1

k=1

≤

m

aj (p(Pm x) + p(Pj−1 x)) ≤ 2p(x),

(3.3)

j=1

for each m ∈ N. Define a linear map Ta : X → X by Ta x :=

∞ k=1

Ak Qk x +

∞ j=2

Pj−1 x, ξaj ej ,

x ∈ X.

(3.4)

10


From (3.3) we obtain, for each p ∈ ΓX with p ≥ p0 , that p(Ta x)

≤

∞ ∞ p( Ak Qk x) + | Pj−1 x, ξ|aj p(ej ) j=2

k=1

≤

2p(x) +

∞

aj p0 (Pj−1 x)p(ej ).

j=2

Note that Mp := supj∈N p(ej ) < ∞, because {ej }∞ j=1 is bounded in X. Moreover, by (3.2) we have p0 (Pj−1 x) ≤ p0 (x) ≤ p(x) for all x ∈ X. Hence, p(Ta x) ≤ (2 + Mp )p(x) for all x ∈ X, where 2 + Mp depends only on p. arbitrary To show that Ta is power bounded, it suffices to show that for ∞ sequences a = {aj }∞ and b = {bj }∞ of positive numbers with j=1 aj = 1 = j=1 j=1 ∞ j=1 bj we have Ta Tb = Tc , with c a sequence of the same type. This is the claim in p. 150 of [10] which is purely algebraic and is proved on p. 151 of [10]. Finally, proceeding as in the final part of the proof of Theorem 1.5 of [1] one shows that Ker(I − Ta ) = {0} and ξ ∈ Ker(I − Tat ), i.e., Ker(I − Tat ) = {0}. Thus, we can apply Theorem 2.12 of [1] to conclude that Ta is not mean ergodic. Recall that a sequence {xn }∞ n=1 in a lcHs X is a basisif, for every x ∈ X, ∞ there is a unique sequence {αn }∞ n=1 ⊆ C such that the series n=1 αn xn converges to x in X. By setting fn (x) := αn we obtain a linear form fn : X → C which is called the nth coefficient functional associated to {xn }∞ n=1 . The functionals fn , ∞ n ∈ N, are uniquely determined by {xn }∞ and {(x , f n n )}n=1 is a biorthogonal n=1 sequence (i.e., xn , fm = δmn for m, n ∈ N). For each n ∈ N, the map Pn : X → X defined by n n Pn : x → fi (x)xi =

x, fi xi , x ∈ X, (3.5) i=1

i=1

is a linear projection with range equal to the finite-dimensional space span(xi )ni=1 . ∞ If, in addition, {fn }∞ n=1 ⊆ X , then the basis {xn }n=1 is called a Schauder basis ∞ for X. In this case, {Pn }n=1 ⊆ L(X) is clearly a Schauder decomposition of X and each dual operator Pnt : x →

n

xi , x fi ,

x ∈ X ,

(3.6)

i=1

for n ∈ N, is a projection with range equal to span(fi )ni=1 . Moreover, for every ∞ x ∈ X the series i=1 xi , x fi converges to f in Xσ . For this reason, {fn }∞ n=1 is also referred to as the dual basis of the Schauder basis {xn }∞ n=1 . The terminology “X has a Schauder basis” will also be abbreviated simply to “X has a basis”. Theorem 3.5. Let X be a complete barrelled lcHs with a Schauder basis and in which every relatively σ(X, X )-compact subset of X is relatively sequentially


11

σ(X, X )-compact. Then X is reflexive if and only if every power bounded operator on X is mean ergodic. Proof. If X is reflexive, then X is mean ergodic by Proposition 2.3. Conversely, if X is not reflexive, then Theorem 1.2 of [1] shows that X admits a non-shrinking Schauder basis. By Theorem 3.4, X is not mean ergodic. Theorem 3.6. Let X be a complete barrelled lcHs which admits a Schauder decomposition without property (M ). Then there exists a power bounded, mean ergodic operator T ∈ L(X) which is not uniformly mean ergodic. Proof. Let {Pj }∞ j=1 ⊆ L(X) denote a Schauder decomposition as given by Lemma 3.2 and define projections Qj := Pj − Pj−1 (P0 := 0) and closed subspaces Xj := Qj (X) for all j ∈ N. By Lemma 3.2 there exist a bounded sequence {zj }∞ j=1 ⊆ X and a continuous seminorm q on X with zj ∈ Xj+1 and q(zj ) > 1/2 for all j ∈ N. Since {Pj }∞ j=1 is an equicontinuous sequence (because X is barrelled), we can apply Remark 3.3 to choose a system ΓX of continuous seminorms generating the topology of X such that p(Pj x) ≤ p(x),

x ∈ X,

(3.7)

for all p ∈ ΓX and j ∈ N. Clearly, there also exists p0 ∈ ΓX such that p0 ≥ q on X. Hence, p0 (zj ) > 1/2 for all j ∈ N. ∞ For any sequence a = {aj }∞ j=1 of positive numbers with j=1 aj = 1 we set n An := j=1 aj and define a linear map Ta : X → X by Ta x :=

∞

Ak Qk x,

x ∈ X.

k=1

As in the proof of Theorem 3.4 one shows that Ta is well defined, satisfies p(Ta x) ≤ 2p(x),

x ∈ X,

(3.8)

for all p ∈ ΓX , and is power bounded. Proceeding as in the proof of Theorem 5.2 of [1], one shows that Ker(I −Ta ) = {0} and Ker(I − Tat ) = {0} and hence, by Theorem 2.12 of [1], Ta is mean ergodic. It remains to show that T := Ta is not uniformly mean ergodic for the choice aj := 2−j . In this case, Ak = 1 − 2−k for all k ∈ N. Moreover, from Qj Qk = 0 whenever j = k and Q2k = Qk it follows that T mx =

∞

Am k Qk x,

x ∈ X,

k=1

for all m ∈ N. Hence, ∞

1 Ak · (1 − Ank )Qk x, T[n] x = n 1 − Ak

x ∈ X, n ∈ N.

k=1

Since T is mean ergodic, there exists P ∈ L(X) with T[n] → P in Ls (X).

(3.9)

12

A.A. Albanese, J. Bonet and W.J. Ricker Next, if x ∈ Xj for a fixed j ∈ N, by (3.9) we have that T[n] x =

1 Aj · (1 − Anj )x, n 1 − Aj

for all n ∈ N, as Qj Qk = 0 whenever j = k and Q2j = Qj . Since 0 < (1 − Anj ) < 1, it follows that 1 Aj p(x) p(T[n] x) ≤ n 1 − Aj for all p ∈ ΓX and n ∈ N. Therefore, p(T[n] x) → 0 as n → ∞ for all p ∈ ΓX . Since T[n] x → P x as n → ∞, we see that P x = 0. That is, P y = 0 for all y ∈ ∪∞ j=1 Xj . Since ∪∞ X is dense in X and P ∈ L(X), we obtain that P = 0 on X, that is, j=1 j T[n] → 0 in Ls (X). Suppose that T is uniformly mean ergodic. Then T[n] → 0 in Lb (X). In particular, since {zj }∞ j=1 is a bounded sequence in X, we have lim sup p(T[n] zj ) = 0

n→∞ j∈N

(3.10)

for all p ∈ ΓX . But, for all j ∈ N, p0 (T[2j ] zj ) >

j 1 [1 − (1 − 2−j )2 ], 4

with limj→∞ (1 − 2−j )2 = e−1 . This is in contradiction with (3.10). j

For Fréchet spaces, the following result occurs in [1, Theorem 1.3]. Theorem 3.7. Let X be a complete barrelled lcHs with a Schauder basis and in which every relatively σ(X, X )-compact subset of X is relatively sequentially σ(X, X )-compact. Then X is Montel if and only if every power bounded operator on X is uniformly mean ergodic, that is, if and only if X is uniformly mean ergodic. Proof. Suppose that X is Montel. Then Proposition 2.4 implies that X is uniformly mean ergodic. Conversely, suppose that X is not Montel. Observe that the Schauder decomposition {Pn }∞ n=1 ⊂ L(X) induced by the basis of X has the property that each space Qn (X) := (Pn − Pn−1 )(X), n ∈ N, is Montel because dim Qn (X) = 1 for all n ∈ N. By [2, Theorem 3.7(iii)], the Schauder decomposition {Pn }∞ n=1 does not satisfy property (M ) and hence, Theorem 3.6 guarantees the existence of a power bounded, mean ergodic operator in L(X) which fails to be uniformly mean ergodic. Theorem 3.8. Let X be a sequentially complete lcHs which contains an isomorphic copy of the Banach space c0 . Then there exists a power bounded operator on X which is not mean ergodic. Proof. Suppose that J is a topological isomorphism from c0 into X. Let {en }∞ n=1 be the canonical basis of c0 . Then the elements yn := Jen form a Schauder basis of Y := J(c0 ).


13

Denote by || ||c0 the norm in c0 and by ΓX a system of continuous seminorms generating the topology of X. Then, for all p ∈ ΓX , there exists Mp > 0 such that p(Jx) ≤ Mp ||x||c0 ,

x ∈ c0 .

There also exist p0 ∈ ΓX and K > 0 such that ||x||c0 ≤ Kp0 (Jx),

x ∈ c0 .

Therefore, we have that ∞ ∞ and p xj yj xj yj ≤ Mp sup |xj | sup |xj | ≤ Kp0 j∈N

j=1

(3.11)

j∈N

j=1

for all x = (xj )∞ j=1 ∈ c0 and p ∈ ΓX . 1 ∞ Let {en }∞ n=1 ⊂ denote the dual basis of {en }n=1 . For each n ∈ N, define −1 ∞ yn ∈ Y by yn := en ◦ J , in which case {yn }n=1 is the dual basis of {yn }∞ n=1 and | y, yn | ≤ Kp0 (y),

y ∈ Y,

as y = Jx for some x ∈ c0 . By the Hahn–Banach theorem, for each n ∈ N we can find fn ∈ X such that fn |Y = yn and Define xn :=

n i=1

| x, fn | ≤ Kp0 (x),

x ∈ X.

(3.12)

yi and gn := fn − fn+1 , for each n ∈ N, and observe that

xk , gn = xk , fn − xk , fn+1 = δkn

for all k, n ∈ N. We can then define projections Pn : X → X via n

x, gk xk , x ∈ X, Pn x := k=1

so that Pn (X) = span {xj }nj=1 = span {yj }nj=1 and Pn Pm = Pmin{n,m} . Set h := f1 and observe that n yj , f1 = 1, n ∈ N.

xn , h = j=1

On the other hand, xn ∈ (Pn − Pn−1 )(X) (with P0 :=0) for all n ∈ N, and n {xn }∞ is a bounded sequence in X because xn = J( j=1 ej ), n ∈ N. Since nn=1 j=1 ej c0 = 1, we have p(xn ) ≤ Mp ,

n ∈ N,

(3.13)

for all p ∈ ΓX . In particular, n 1 1 ej c0 = , p0 (xn ) ≥ K j=1 K

Moreover, the identities n Pn x = ( x, fk − x, fn+1 )yk , k=1

n ∈ N.

x ∈ X, n ∈ N,

14


together with (3.11) and (3.12) imply that p(Pn x) ≤ Mp sup | x, fk − x, fn+1 | ≤ 2Mp Kp0 (x)

(3.14)

1≤k≤n

for all p ∈ ΓX , n ∈ N and x ∈ X. Accordingly, {Pn }∞ equicontinuous. n=1 ⊂ L(X) is ∞ ∞ Let a = {aj }j=1 be any sequence of positive numbers with j=1 aj = 1 and n set An := j=1 aj for n ∈ N. As in the statement of Theorem 3 of [10], we define Sa x := x −

∞

an Pn−1 x +

n=2

∞

Pn−1 x, hxn ,

x ∈ X.

n=2

Then by (3.13), (3.12) and (3.11) we have, for each x ∈ X, that p(Sa x)

≤ p(x) + 2Mp Kp0 (x) + Mp sup | Pn−1 x, h| n≥2

≤ p(x) + 2Mp Kp0 (x) + Mp Kp0 (Pn−1 x) ≤ p(x) + 2Mp Kp0 (x) + Mp K 2 Mp0 p0 (x) = (1 + 2Mp K + Mp K 2 Mp0 )p(x) for all p ∈ ΓX with p ≥ p0 . So, Sa ∈ L(X). The fact that Sa is power bounded follows from the Claim on p. 156 of [10], stating that Sa Sb = Sc for an appropriate c. It remains to show that Sa is not mean ergodic. For this, we can now proceed exactly as in the final part of the proof of Theorem 1.6 of [1].

4. Mean ergodicity of co-echelon spaces We wish to give an application of the previous results to K¨ othe co-echelon spaces. Let I be a countable index set. A K¨ othe matrix A = (an )∞ n=1 is an increasing sequence of strictly positive functions on I. Let V = (vn )∞ n=1 denote the associated decreasing sequence of functions vn := 1/an , n ∈ N. Define the inductive limits kp (V ) = kp (I, V ) = ind p (vn ), 1 ≤ p ≤ ∞, and k0 (V ) = k0 (I, V ) = ind c0 (vn ) , n

n

generated by the (weighted) Banach spaces

1/p I p p (vn ) = {x = (xi )i∈I ∈ C : qp,n (x) = (vn (i)|xi |) < ∞}, if 1 ≤ p < ∞ , i∈I

and ∞ (vn ) = {x = (xi )i∈I ∈ CI : q∞,n (x) = sup vn (i)|xi | < ∞} , i∈I

c0 (vn ) = {x = (xi )i∈I ∈ C : (vn (i)|xi |)i∈I converges uniformly to 0 in I} . I

That is, kp (V ) is the increasing union of the Banach spaces p (vn ), respectively c0 (vn ), for n ∈ N, endowed with the strongest locally convex topology under which


15

the inclusion of each of these Banach spaces is continuous, i.e., kp (V ) is an (LB)space and so a barrelled, ultrabornological (DF)-space. The spaces kp (V ) are called co-echelon spaces of order p. Given a decreasing sequence V = (vn )∞ n=1 of strictly positive functions on I, set v¯(i) I ¯ 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorem 4.7]. Then, in the sectional subspace E0 of k1 (V ) defined by E0 := {x ∈ k1 (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of 1 (vm ) coincides with that of 1 (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have q1,m (x) ≤ q1,n (x) = vn (i)|xi | ≤ c−1 vm (i)|xi | = c−1 m m q1,m (x) . i∈I0

i∈I0

Consequently, the topology of k1 (V ) also coincides with that of 1 (vn ) in E0 . Hence, k1 (V ) contains an isomorphic copy of 1 , which is a contradiction. Proposition 4.3. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I. Then the following assertions are equivalent. (i) k∞ (V ) is mean ergodic. (ii) k∞ (V ) is uniformly mean ergodic. (iii) k∞ (V ) is a Montel space (hence, a (DFM)-space). (iv) k∞ (V ) does not contain an isomorphic copy of ∞ . (v) K0 (V¯ ) = K∞ (V¯ ) = k∞ (V ) algebraically and topologically. Proof. By the discussion just prior to Proposition 2.3, together with Proposition 2.4, it is clear that (iii) ⇒ (ii). That (ii) ⇒ (i) is obvious. Since ∞ is an infinitedimensional Banach space (i.e., its closed unit ball is not compact), it is clear that (iii) ⇒ (iv). Moreover, (iii) ⇔ (v) by [3, Theorem 4.7]. (iv) ⇒ (iii): Suppose that k∞ (V ) is not a Montel space. Then there exist an infinite set I0 ⊂ I and n ∈ N such that vm (i) = cm > 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorem 4.7]. Then, in the sectional subspace E0 of k∞ (V ) defined by E0 := {x ∈ k∞ (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of ∞ (vm ) coincides with that of ∞ (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have −1 q∞,m (x) ≤ q∞,n (x) = sup vn (i)|xi | ≤ c−1 m sup vm (i)|xi | = cm q∞,m (x) . i∈I0

i∈I0

Consequently, the topology of k∞ (V ) also coincides with that of ∞ (vn ) in E0 . Hence, k∞ (V ) contains an isomorphic copy of ∞ . This is a contradiction. (i) ⇔ (iv): Suppose that k∞ (V ) contains an isomorphic copy of ∞ . This implies that k∞ (V ) is not mean ergodic by [1, Remark 2.14(i)].


17

Proposition 4.4. Let V = (vn )∞ n=1 be a decreasing sequence of strictly positive functions on I. Suppose that the (LB)-space k0 (V ) is complete. Then the following assertions are equivalent. (i) k0 (V ) is mean ergodic. (ii) k0 (V ) is uniformly mean ergodic. (iii) k0 (V ) is a Schwartz space (hence, a (DFS)-space). (iv) k0 (V ) does not contain an isomorphic copy of c0 . (v) k0 (V ) = k∞ (V ) algebraically and topologically. Proof. The complete, barrelled (LB)-space k0 (V ) admits a Schauder basis and every relatively σ(k0 (V ), (k0 (V )) )-compact subset of k0 (V ) is relatively sequentially σ(k0 (V ), (k0 (V )) )-compact, [6, Theorem 11, Examples 1, 2]. So, by Theorem 3.7 above and [3, Theorem 4.9] we have (ii) ⇔ k0 (V ) is a Montel space ⇔ (iii) ⇔ (v). Next, (ii) ⇒ (i) is obvious. Also, (iii) ⇒ (iv) is obvious, because a Schwartz space cannot contain an isomorphic copy of any infinite-dimensional Banach space. (iv) ⇒ (iii): Suppose that k0 (V ) is not a Schwartz space. Since k0 (V ) is complete, there exist an infinite set I0 ⊂ I and n ∈ N such that vm (i) = cm > 0 , ∀m ≥ n, inf i∈I0 vn (i) [3, Theorems 3.7, 4.9]. Then, in the sectional subspace E0 of k0 (V ) defined by E0 := {x ∈ k0 (V ) : xj = 0 for all j ∈ I \ I0 }, the topology of c0 (vm ) coincides with that of c0 (vn ) for all m ≥ n. Indeed, for every x ∈ E0 and m ≥ n we have −1 q∞,m (x) ≤ q∞,n (x) = sup vn (i)|xi | ≤ c−1 m sup vm (i)|xi | = cm q∞,m (x) . i∈I0

i∈I0

Consequently, the topology of k0 (V ) also coincides with that of c0 (vn ) in E0 . Hence, k0 (V ) contains an isomorphic copy of c0 . This is a contradiction. (i) ⇒ (ii): By Theorem 3.5 we have (i) ⇔ k0 (V ) is reflexive. Since k0 (V ) is complete, it is then also Montel by [3, Theorems 3.7, 4.7], thereby implying that (ii) holds via Theorem 3.7. Example. Every (LF)-space X (hence, every (LB)-space) satisfies the hypothesis of Proposition 2.5, because every linear continuous surjective map between two (LF)-spaces is necessarily open. But, in this setting, condition (iii) of Proposition 2.5 does not imply the condition (ii). Hence, also (iii) does not imply condition (i) as the following example illustrates. Let (an )∞ n=1 be a sequence of real numbers satisfying 1 < an+1 < an < a for ∞ some a ∈ R and for all n ∈ N. For each n ∈ N set vn := (ain )∞ i=1 and V := (vn )n=1 , where i ∈ I := N. Consider the co-echelon space k1 (V ) which is a Montel space (i) (hence, a (DFM)-space) because, for all n, m ∈ N with m > n, we have vvm = n (i) i am → 0 as i → ∞, [3, Theorem 4.7]. In particular, its (strong) topological an

18


dual is the K¨ othe echelon Fréchet space λ∞ (A) = λ0 (A), with A := (vn−1 )∞ n=1 , [3, Theorem 4.7]. Define T ∈ L(k1 (V )) by T x := ((1 − a−i )xi )∞ i=1 ,

x ∈ k1 (V ) .

It is easy to verify that Ker(I − T ) = {0} and that y := (a−i )∞ i=1 ∈ k1 (V ) does not belong to Im(I − T ), i.e., I − T is not surjective. So, condition (ii) of Theorem 2.5 does not hold. Since T m x = ((1−a−i )m xi )∞ i=1 for x ∈ k1 (V ) and for all m ∈ N, the sequence is bounded for all x ∈ k1 (V ). Indeed, given any x ∈ k1 (V ) there is {T m x}∞ m=1 n ∈ N such that x ∈ 1 (vn ), thereby implying that q1,n (T m x) = |(1 − a−i )m | · |xi |vn (i) ≤ |xi |vn (i) = q1,n (x) i∈N

i∈N

for all m ∈ N. So, the barrelledness of k1 (V ) implies that the sequence {T m}∞ m=1 ⊂ L(k1 (V )) is equicontinuous, i.e., for every p ∈ Γk1 (V ) the exists q ∈ Γk1 (V ) for which p(T m x) ≤ q(x) for all m ∈ N and x ∈ k1 (V ). In particular, T is power bounded. Since k1 (V ) is a complete (LB)-space and hence, a regular (LB)-space, given any bounded set B ⊂ k1 (V ) there exist k, n ∈ N such that B ⊂ kBn (Bn denotes the unit ball of 1 (vn )). On the other hand, since the inclusion map 1 (vn ) → k1 (V ) is continuous, given any p ∈ Γk1 (V ) there exists c > 0 such that p(x) ≤ cq1,n (x),

x ∈ 1 (vn ).

Therefore, for every m ∈ N we have 1 m 1 1 1 T x ≤c sup q1,n (T m x) ≤ c sup q1,n (x) ≤ ck sup p m m x∈kBn m x∈kBn m x∈B 1 m 1 m T x) → 0 as m → ∞. This shows that m T → 0 in and hence, supx∈B p( m Lb (k1 (V )). It remains to establish condition (iii) of Proposition 2.5. For this, we observe that ξ ∈ λ∞ (A) , T t ξ = ((1 − a−i )ξi )∞ i=1 ,

so that Ker(I − T t ) = {0}. Since T is power bounded and both Ker(I − T ) = {0} and Ker(I − T t) = {0}, we can apply [1, Theorem 2.12] to conclude that T is mean ergodic. But, k1 (V ) is a Montel space whose relatively σ(k1 (V ), (k1 (V )) )-compact subsets are relatively sequentially σ(k1 (V ), (k1 (V )) )-compact (see the discussion prior to Proposition 2.3). So, by Proposition 2.4, T is also uniformly mean ergodic. Hence, there is P ∈ L(k1 (V )) such that T[n] → P in Lb (k1 (V )). For each r ∈ N, let er be the element of k1 (V ) with 1 in the rth coordinate and 0’s elsewhere (we point out that {er }∞ r=1 is a Schauder basis for k1 (V )). Then, for all r ∈ N, T m er = (1 − a−r )m er → 0 as m → ∞ ,

On Mean Ergodic Operators so that T[n] er =

μ(1 − μ) er → 0 n(1 − μ)

as

19

m → ∞,

with μ := (1−a−r ). This implies that P = 0 and hence, that T[n] → 0 in Lb (k1 (V )), i.e., condition (iii) is satisfied. ∞ We remark that the operator T x := ((1 − 2−i )xi )∞ i=1 , for x = (xi )i=1 ∈ s (here s is the strong dual of the Fréchet space s of all rapidly decreasing sequences, so that s is an (LB)-space), also satisfies condition (iii) of Proposition 2.5, but, fails condition (ii); the proof is similar to the previous one for T in k1 (V ).

References [1] A.A. Albanese, J. Bonet, W.J. Ricker, Mean ergodic operators in Fr´ echet spaces. Ann. Acad. Sci. Fenn. Math. 34 (2009), 1–37. [2] A.A. Albanese, J. Bonet, W.J. Ricker, Grothendieck spaces with the Dunford–Pettis property. Positivity, in press. [3] K.D. Bierstedt, R.G. Meise, W.H. Summers, K¨ othe sets and K¨ othe sequence spaces. In: “Functional Analysis, Holomorphy and Approximation Theory”, J.A. Barroso (Ed.), North-Holland, Amsterdam, 1982, pp. 27–91. [4] J. Bonet, W.J. Ricker, Schauder decompositions and the Grothendieck and Dunford– Pettis properties in K¨ othe echelon spaces of infinite order. Positivity 11 (2007), 77–93. [5] J.M.F. Castillo, J.C. D´ıaz, J. Motos, On the Fréchet space Lp− . Manuscripta Math. 96 (1998), 219–230. [6] B. Cascales, J. Orihuela, On compactness in locally convex spaces. Math. Z. 195 (1987), 365–381. [7] J.C. D´ıaz, M.A. Mi˜ narro, Distinguished Fr´ echet spaces and projective tensor product. Doˇ ga-Tr. J. Math. 14 (1990), 191–208. [8] N. Dunford, J.T. Schwartz, Linear Operators I: General Theory. 2nd Edition, Wiley– Interscience, New York, 1964. [9] R.E. Edwards, Functional Analysis. Reinhart and Winston, New York, 1965. [10] V.P. Fonf, M. Lin, P. Wojtaszczyk, Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187 (2001), 146–162. [11] N.J. Kalton, Schauder decompositions in locally convex spaces. Proc. Camb. Phil. Soc. 68 (1970), 377–392. [12] G. K¨ othe, Topological Vector Spaces I. 2nd Rev. Edition, Springer Verlag, Berlin– Heidelberg–New York, 1983. [13] G. K¨ othe, Topological Vector Spaces II. Springer Verlag, Berlin–Heidelberg–New York, 1979. [14] U. Krengel, Ergodic Theorems. Walter de Gruyter, Berlin, 1985. [15] M. Lin, On the uniform ergodic theorem. Proc. Amer. Math. Soc. 43 (1974), 337–340. [16] H.P. Lotz, Uniform convergence of operators on L∞ and similar spaces. Math. Z. 190 (1985), 207–220.

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[17] R.G. Meise, D. Vogt, Introduction to Functional Analysis. Clarendon Press, Oxford, 1997. [18] G. Metafune, V.B. Moscatelli, On the space p+ = q>p q . Math. Nachr. 147 (1990), 7–12. [19] S. Okada, Spectrum of scalar–type spectral operators and Schauder decompositions. Math. Nachr. 139 (1988), 167–174. [20] P. Pérez Carreras, J. Bonet, Barrelled Locally Convex Spaces. North Holland Math. Studies 131, Amsterdam, 1987. [21] K. Yosida, Functional Analysis. Springer Verlag, Berlin–Heidelberg, 1965. Angela A. Albanese Dipartimento di Matematica “E. De Giorgi” Universit` a del Salento Via Provinciale per Arnesano P.O. Box 193 I-73100 Lecce, Italy e-mail: [email protected] José Bonet Instituto Universitaro de Matem´ atica Pura y Aplicada IUMPA Edificio ID15 (8E), Cubo F, Cuarta Planta Universidad Politécnica de Valencia E-46071 Valencia, Spain e-mail: [email protected] Werner J. Ricker Math.-Geogr. Fakult¨ at Katholische Universit¨ at Eichst¨ att–Ingolstadt D-85072 Eichst¨ att, Germany e-mail: [email protected]


Fourier Series in Banach spaces and Maximal Regularity Wolfgang Arendt and Shangquan Bu Abstract. We consider Fourier series of functions in Lp (0, 2π; X) where X is a Banach space. In particular, we show that the Fourier series of each function in Lp (0, 2π; X) converges unconditionally if and only if p = 2 and X is a Hilbert space. For operator-valued multipliers we present the Marcinkiewicz theorem and give applications to differential equations. In particular, we characterize maximal regularity (in a slightly different version than the usual one) by Rsectoriality. Applications to non-autonomous problems are indicated. Mathematics Subject Classification (2000). Primary 42B15; Secondary 34G10. Keywords. Operator-valued multiplier, maximal regularity, non-autonomous problems.

0. Introduction The study of vector-valued Fourier transforms is on one hand motivated by the structure theory of Banach spaces where the validity of certain classical properties reflects geometric properties of the Banach spaces; on the other hand, it has fundamental applications in PDE. Of particular importance is the subject of operator-valued Fourier multipliers which have immediate applications to properties of maximal regularity for evolution equations. The aim of this article is to introduce into this subject and to show how it can be applied. The approach we use here is based on Fourier series. Given a Banach space X, the Fourier series of each f ∈ Lp (0, 2π; X) converges in Lp (0, 2π; X) if and only if X is a UMD-space and 1 < p < ∞. We show here that the Fourier series converges unconditionally in Lp (0, 2π; X) if and only if p = 2 and X is a Hilbert space (Theorem 1.5). Even though this result is known to specialists of unconditional The second author is supported by the NSF of China (Grant no. 10731020), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 200800030059) and the Alexander von Humboldt Foundation.

22

W. Arendt and S. Bu

structures in Banach spaces, it seems not be contained in the literature. The result can be reformulated by saying that each bounded sequence of scalar operators (λk I)k∈Z is a multiplier for Lp (0, 2π; X) if and only if p = 2 and X is a Hilbert space. The phenomenon that operator-valued versions of certain classical multiplier theorems are only valid in Hilbert spaces was first observed by Pisier (unpublished) as a consequence of Kwapien’s deep characterization of Hilbert spaces. In recent years the subject saw a spectacular revival. A break-through was the operator-valued Michlin multiplier theorem proved by Weis [W01] in 2001. Here we will concentrate on periodic multipliers, i.e., operator-valued versions of the Marcinkiewicz multiplier Theorem. If 1 < p < ∞ and X is a Hilbert space, we present the Marcinkiewicz multiplier Theorem for operator-valued sequences (Mk )k∈Z ∈ L(X) (Theorem 1.7 and Corollary 1.8). This result has a version which also holds in UMD-spaces but the notion of R-boundedness is needed. This is the content of Section 2. Then we show how the operator-valued Marcinkiewicz multiplier Theorem (Theorem 2.3) can be applied to vector-valued differential equations. Most natural is the periodic case (Section 3), but our emphasis is on the classical maximal regularity problem: u(t) ˙ = Au(t) + f (t), t ∈ (0, τ ) a.e. P0 (τ, p) u(0) = 0 . Here we consider an operator A ∈ L(D, X), where D is a Banach space which is continuously embedded in X. If for 1 ≤ p < ∞, P0 (τ, p) is well posed (i.e., for all f ∈ Lp (0, τ ; X) there is a unique solution u ∈ W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) of P0 (τ, p)), then we show that the operator A is closed and R-sectorial without any assumptions on the spaces. This defers somehow from the usual setting since a priori, we do not consider A as an unbounded operator on X. To do so is motivated by the recent interesting applications of maximal regularity to the non-autonomous problem which we explain briefly in Section 5. In Section 4, we also show that conversely, problem P0 (τ, p) is well posed whenever 1 < p < ∞, X is a UMD-space and A is R-sectorial. In the present paper, we explain and complement results of [AB02] and the approach to maximal regularity via periodic multipliers chosen there. There is an alternative way based on the Michlin multiplier theorem, and we refer to [W01] and [KW04] for further information.

1. Vector-valued Fourier series and operator-valued Fourier multipliers

Let X be a Banach space and let Lp2π (X) := f : R → X measurable, f (t + 2π) = 2π f (t) a.e. and 0 f (t)p dt < ∞ the space of all X-valued 2π-periodic locally pintegrable functions on R, 1 ≤ p < ∞. Then Lp2π (X) is a Banach space for the

Fourier Series in Banach Spaces and Maximal Regularity norm

⎛ 1 f p := ⎝ 2π

2π

23

⎞1/p f (t)p dt⎠

.

0

If f ∈ Lp2π (X), then we denote by 1 fˆ(k) := 2π

2π

f (t)e−ikt dt

0

the kth Fourier coefficient of f , where k ∈ Z. The Fourier series of f +∞ ek ⊗ fˆ(k) k=−∞

converges in the sense of Cesàro to f in Lp2π (X); i.e., σn − f p → 0 (n → ∞), m 1 n ikt ˆ where σn = n+1 (t ∈ R) and for m=0 k=−m ek ⊗ f (k). Here we let ek (t) = e ikt x ∈ X we define ek ⊗ x by (ek ⊗ x)(t) = e x (t ∈ R), where k ∈ Z. In particular, the space of all X-valued trigonometric polynomials n T (X) := ek ⊗ xk : n ∈ N, x−n , . . . , xn ∈ X k=−n

is dense in Lp2π (X) for all 1 ≤ p < ∞. Moreover, the Uniqueness Theorem holds: If f ∈ Lp2π (X) is such that fˆ(k) = 0 for all k ∈ Z, then f = 0 a.e. This allows us to define operator-valued multipliers in the following way. If Y is another Banach space, we denote by L(X, Y ) the space of all bounded linear operators from X to Y . When X = Y , we will simply denote it by L(X). Definition 1.1. Let X, Y be Banach spaces and let 1 ≤ p < ∞. A sequence (Mk )k∈Z ⊂ L(X, Y ) is an Lp -multiplier if for each f ∈ Lp2π (X), there exists g ∈ Lp2π (Y ) such that gˆ(k) = Mk fˆ(k) for all k ∈ Z. Now the Uniqueness Theorem guarantees that g is unique. It follows that the mapping f → g is linear. Thus, by the Closed Graph Theorem there exists a unique linear operator M ∈ L(Lp2π (X), Lp2π (Y )) such that (M f )∧ (k) = Mk fˆ(k) (k ∈ Z) (1.1) for all f ∈ Lp2π (X). The operators obtained in this way are exactly the translation invariant operators. To make this precise we consider the C0 -group U on Lp2π (X) given by (U(τ )f )(t) = f (t + τ ) (t ∈ R). The same group is also considered on Lp2π (Y ) without changing the name. Then an operator T ∈ L Lp2π (X), Lp2π (Y ) is called translation invariant, if U(t)T = T U(t) for all t ∈ R.

24

W. Arendt and S. Bu

Proposition 1.2. Let T ∈ L Lp2π (X), Lp2π (Y ) , where X, Y are Banach spaces and 1 ≤ p < ∞. The following assertions are equivalent (i) T is translation invariant; (ii) there exist Mk ∈ L(X, Y ), such that (T f )∧ (k) = Mk fˆ(k), (k ∈ Z) for all f ∈ Lp2π (X). The following lemma is needed for the proof of Proposition 1.2. Lemma 1.3. Let g ∈ Lp2π (Y ), k ∈ Z be such that g(t + τ ) = eikτ g(t)

t-a.e.

for all τ ∈ R. Then there exists a unique y ∈ Y such that g = ek ⊗ y. 1,p (Y ) := u ∈ C(R; Y ) : Proof. Let B be the generator of U. Then D(B) = W2π u(t + 2π) = u(t) for all t ∈ R, and u ∈ Lp2π (Y ) and Bu = u for all u ∈ D(B). Here u is understood in the sense of distributions. Now the assumption on g says that U(τ )g = eikτ g for all τ ∈ R. Thus g ∈ D(B) and g = ikg. It follows that g is continuous, hence g ∈ C 1 (R; Y ). Consequently, g(t) = eikt g(0) for all t ∈ R. Proof of Proposition 1.2. (i)⇒(ii). Assume that T is translation invariant. Let k ∈ Z. For x ∈ X consider g := T (ek ⊗ x). Then U(τ )g = T U(τ )(ek ⊗ x) = eikτ T (ek ⊗ x) = eikτ g for all τ ∈ R. By Lemma 1.3 there exists a unique y ∈ Y such that T (ek ⊗ x) = ek ⊗ y.

(1.2)

We let Mk x := y. Then Mk : X → Y is linear and continuous. Moreover, by (1.2) one has (T f )∧ (k) = Mk fˆ(k) (1.3) for all f ∈ T (X). Since T (X) is dense in Lp2π (X) and T is continuous from Lp2π (X) to Lp2π (Y ), the identity (1.3) remains true for all f ∈ Lp2π (X). (ii)⇒(i). Let τ ∈ R and f ∈ Lp2π (X). Then for k ∈ Z one has ∧ U(τ )T f (k) = eikτ (T f )∧ (k) = eikτ Mk fˆ(k) ∧ = Mk eikτ fˆ(k) = Mk U(τ )f (k) ∧ = T U(τ )f (k). The Uniqueness Theorem implies that U(τ )T f = T U(τ )f .

Next we want to describe criteria which insure that a given sequence (Mk )k∈Z in L(X, Y ) is an Lp -multiplier. We say that a Banach space X is a Hilbert space, if there exists a scalar product · , · on X such that x, x1/2 defines an equivalent norm on X. If p = 2, then even in the scalar case there are bounded sequences which are not Lp -multipliers. If both X, Y are Hilbert spaces, then each bounded

Fourier Series in Banach Spaces and Maximal Regularity

25

sequence (Mk )k∈Z is an L2 -multiplier. The following result shows that the converse remains true. Theorem 1.4. Let X, Y be Banach spaces. Assume that each bounded sequence (Mk )k∈Z ⊂ L(X, Y ) is an L2 -multiplier. Then both spaces X, Y are Hilbert spaces. Proof. It follows from the assumption that there exists C > 0 such that for every finite sequence (xk )k∈Z in X and Mk ∈ L(X, Y ) satisfying Mk ≤ 1, we have ek ⊗ Mk xk ≤ C ek ⊗ xk . (1.4) 2 2 L (0,2π;Y )

k∈Z

k∈Z

L (0,2π;X)

Let (xk )k∈Z be a finite sequence in X. There exist fk ∈ X such that fk (xk ) = xk and fk = 1. Let u ∈ X, u = 1 be fixed. Consider the linear operators Mk ∈ L(X, Y ) given by Mk (x) := fk (x)u (x ∈ X). Then Mk L(X,Y ) = 1. Moreover, Mk xk = fk (xk )u = xk u. It follows from (1.4) that 1/2 xk 2 ≤ C ek ⊗ xk . k∈Z

L2 (0,2π;X)

k∈Z

Thus X is of Fourier type 2. Hence X is a Hilbert space [Pie07, page 317]. Now we are going to show that Y is also a Hilbert space. For this we let (yk )k∈Z be a finite sequence in Y and let u ∈ X and f ∈ X be such that u = f = f (u) = 1. Consider the linear operators Nk ∈ L(X, Y ) given by Nk (x) := f (x)yk /yk (x ∈ X) when yk = 0, and Nk = 0 when yk = 0. Then Nk L(X,Y ) ≤ 1. It follows from (1.4) that ek ⊗ Nk xk ≤ C ek ⊗ xk 2 2 L (0,2π;Y )

k∈Z

k∈Z

L (0,2π;X)

for xk ∈ X. Taking xk = yk u. Then Nk xk = yk . It follows that 1/2 ek ⊗ yk ≤C yk 2 . 2 k∈Z

L (0,2π;Y )

k∈Z

We have shown that Y is of Fourier cotype 2. Thus Y is of Fourier type 2 and hence Y is a Hilbert space [Pie07, p. 316]. When X = Y , the preceding result can be improved. Instead of considering all bounded linear operator sequences, we only need to consider (λk I)k∈Z where (λk )k∈Z is a bounded scalar sequence. For the proof we need to introduce Rademacher functions. by rk the kth Rademacher function on [0, 1] We denote given by rk (t) = sgn sin(2k πt) , k = 1, 2, 3, . . . . Here we recall two fundamental properties of Rademacher functions which will be used in the proof of the next result. Let π : N → N be a bijection and let (xk )k≥1 be a finite sequence in X. Then it follows easily from the definition hat rk xk 2 = rπ(k) xk 2 . k

L (0,1;X)

k

L (0,1;X)

26

W. Arendt and S. Bu

Kahane’s contraction principle states that for every finite sequence (xk )k≥1 in X and λk ∈ C with |λk | ≤ 1, λk rk xk 2 ≤ 2 rk xk 2 . L (0,1;X)

k

L (0,1;X)

k

(see, e.g., [LT79]). Let (Ω1 , Σ1 , μ1 ) and (Ω2 , Σ2 , μ2 ) be two measure spaces. We consider subsets J1 ⊂ L2 (Ω1 ), J2 ⊂ L2 (Ω2 ). Then by fk xk 2 gk xk 2 k

L (Ω1 ;X)

L (Ω2 ;X)

k

we mean that there exists a constant C > 0 depending only on X such that for every finite number of xk ∈ X, and every sequence (fk ) ⊂ J1 and (gk ) ⊂ J2 (satisfying possibly some further restriction to be made precise), one has 1 fk xk 2 ≤ gk xk 2 ≤ C fk xk 2 . C L (Ω1 ;X) L (Ω2 ;X) L (Ω1 ;X) k

k

k

Theorem 1.5. Let X be a Banach space. Assume that each bounded sequence (λk I)k∈Z defines an L2 -multiplier. Then X is a Hilbert space. Proof. We claim that for every finite number of trigonometric polynomials fk ∈ L2 (0, 2π) and every finite number of xk ∈ X, we have rk fk xk rmn,k fˆk (n)xk , (1.5) 2 2 L ([0,1]×[0,2π];X)

k

L ([0,1];X)

n,k

where mn1 ,k1 = mn2 ,k2 when (n1 , k1 ) = (n2 , k2 ). Indeed, it follows from the Kahane’s contraction principle that for every finite number of xk ∈ X, rk ek ⊗ xk rk xk . (1.6) 2 2 k

L ([0,1]×[0,2π];X)

k

L ([0,1];X)

On the other hand, since every bounded scalar sequence defines an L2 -multiplier by assumption, it follows that rk ek ⊗ xk 2 ek ⊗ xk 2 . (1.7) k

L ([0,1]×[0,2π];X)

k

L ([0,2π];X)

If f ∈ L2 (0, 2π), we let Δ(f ) := {n ∈ Z : fˆ(n) = 0} be the Fourier spectrum of f . Then there exist N1 , N2 , . . . ∈ N sufficiently large, so that {0} < Δ(eNk fk ) < Δ(eNl fl ) whenever k < l. Here by M1 < M2 we mean that k < l for every k ∈ M1 and l ∈ M2 , where M1 and M2 are subsets of N.


27

It follows from Kahane’s contraction principle, (1.6) and (1.7) that rk fk xk 2 L ([0,1]×[0,2π];X)

k

rk eNk en ⊗ fˆk (n) xk

L2 ([0,1]×[0,2π];X)

n

k

= rk eNk +n ⊗ fˆk (n)xk

L2 ([0,1]×[0,2π];X)

k,n

rmn,k rk eNk +n ⊗ fˆk (n)xk

L2 ([0,1]×[0,1]×[0,2π];X)

k,n

rmn,k eNk +n ⊗ fˆk (n)xk

L2 ([0,1]×[0,2π];X)

k,n

rmn,k fˆk (n)xk

L2 ([0,1];X)

k,n

.

We have shown that our claim (1.5) is true. Now under the assumption of the theorem, X is a UMD-space as the Hilbert transform is bounded on L2 (0, 2π; X). It follows that X has a non trivial cotype q < ∞. This implies that for every finite number of xk ∈ X, one has rk xk 2 gk xk 2 (1.8) L ([0,1];X)

k

L (Ω;X)

k

where the gk ’s are independent complex standard Gaussian variables on some probability space (Ω, Σ, p) [LeTa91, p. 253]. It follows from this and (1.5) that if fk ∈ L2 (0, 2π) are trigonometric polynomials, then gk fk xk 2 gn,k fˆk (n)xk 2 (1.9) L ([0,2π]×Ω;X)

k

L (Ω ;X)

n,k

where the gn,k ’s are independent complex standard Gaussian variables on some probability space (Ω , Σ , p ). We remark that (1.9) remains true for arbitrary fk ∈ L2 (0, 2π) by an approximation argument. Now assume that fk ∈ L2 (0, 2π) satisfy fk L2 (0,2π) = 1 and let hk := n gn,k fˆk (n). Then the hk ’s are also independent 2 1/2 ˆ = fk 2 = complex standard Gaussian variables on (Ω , Σ , p ) as n |fk (n)| 1. Thus we have by (1.8) gn,k fˆk (n)xk 2 = hk xk 2 rk xk 2 . n,k

L (Ω ;X)

k

On the other hand, if we let fk := and gk fk xk 2 k

L (Ω ;X)

k

L ([0,1];X)

√ 2N πχ[ k−1 , k ] for 1 ≤ k ≤ N . Then fk 2 = 1 N

L ([0,2π]×Ω;X)

=

N

1/2 xk 2 . k

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W. Arendt and S. Bu

It follows from (1.9) that

xk 2

1/2

rk xk

k

L2 ([0,1];X)

k

.

It follows that X is of cotype 2 and type 2. Thus X is a Hilbert space by a result of S. Kwapien [Kw72]. Recall that a series

+∞

xk in a Banach space X is called unconditionally

k=−∞

convergent if

n

lim

n→∞

xπ(k)

k=−n

exists for each bijection π : Z → Z. This is equivalent to the fact that n

lim

n→∞

λk xk

k=−n

exists for each (λk )k∈Z ∈ ∞ (Z). Thus if the Fourier series +∞

ek ⊗ fˆ(k)

k=−∞

Lp2π (X),

converges unconditionally in then each bounded sequence (λk I)k∈Z with I the identity operator on X, is an Lp -multiplier. This implies that p = 2 (as a consequence of the scalar result mentioned above) and X is a Hilbert space by Theorem 1.5. Next we discuss convergence of the Fourier series. For f ∈ Lp2π (X) and n ∈ N, we let Sn (f ) := |k|≤n ek ⊗ fˆ(k) be the partial sum of the Fourier series; hence Sn ∈ L(Lp2π (X)) for n ∈ N. Theorem 1.6. Let X be a Banach space. The following conditions are equivalent. (i) X is a UMD-space; (ii) supn∈N Sn < ∞ for some 1 < p < ∞ (equivalently for all 1 < p < ∞); +∞ (iii) for each 1 < p < ∞ and each f ∈ Lp2π (X), the Fourier series ek ⊗ fˆ(k) converges to f in Lp2π (X);

k=−∞

(iv) there exists 1 < p < ∞ and for each f ∈ Lp2π (X), the Fourier series

+∞

ek ⊗

k=−∞

fˆ(k) converges to f in Lp2π (X); (v) the sequence (Mk )k∈Z given by Mk = I for k ≥ 0, Mk = −I for k < 0, is an Lp -multiplier for some 1 < p < ∞ (equivalently for all 1 < p < ∞). We refer to the literature (e.g., [Bur01]) for definition of UMD-spaces. Here we just mention that each Lp -space is a UMD-space whenever 1 < p < ∞. Moreover,


29

each UMD-space is reflexive and closed subspaces and quotient spaces of UMDspaces are UMD-spaces. The operator associated with the multiplier Mk = sgn(k)I on Lp2π (X) is called the Hilbert transform. Thus the Hilbert transform is bounded on Lp2π (X) if and only if 1 < p < ∞ and X is a UMD-space. For the operator-valued multiplier theorems we want to present here we need the notion of Rademacher type (or briefly type) and Rademacher cotype (or briefly cotype) of a Banach space. We refer to [LT79] [Pie07, p. 308] for the definitions and further properties of these notions. We just recall that every Lq -space with 1 ≤ q ≤ 2 is of type q and cotype 2. Every Lq -space with 2 ≤ q < ∞ is of cotype q and type 2. A Banach space X is of type 2 and of cotype 2 if and only if X is a Hilbert space [Kw72]. Next we formulate the variational form of the Marcinkiewicz multiplier theorem. Theorem 1.7. Let X, Y be UMD-spaces. Assume that X is of cotype 2 and Y is of type 2. Let (Mk )k∈Z ⊂ L(X, Y ) be a bounded sequence such that sup n∈N

n n−1 −1 −1 −2 2 M + − M k+1 Mk+1 − Mk < ∞. k

(1.10)

k=−2n

k=2n−1

Then (Mk )k∈Z is an Lp -multiplier for 1 < p < ∞. The hypothesis on X, Y are satisfied in particular when both spaces X and Y are Hilbert spaces. In that case Theorem 1.7 was proved by J. Schwartz [Sch61] in 1961. The scalar case is due to J. Marcinkiewicz and appeared in 1939 in Studia Mathematica. The more general case we present here can be obtained by an inspection of the proof of [AB02, Theorem 1.3] stopping on page 318 line 9 and [AB02, Proposition 1.13]. We mention a special case which turns out to the most suitable for generalizations Corollary 1.8 ([AB02, Theorem 1.3]). Let X, Y be UMD-spaces, X is of cotype 2 and Y is of type 2. Let (Mk )k∈Z ⊂ L(X, Y ) be a bounded sequence such that τ := sup |k|Mk+1 − Mk < ∞.

(1.11)

k∈Z

Then (Mk )k∈Z is an Lp -multiplier whenever 1 < p < ∞. Proof. For n ∈ N one has n n 2 −1 2 −1 1 2n − 2n−1 ≤τ = τ, Mk+1 − Mk ≤ τ k 2n−1 n−1 n−1 k=2

and similarly

k=2

n−1 −2 −1

Mk+1 − Mk ≤ τ.

k=−2n

Now the result follows from Theorem 1.7.

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W. Arendt and S. Bu

Concerning operator-valued versions of the variational version of Marcinkiewicz Theorem (Corollary 1.8) on UMD-spaces instead of Hilbert spaces we refer to Z. Strkalj and L. Weis [SW07] (see also [CPSW00] for results on operator-valued Fourier multipliers with respect to more general Schauder decompositions).

2. The Marcinkiewicz multiplier theorem in the general case In order to describe the general periodic multiplier theorem we need the notion of R-boundedness. We recall that rk is the kth Rademacher function on [0, 1]. Definition 2.1. Let X, Y be Banach spaces. A set T ⊂ L(X, Y ) is called R-bounded if for all (equivalently for one) q ∈ [1, ∞) there exists a constant C > 0 such that n n rk Tk xk q ≤ C rk xk q k=1

L (0,1;Y )

k=1

L (0,1;X)

for all n ∈ N, x1 , . . . , xn ∈ X, T1 , . . . , Tn ∈ T . This notion of unconditional boundedness of a family of operators implies boundedness but is stronger in general. More precisely the following holds. Proposition 2.2 ([AB02, Proposition 1.13]). Let X, Y be Banach spaces. The following assertions are equivalent. (i) Each bounded T ⊂ L(X, Y ) is R-bounded; (ii) X is of cotype 2 and Y is of type 2. One immediate consequence of Proposition 2.2 is that each bounded subset of L(X) is R-bounded if and only if X is a Hilbert space. We note however that for an arbitrary Banach space a family T ⊂ L(X) of scalar operators is R-bounded if and only if it is bounded. Here we call T ∈ L(X) scalar, if it is of the form λI for some λ ∈ C. Now we can formulate the operator-valued Marcinkiewicz Theorem. Theorem 2.3 ([AB02, Theorem 1.3]). Let X, Y be UMD-spaces, 1 < p < ∞. Assume that (Mk )k∈Z ⊂ L(X, Y ) is such that both Mk : k ∈ Z , k(Mk+1 − Mk ) : k ∈ Z are R-bounded in L(X, Y ). Then (Mk )k∈Z is an Lp -multiplier. p It is known that when (Mk )k∈Z ⊂ L(X, Y ) is an L -multiplier, then the set Mk : k ∈ Z must be R-bounded [AB02, Proposition 1.11]. In general, it is not possible to replace the R-boundedness in Theorem 2.3 by norm boundedness. More precisely, the following is true.

Proposition 2.4 ([AB02, Proposition 1.17]). Let X, Y be Banach space and 1≤ p < Mk < ∞ ∞. Assume that every sequence (M ) ⊂ L(X, Y ) satisfying sup k k∈Z k∈Z p and supk∈Z k(Mk+1 − Mk ) < ∞, is an L -multiplier. Then X is of cotype 2 and Y is of type 2. In particular, when X = Y , then X is a Hilbert space.


31

3. The periodic non-homogeneous problems We are now going to show how the multiplier theorem can be applied. Let X be a UMD-space and let D be a Banach space which is continuously embedded into X; we write D → X for short. Let A ∈ L(D, X), 1 < p < ∞. We consider the following problem. Given f ∈ Lp (0, 2π; X) we want to find a solution u of the problem ⎧ ⎨ u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) u(t) ˙ = Au(t) + f (t) a.e. Pper ⎩ u(0) = u(2π). Here W 1,p (0, 2π; X) consists of those continuous functions u : [0, 2π] → X for which there exists u ∈ Lp (0, 2π; X) such that t u (s)ds (t ∈ [0, 2π]). u(t) = u(0) + 0

Equivalently, W (0, 2π; X) consists of those functions u ∈ Lp (0, 2π; X) for which u ∈ Lp (0, 2π; X), where u is defined in the sense of distributions. We say that problem Pper is well posed if for each f ∈ Lp (0, 2π; X), there exists a unique solution u of Pper . The following result characterizes well-posedness of the problem Pper . 1,p

Theorem 3.1. Let 1 < p < ∞. The following assertions are equivalent. (i) For each f ∈ Lp (0, 2π; X) there exists a unique solution of Pper ; (ii) for each k ∈ Z the operator ik − A ∈ L(D, X) is invertible and the family (ik − A)−1 : k ∈ Z is R-bounded in L(X, D). Theorem 3.1 is a consequence of the multiplier theorem. It is similar to [AB02, Theorem 2.3], where a stronger hypothesis on A is imposed, namely that A is closed as an unbounded operator on X. This means by definition that the graph of A G(A) := (x, Ax) : x ∈ D is closed in X × X. Condition (ii) does imply closedness of A. In fact, taking k = 0, (ii) implies that A is invertible. Now let (x, y) ∈ G(A). Then there exist xn ∈ D such that xn → x and Axn → y in X. It follows that xn = A−1 (Axn ) → A−1 y in D. Hence x = A−1 y ∈ D and Ax = y. We now give a proof of Theorem 3.1. Proof of Theorem 3.1. (ii)⇒(i). Assume condition (ii). Then A is invertible. It follows that the graph norm xA := Ax

(x ∈ D)

defines an equivalent norm on D. In fact, since A is an isometric isomorphism from (D, · A ) to X, it follows that (D, · A ) is complete. Since xA ≤ AxD , it follows from the Open Mapping Theorem that both norms on D are equivalent.

32

W. Arendt and S. Bu

Thus A is an isomorphism from D to X. It follows from (ii) that the family A(ik − A)−1 : k ∈ Z is R-bounded in L(X). Since ik(ik − A)−1 − A(ik − A)−1 = IX , we conclude that the family

(3.1)

k(ik − A)−1 : k ∈ Z

is R-bounded in L(X). Now [AB02, Theorem 2.3] implies that Pper is well posed. (i)⇒(ii). Assume that Pper is well posed. a) We claim that A is bijective. Let y ∈ X be given. Then for f (t) ≡ −y, there exists u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) such that u(t) ˙ = Au(t) − y a.e. and 2π 1 u(0) = u(2π). Then x := 2π u(t)dt ∈ D and 0

2π 2π 1 1 Au(t)dt = u(t)dt ˙ − f (t)dt 2π 2π 0 0 0 1 u(2π) − u(0) + y = y. = 2π We have shown that A is surjective. Injectivity can be seen as follows. Let x ∈ D be such that Ax = 0. Then u(t) := x satisfies 0 = u(t) ˙ = Au(t) + 0. Hence u is a solution of Pper for f ≡ 0. Since also v ≡ 0 is a solution, it follows that x ≡ u ≡ 0. b) It follows from a) that A is invertible and hence that A is closed. Now [AB02, Theorem 2.3] implies that ik− A is bijective and the set k(ik − A)−1 : k ∈ Z is R-bounded in L(X), and so (ik − A)−1 : k ∈ Z is R-bounded in L(X, D) since A−1 : X → D is an isomorphism. Two consequences of Theorem 3.1 are remarkable. First of all it follows that well-posedness of Pper is independent of p ∈ (1, ∞). Secondly, D is dense in X. This follows from condition (ii) of Theorem 3.1, [ABHN01, Proposition 3.3.8] and the fact that X is reflexive. 1 Ax = 2π

2π

4. Maximal regularity Instead of Dirichlet boundary conditions we consider now the initial value problem. Again X is a UMD-space and D is a Banach space such that D → X. Let A ∈ L(D, X) be an operator, 1 0. Given f ∈ Lp (0, τ ; X) we want to find a solution of the problem ⎧ ⎨ u ∈ W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) u(t) ˙ = Au(t) + f (t) a.e. P0 (τ, p) ⎩ u(0) = 0 . We say that problem P0 (τ, p) is well posed if for every f ∈ Lp (0, τ ; X), there exists a unique solution u of P0 (τ, p). This can be characterized as follows.


33

Theorem 4.1. Let 1 0 be fixed. The following assertions concerning the operator A are equivalent. (i) Problem P0 (τ, p) is well posed; (ii) there exists ω ∈ R such that (λ − A) is invertible whenever Re λ > ω and the set {(λ − A)−1 : Re λ > ω} is R-bounded in L(X, D). If these equivalent conditions are satisfied, then D is dense in X and A (with domain D) generates a holomorphic C0 -semigroup on X. Implication (ii)⇒(i) is due to L. Weis [W01] who uses an operator-valued multiplier theorem on Lp (R, X). Here we will deduce this implication from the periodic multiplier theorem, Theorem 3.1. The proof of the implication (i)⇒(ii) is given in two steps. At first we show that A generates a holomorphic C0 -semigroup. This result is due to G. Dore [Do93, Theorem 2.2], who states it under the additional hypothesis that A is closed and D is dense in X. However, the proof sketched in [Do93] can be carried over to the more general situation. We give all the details to convince the reader. Once it is known that A generates a holomorphic C0 -semigroup one may again use Theorem 3.1 (or use [CP00]). Proof of Theorem 4.1. (i)⇒(ii). By the Closed Graph Theorem there exists a constant c1 ≥ 0 such that uW 1,p (0,τ ;X) ≤ c1 f Lp(0,τ ;X)

(4.1)

for every f ∈ Lp (0, τ ; X), where u denotes the unique solution of P0 (τ, p). a) First we show that there exists ω1 ∈ R such that λ − A is injective whenever Re λ ≥ ω1 . In fact, let λ ∈ C, x ∈ D, such that λx − Ax = 0. Then u(t) = 1 λt − 1)x is the unique solution of P0 (τ, p) for f ≡ x. Hence uW 1,p (0,τ ;X) ≤ λ (e c1 f Lp(0,τ ;X) = c1 τ 1/p x. Hence by (4.1) x

1 p 1 Re λpτ e −1 = u ˙ Lp (0,τ ;X) pRe λ ≤ c1 f Lp(0,τ ;X) = c1 τ 1/p x.

Thus if we choose ω1 > 0 so large that 1 Re λpτ e − 1 > cp1 τ, pRe λ whenever Re λ ≥ ω1 , then x = 0. b) Since W 1,p (0, τ ; X) → C([0, τ ]; X), there exists c2 ≥ 0 such that u(τ )X ≤ c2 uW 1,p (0,τ ;X) for all u ∈ W 1,p (0, τ ; X). Let fλ (t) :=

eλt 0

if t ≤ Re1 λ if Re1 λ < t ≤ τ

(4.2)

34

W. Arendt and S. Bu

where Re λ > max

1

τ , ω1

=: ω2 . Then for x ∈ X one has

fλ ⊗ xLp (0,τ ;X) =

c3 x 1

,

(Re λ) p

p1 − 1) . Let u be the solution of P0 (τ, p) for the inhomogeneity τ fλ ⊗ x. Define R(λ)x := Re λ e−λt u(t) dt ∈ D. Then where c3 =

1 p p (e

0

τ (λ − A)R(λ)x

=

=

−Re λ

(e

−λt

τ

) u(t) dt − Re λ

0

0

τ

τ

−Re λ

(e−λt ) u(t) dt − Re λ

0

e−λt Au(t) dt

e−λt u (t) dt

0

1 Re λ

+Re λ =

−Re λe

e−λt fλ (t) dt x

0 −λτ

u(τ ) + x.

Define Sx := Re λe−λτ u(τ ). Then S : X → X is linear and by (4.1) and (4.2) Sx

≤ c2 Re λe−Re λτ uW 1,p (0,τ ;X) ≤ c1 c2 Re λe−Re λτ fλ Lp (0,τ ;X) x 1

= c1 c2 c3 (Re λ) p e−Re λτ x, where p1 + p1 = 1. Thus there exists ω ≥ ω2 such that SL(X) ≤ Re λ ≥ ω. Let Re λ ≥ ω. Since (λ − A)R(λ)x = x − Sx

1 2

whenever (4.3)

for all x ∈ X, and since I −S is surjective, it follows that (λ−A) is surjective. Hence by a) the operator (λ − A) : D → X is bijective. By using a similar argument used before the proof of Theorem 3.1, this implies already that the unbounded operator λ − A on X with domain D is closed. Thus the unbounded operator A on X with domain D is also closed. Moreover, by (4.3) (λ − A)−1 x = R(λ)x + (λ − A)−1 Sx . Hence 1 (λ − A)−1 L(X) ≤ R(λ)L(X) + (λ − A)−1 L(X) . 2 Consequently (λ − A)−1 L(X) ≤ 2R(λ)L(X) .

(4.4)


35

Let x ∈ X, then for u as above, R(λ)x

=

Re λ λ

τ 0

=

Re λ λ

−(e−λt ) u(t) dt

τ

e−λt u (t) dt − e−λτ u(τ ) .

0

Hence by (4.1) and (4.2) 1 τ p e−Re λp t dt u Lp (0,τ ;X) + c2 e−Re λτ uW 1,p (0,τ ;X) λR(λ)x ≤ Re λ 0

1/p 1 −Re λτ c1 xfλ Lp (0,τ ;X) + c2 e p Re λ 1/p 1 1/p −Re λτ x ≤ c1 c 3 + c2 (Re λ) e p ≤ c4 x

≤ Re λ

for some constant c4 > 0 whenever Re λ ≥ ω. Thus it follows from (4.4) that sup λ(λ − A)−1 L(X) < ∞ .

Re λ>ω

By [ABHN01, Proposition 3.3.8] this implies that D is dense in X (since X is reflexive) and that A generates a holomorphic C0 -semigroup (by [ABHN01, Corollary 3.7.17]). Now assertion (ii) can be deduced from Theorem 3.1 as in [AB02, Corollary 5.2 and the following lines]. Alternatively one can use [CP00]. (ii)⇒(i). Let 1 0. We have to show that problem P0 (τ, p) is well posed. The assumption (ii) implies that A generates a holomorphic C0 -semigroup T (keep in mind that X is reflexive, so the domain D of A is dense in X by [ABHN01, Proposition 3.3.8]). This shows in particular that P0 (τ, p) has at most one solution. In fact, if u ∈ W 1,p (0, τ ; X)∩Lp (0, τ ; D) such that u(t) ˙ = Au(t) t and u(0) = 0, consider v(t) = u(s) ds. Then v ∈ C 1 ([0, τ ]; X) ∩ C([0, τ ]; D) is 0

a classical solution of v(t) ˙ = Av(t) and v(0) = 0. Hence v ≡ 0 by [ABHN01, Theorem 3.1.12] and so u ≡ 0. Thus it remains to show existence which we do now. 1st case: ω < 0, τ = 2π. If the constant ω of condition (ii) is negative, then we can apply Theorem 3.1. Thus, given f ∈ Lp (0, 2π; X) there exists w ∈ W 1,p (0, 2π; X)∩ Lp (0, 2π; D) satisfying w(t) ˙ = Aw(t) + f (t) a.e. Let v(t) := T (t)w(0). Then by [Lun95, 1.2.2 and 2.2.1], one has v ∈ W 1,p (0, 2π; X)∩Lp (0, 2π; D) and v(t) ˙ = Av(t) a.e., v(0) = w(0). Thus u := w − v is a solution of P0 (2π, p). 2nd case: ω < 0, τ > 0 arbitrary. Let f ∈ Lp (0, τ ; X). Define g(t) = rf (rt) where τ r := 2π . The operator rA satisfies condition (ii) as well. Then g ∈ Lp (0, 2π; X).

36

W. Arendt and S. Bu

˙ = By the first case there exists v ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) such that v(t) 1,p rAv(t) + g(t) a.e. and v(0) = 0. Let u(t) := v(t/r). Then u ∈ W (0, τ ; X) ∩ Lp (0, τ ; D), u(0) = 0 and u(t) ˙ = 1r v˙ rt = Av rt + 1r g rt = Au(t) + f (t) a.e. Thus u is a solution of P0 (τ, p). 3rd case: The constant ω ∈ R and τ > 0 are arbitrary. Let ω1 > ω. Then the operator A−ω1 satisfies the assumptions of the 2nd case. Let f ∈ Lp (0, τ ; X). Then there exists v ∈ W 1,p (0, τ ; X)∩Lp (0, τ ; D) satisfying v(t) ˙ = Av(t)− w1 v(t)+ g(t), v(0) = 0 where g(t) = e−ω1 t f (t). Then u(t) = eω1 t v(t) is a solution of P0 (τ, p). An immediate consequence of Theorem 4.1 is the following. Corollary 4.2. If P0 (τ, p) is well posed for some τ > 0, 1 0, 1 < p < ∞. Remark 4.3. It can be seen from the proof of Theorem 4.1 that the implication (i)⇒(ii) is always true, without any assumption on the Banach space X. We know that (i) does not imply that D is dense in X, in general. If X is reflexive, though, then (ii) implies density of D.

5. The non-autonomous equations Let X be a UMD-space and D a Banach space such that D → X. Given an operator A ∈ L(D, X) we say that A satisfies maximal regularity if condition (i) of Theorem 4.1 is satisfied. We know that this is independent of the choice of 1 0 (Corollary 4.2). Given 1 0 we define the maximal regularity space M Rp (0, τ ) := W 1,p (0, τ ; X) ∩ Lp (0, τ ; D) which is a Banach space for the sum norm uMR := uW 1,p (0,τ ;X) + uLp(0,τ ;D) . By T rp := u(0) : u ∈ M Rp (0, τ ) we define the trace space. Then the following result on well-posedness can be obtained by a simple perturbation argument (see [Am04] or [ACFP07]). Theorem 5.1. Let A : [0, τ ] → L(D, X) be continuous and assume that A(t) satisfies maximal regularity for each t ∈ [0, τ ]. Then for each f ∈ Lp (0, τ ; X) and each initial value u0 ∈ T rp , there exists a unique u ∈ M Rp (0, τ ) satisfying u(t) ˙ = A(t)u(t) + f (t) a.e. u(0) = u0 . The non-autonomous problem with periodic boundary conditions is not as simple. Let A : [0, 2π] → L(D, X) be continuous such that A(0) = A(2π). Furthermore we want to assume that the injection D → X is compact. Then there are two type of results. In the first one we assume the same condition on A(t) as in Theorem 5.1 for Dirichlet boundary conditions. We define for 1 < p < ∞ the


37

periodic maximal regularity space M Rper,p := u ∈ W 1,p (0, 2π; X) ∩ Lp (0, 2π; D) : u(0) = u(2π) which is a Banach space for the natural norm uMR := uW 1,p (0,2π;X) + uLp(0,2π;D) . Theorem 5.2 ([AR09, Corollary 9.3]). Assume that (a) A(t) + is dissipative for all t ∈ [0, 2π] and some > 0 and that (b) A(t) satisfies maximal regularity (as defined above) for all t ∈ [0, 2π]. Then for 1 < p < ∞ and each f ∈ Lp (0, 2π; X), there exists a unique solution of u ∈ M Rper,p Pper u(t) ˙ = A(t)u(t) + f (t) a.e. If Pper is well posed for A then it is so for −A as well. But conditions (a) and (b) are not invariant by this inversion. This shows that conditions (a) and (b) are too strong in general. In the autonomous case, condition (ii) of Theorem 3.1 is equivalent to well-posedness of Pper . So also in the non-autonomous case, it is natural to assume that each A(t) satisfies this condition. However, this condition alone is too weak to deduce that Pper is well posed in this case, and in the finitedimensional case Floquet Theory is available. Still some significant results on the solutions of problem Pper are valid. In order to formulate them, we consider the operator DA : M Rper,p → Lp (0, 2π; X) given by DA u := u˙ − A(·)u(·). This is a bounded linear operator between the two Banach spaces M Rper,p and Lp (0, 2π; X). Note that M Rper,p is continuously (and compactly) embedded in Lp (0, 2π; X). So we may consider DA as an unbounded operator on the Banach space Lp (0, 2π; X). If each A(t) satisfies condition (ii) of Theorem 3.1, then this operator is indeed closed (as unbounded operator on Lp (0, 2π; X)). This is not obvious but a particular result of maximal regularity. Some more can be said. The operator DA is a Fredholm operator, that is, DA has finite-dimensional kernel and closed image R(DA ) with finite codimension in Lp (0, 2π; X) for each 1 < p < ∞. To say that Pper is well posed means that DA is invertible. This is stronger than Fredholm, but knowing the Fredholm property helps a lot to prove invertibility (under further assumptions as in Theorem 5.2 for example). We collect the information given here in the following concluding theorem. Theorem 5.3 ([ AR09, Theorem 4.1 and Corollary 3.7]). Assume that A(t) satisfies condition (ii) of Theorem 3.1 for each t ∈ [0, 2π]. Let 1 < p < ∞. Then the following holds. (a) DA is closed seen as unbounded operator on Lp (0, 2π; X); (b) DA is a Fredholm operator; (c) if σ(DA ) = C, then DA has compact resolvent and in particular Fredholm’s alternative holds: Pper is well posed if and only if DA is injective.

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W. Arendt and S. Bu

Acknowledgement We are most grateful to Quanhua Xu (Besan¸con) for some helpful comments in the context of Theorem 1.5.

References [Am04]

H. Amann: Maximal regularity for nonautonomous evolution equations. Adv. Nonlinear Stud. 4 (2004), 417–430. [ABHN01] W. Arendt, C. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Birkh¨ auser, Basel, 2001. [ACFP07] W. Arendt, R. Chill, S. Fornaro, C. Poupaud: Lp -maximal regularity for nonautonomous evolution equations. J. Diff. Equ. 237 (2007), 1–26. [AB02] W. Arendt, S. Bu: The operator-valued Marcinkiewicz multiplier theorems and maximal regularity. Math. Z. 240 (2002), 311–343. [AR09] W. Arendt, P. Rabier: Linear evolution operators on spaces of periodic functions. Comm. Pure and Applied Analysis 8 (2009), 5–36. [Bur01] D. Burkholder: Martingales and singular integrals in Banach spaces. In: Handbook of the Geometry of Banach Spaces, Vol. I (W.B. Johnson and J. Lindenstrauss Eds.), Elsevier, 2001, 233–269. [CPSW00] Ph. Clément, B. de Pagter, F. A. Sukochev, H. Witvliet: Schauder decomposition and multiplier theorems. Studia Math. 138 (2000), 135–163. [CP00] Ph. Clément, J. Pr¨ uss: An operator-valued transference principle and maximal regularity on vector-valued Lp -spaces. In: Evolution Equations and Their Applications in Physical and Life Sciences (G. Lumer and L. Weis Eds.), Marcel Dekker, 2000, 67–78. [Do93] G. Dore: Lp -regularity of abstract differential equations. In: Functional Analysis and Related Topics (H. Komatsu Ed.), Springer LNM 1540 (1993), 25–38. [KW04] P. Kunstmann, L. Weis: Maximal Lp -regularity for parabolic equations, Fourier multipliers theorems and H ∞ -functional calculus. In: Functional Analytic Methods for Evolution Equations. Springer LNM 1855, 2004, 65–311. [Kw72] S. Kwapien: Isomorphic characterization of inner product spaces by orthogonal series with vector-valued coefficients. Studia Math. 44 (1972), 583–595 [LeTa91] M. Ledoux, M. Talagrand: Probability in Banach Spaces, Isoperimetry and Processes. Springer-Verlag, Berlin, 1991. [LT79] J. Lindenstrauss and L. Tzafriri: Classical Banach Spaces II, Springer, Berlin, 1979. [Lun95] A. Lunardi: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh¨ auser, Basel, 1995. [Pie07] A. Pietsch: History of Banach Spaces and Linear Operators. Birkh¨ auser, Basel, 2007. [Sch61] J. Schwartz: A remark on inequalities of Calderon-Zygmund type for vectorvalued functions. Comm. Pure Appl. Math. 14 (1961), 785–799. [SW07] Z. Strkalj, L. Weis: On operator-valued Fourier multiplier theorem. Trans. Amer. Math. Soc. 359 (2007), 3529–3547.

Fourier Series in Banach Spaces and Maximal Regularity [W01]

39

L. Weis: Operator-valued Fourier multiplier theorems and maximal Lp regularity. Math. Ann. 319 (2001), 735–758.

Wolfgang Arendt Institute of Applied Analysis University of Ulm D–89069 Ulm, Germany e-mail: [email protected] Shangquan Bu Department of Mathematical Science University of Tsinghua Beijing 100084, China e-mail: [email protected]


Spectral Measures on Compacts of Characters of a Semigroup Dragu Atanasiu Abstract. In this note we give integral representations for some ∗-representations of the type U : S → B(H) where S is a commutative semigroup with involution and neutral element and B(H) are the bounded operators of the Hilbert space H. Mathematics Subject Classification (2000). Primary 47B15; Secondary 43A35. Keywords. Radon spectral measure, positive definite function.

1. Introduction In this paper we give, in Section 2, an integral representation for positive definite functions, defined on a commutative semigroup with involution and neutral element, which includes the Berg-Maserick theorem [3, p. 169, Theorem 2.1] (see also [2, p. 93, Theorem 2.5]) and the extension of this theorem from [5, p. 96, Corollary 1]. The integral representation proved in Section 2 is similar to the representation given in [1, p. 96, Theorem 2.5] but the proof is shorter. In [6, p. 2950, Theorem 2] it is shown, independent of the Banach algebra theory, that every ∗-representation U : S → B(H), where S is a commutative semigroup with involution and neutral element and B(H) are the bounded operators of the Hilbert space H has an integral representation with respect to a unique selfadjoint Radon spectral measure defined on the Borel sets of the space of characters of S with the pointwise convergence topology. This integral representation is used to give a new proof of the Gelfand-Naimark theorem for abelian C∗ -algebras. In Section 3 we obtain, using the result proved in Section 2, an integral representation which extends [6, p. 2950, Theorem 2]. In Section 4 are obtained conditions such that the support of the spectral measure considered in Section 3 is contained in sets which generalize the ball, the torus and the rectangle.

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In Section 5 we give a construction of the spectral measure from Section 3. This construction follows the way of the proof in [4, p. 435, Section 3]. In the last section we use the construction of the spectral measure from Section 5 to prove the Gelfand-Naimark theorem for abelian C∗ -algebras.

2. A Berg-Maserick type theorem 2.1. Definitions and notations In this paper (S, ·) is a commutative semigroup with involution and neutral element e (see [2, p. 86]). A function ϕ : S → C is positive definite if for every natural number n ≥ 1 and every choice of elements s1 , . . . , sn of S and complex numbers c1 , . . . , cn we have n cj ck ϕ(sj s∗k ) ≥ 0. j,k=1

A function v : S → [0, ∞) is an absolute value on S if v(e) = 1,v(s∗ ) = v(s) for all s ∈ S and v(st) ≤ v(s)v(t) for all s, t ∈ S. A function ϕ : S → C is v-bounded if for every s ∈ S we have |ϕ(s)| ≤ Cϕ v(s) where Cϕ is a positive real number. In [2, p. 90, Proposition 1.12] it is shown that we can always take Cϕ = ϕ(e) if the function ϕ is positive definite. Let Γ be a set. Let (aγ,t )γ∈Γ,t∈S be a family of complex numbers such that for every γ ∈ Γ we have aγ,t = 0 only for a finite number of t. For a function ϕ : S → C and γ ∈ Γ we denote by ϕγ : S → C the function defined by ϕγ (s) = aγ,t ϕ(ts), s ∈ S. t∈S

Let v be an absolute value on S. We denote by P the set {ϕ : S → C|ϕ and (ϕγ )γ∈Γ are positive definite ; ϕ v-bounded}. We also denote by C = {ρ : S → C|ρ(e) = 1, ρ(s∗ ) = ρ(s), ρ(st) = ρ(s)ρ(t), s, t ∈ S} the set of characters of S. We equip C with the topology of pointwise convergence. We consider the following subsets of C: V = {ρ ∈ C||ρ(s)| ≤ v(s), s ∈ S} and M = {ρ ∈ V|ργ (e) ≥ 0, ∀γ ∈ Γ}.

Spectral Measures on Compacts of Characters of a Semigroup

43

2.2. The Berg-Maserick type theorem Theorem 2.1. For a function ϕ : S → C the following conditions are equivalent a) ϕ ∈ P; b) there is a unique positive Radon measure on M such that ρ(s)dμ(ρ). ϕ(s) = M

Proof. We only have to prove a ⇒ b. The uniqueness of the measure μ can be proved as in [2, p. 95]. We prove the existence of the measure μ. According to the Berg-Maserick theorem [2, p. 93, Theorem 2.5] there is a positive Radon measure ν on V such that ϕ(s) =

ρ(s)dν(ρ). V

For every γ ∈ Γ we obtain n

cj ck ϕγ (sj s∗k ) =

j,k=1

V t∈S

aγ,t ρ(t)|

n

cj ρ(sj )|2 ≥ 0.

j=1

Because for every continuous function f : V → C such that f ≥ 0 the function can be uniformly approximated on V by functions of the type n

√ f

cj ρ(sj )

j=1

this means that we have

V t∈S

aγ,t ρ(t)f (ρ)dν(ρ) ≥ 0

for every continuous function f : V → C such that f ≥ 0. Consequently for every γ ∈ Γ the measure aγ,t ρ(t)) · ν (ρ → t∈S

is positive which means that ν{ρ ∈ V|

aγ,t ρ(t) < 0} = 0.

t∈S

This finishes the proof.

Remark 2.2. If for some γ0 ∈ Γ we have ϕγ0 = 0 where ϕ ∈ P has the representing measure μ, then μ({ρ|ργ0 (e) = 0}) = 0.

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D. Atanasiu

Corollary 2.3. For every ϕ ∈ P the following inequality holds |

n

cj ϕ(sj )| ≤ ϕ(e) sup |

n

cj ρ(sj )|

ρ∈M j=1

j=1

for every natural number n ≥ 1 and every choice of elements s1 , . . . , sn of S and of complex numbers c1 , . . . , cn . Proof. This corollary is an immediate consequence of Theorem 2.1.

3. An integral representation via spectral measures In this section H is a Hilbert space and B(H) is the space of bounded operators on H. A function U : S → B(H) is a ∗-representation if we have U(e) = I; U(s∗ ) = U(s)∗ , s ∈ S; U(st) = U(s)U(t), s, t ∈ S. With the theorem from the preceding section we can obtain as in [6, p. 2951] the following result Theorem 3.1. Let U : S → B(H) be a ∗-representation. We suppose that for every γ ∈ Γ the operator aγ,t U(t) t∈S

is positive. There exists a unique selfadjoint Radon spectral measure E : Bor(M) → B(H), where Bor(M) are the Borel sets of M, such that ρ(s)dE(ρ), s ∈ S, U(s) = M

where M = {ρ ∈ C|ργ (e) ≥ 0, ∀γ ∈ Γ; |ρ(s)| ≤ U(s) , s ∈ S}. Corollary 3.2. Suppose that (S, ·, +) is a commutative algebra equipped with a involution and neutral element e. Let U : S → B(H) be a linear ∗-representation. There exists a unique selfadjoint Radon spectral measure E : Bor(M) → B(H) such that U(s) = ρ(s)dE(ρ), s ∈ S M

where M is the set {ρ : S → C|ρ linear, ρ(e) = 1,ρ(s∗ ) = ρ(s),ρ(st) = ρ(s)ρ(t),|ρ(s)| ≤ U(s),s,t ∈ S}


45

Proof. The equalities U((at1 + bt2 )s) − aU(t1 s) − bU(t2 s) = 0, a, b ∈ C; s, t1 , t2 ∈ S, and the way of proof in [6, p. 2951] imply that μ-almost every ρ ∈ M is linear (see Remark 2.2). Remark 3.3. We note that [6, p. 2949, Theorem 1] is a consequence of Corollary 3.2.

4. Examples of ∗-representations Definition 4.1. We say that a ∗-representation U : S → B(H) has an integral ∗-representation on a compact set M ⊂ C if there is a selfadjoint spectral Radon measure (necessarily unique) E : Bor(M) → B(H) such that U(s) = ρ(s)dμ(ρ), s ∈ S. M

A set G ⊂ S is a generator set if each element of S is a product of elements from G ∪ {g ∗ |g ∈ G}. Proposition 4.2. Let H be a Hilbert space. For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y, U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that

(Mg2 I − U(gg ∗ ))x, x ≥ 0 , g ∈ G, x ∈ H, where Mg ≥ 0, the operator U(s) is continuous for every s ∈ S, and U has an integral ∗-representation on the compact M = {ρ ∈ C||ρ(g)| ≤ Mg , g ∈ G}. Proof. We have

U(g)x, U(g)x = U(gg ∗ )x, x ≤ Mg2 x, x g ∈ G, x ∈ H, and consequently the operator U(g) is continuous for every g ∈ G and we get U(g) ≤ Mg . From the fact that G is a generator set it results that for every s ∈ S the linear application U(s) is bounded. Now we can apply Theorem 3.1 to finish the proof. Corollary 4.3. Let H be a Hilbert space.For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y, U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that

(Mg2 I − U(gg ∗ ))x, x = 0 , g ∈ G, x ∈ H,

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D. Atanasiu

the operator U(s) is continuous for every s ∈ S and U has an integral ∗-representation on the compact M = {ρ ∈ C||ρ(g)| = Mg , g ∈ G}. Proof. This result is a consequence of the preceding proposition and of Remark 2.2 Proposition 4.4. Let H be a Hilbert space. For every s ∈ S let U(s) : H → H be a linear application such that U(st) = U(s)U(t), U(s)x, y = x, U(s∗ )y, U(e) = I, s, t ∈ S, x, y ∈ H. If we suppose that G is a generator set of the semigroup S and that 1 1 Mg,τ I − U(g) − U(g ∗ ) x, x ≥ 0 , g ∈ G, τ ∈ {±1, ±i}, x ∈ H, 2τ 2τ where (Mg,τ )g∈G,τ ∈{±1,±i} is a family of real numbers such that Mg,τ > 0 τ ∈{±1,±i}

and −Mg,−1 ≤ Mg,1 , −Mg,−i ≤ Mg,i , g ∈ G, the operator U(s) is continuous for every s ∈ S and U has an integral ∗-representation on the compact M = {ρ ∈ C| − Mg,−1 ≤ Re ρ(g) ≤ Mg,1 , −Mg,−i ≤ Im ρ(g) ≤ Mg,i , g ∈ G}. Proof. From the equality 1 1 U(g) − U(g ∗ ) = Mg,τ I − 2τ 2τ τ ∈{±1,±i}

we obtain that

Mg,τ I

τ ∈{±1,±i}

1 1 ∗ U(g) − U(g ) x, x 0≤ Mg,τ I − 2τ 2τ " ! Mg,τ x, x , g ∈ G, x ∈ H, ≤ τ ∈{±1,±i}

which means that all the operators 1 1 Mg,τ I − U(g) − U(g ∗ ) , g ∈ G, 2τ 2τ are continuous. Now using the relations ⎛ ⎞ 1 1 U(g) − U(g ∗ ) , g ∈ G, 2U(g) = ⎝ τ Mg,τ ⎠ I − τ Mg,τ I − 2τ 2τ τ ∈{±1,±i}

τ ∈{±1,±i}


47

and the fact that G is a generator set it results that for every s ∈ S the linear application U(s) is bounded. We finish the proof using Theorem 3.1.

5. A construction of the spectral measure From the following result we can obtain the selfadjoint Radon spectral measure from Theorem 3.1. Theorem 5.1. Let U : S → B(H) be a ∗-representation as in Theorem 3.1. Defining M as in Theorem 3.1 there is a ∗-representation μ : C(M) → B(H) where C(M) is the algebra {f : M → C|f

continuous}

with the involution f ∗ (ρ) = f (ρ), such that μ(f ) ≤ sup |f (ρ)|, μ(ρ → ρ(s)) = U(s)

and

ρ∈M

U(s) = sup |ρ(s)|. ρ∈M

Proof. Let U : S → B(H) be a ∗-representation as in Theorem 3.1. For every x ∈ H the function s → U(s)x, x is in P (see Section 2) if we take v(s) = U(s), s ∈ S. According to Corollary 2.3 we have # # # !⎛ ⎞ "## # # n # n # # # # # # # ⎝ # ⎠ c U(s ) x, x ≤

x, x sup c ρ(s ) j j j j # # # # ρ∈M # j=1 # # # j=1 It results because (U(s))s∈S are normal operators that # # # # n # # n # cj U(sj ) ≤ sup # cj ρ(sj )## j=1 ρ∈M # j=1 # and consequently the function μ(ρ →

n

cj ρ(sj )) =

j=1

n

cj U(sj )

j=1

is well defined. If we denote by A the algebra of functions M → C of the form ρ →

n j=1

cj ρ(sj )

48

D. Atanasiu

it is immediate to verify that we have μ(f + g) = μ(f ) + μ(g),

μ(f ) = (μ(f ))∗ , f, g ∈ A.

μ(f g) = μ(f )μ(g) and μ(1) = I,

Now we construct our representation using the fact that according to the Stone-Weierstrass theorem the set A is dense in C(M) (see [2, p. 95]). Because from the definition of M in this theorem we have |ρ(s)| ≤ U(s), s ∈ S, for every ρ ∈ M and we also have | U(s)x, x| ≤ x, x sup |ρ(s)|, s ∈ S, x ∈ H, ρ∈M

it results that U(s) = sup |ρ(s)|, s ∈ S, ρ∈M

which finishes the proof.

6. The Gelfand-Naimark theorem for abelian C∗ -algebras Lemma 6.1. Let S ⊂ B(H) be a commutative algebra with involution and neutral element e. If M is the set {ρ : S → C|ρ linear, ρ(e) = 1, ρ(st) = ρ(s)ρ(t), ρ(s∗ ) = ρ(s), |ρ(s)| ≤ s, s, t ∈ S} there is a ∗-representation μ : C(M) → B(H) where C(M) is the algebra {f : M → C|f

continuous}

with the involution f ∗ (ρ) = f (ρ), such that μ(f ) ≤ sup |f (ρ)|, μ(ρ → ρ(s)) = s ρ∈M

and

s = sup |ρ(s)|. ρ∈M

Proof. Defining U : S → B(H) by U(s) = s this lemma is a consequence of Theorem 5.1. Theorem 6.2 (Gelfand-Naimark theorem). Let S ⊂ B(H) be an abelian C ∗ -algebra and M as in the preceding lemma. The function defined by s → G(s), where G(s) is the function M → C defined by G(s)(ρ) = ρ(s), is a ∗-isomorphism S → C(M) such that s = sup |ρ(s)|. ρ∈M

Proof. This result is a consequence of Lemma 6.1 if we notice that S is a complete space and that the linear subspace generated by {G(s)|s ∈ S} is dense in C(M).


49

Acknowledgment I am indebted to the referee for valuable suggestions.

References [1] D. Atanasiu, Un théorème du type Bochner-Godement et le problème des moments , J. Funct. Anal. 92(1990), 92–103. [2] C. Berg, J.P.R. Christensen and P. Ressel, Harmonic analysis on semigroups. Theory of positive definite and related functions, Springer-Verlag, 1984. [3] C. Berg and P.H. Maserick, Exponentially bounded positive definite functions, Illinois J. Math. 28(1984), 162–179. [4] G. Maltese, A representation theorem for positive functionals on involution algebras (Revised), Boll. U.M.I. (7)8-A(1994), 431–438. [5] P. Ressel, Integral representations on convex semigroups, Math. Scand. 61(1987), 93–101. [6] P. Ressel and W.J. Ricker, Semigroup representations, positive definite functions and abelian C∗ -algebras, Proc. Amer. Math. Soc. 126(1998), 2949–2955. Dragu Atanasiu Bor˚ as University S-501 90 Bor˚ as, Sweden e-mail: [email protected]


On Vector Measures, Uniform Integrability and Orlicz Spaces Diomedes Barcenas and Carlos E. Finol Abstract. Given a Banach space X and a probability space (Ω, Σ, μ), we characterize the countable additivity of the Dunford integral for Dunford integrable functions taking values in X as those weakly measurable functions f : Ω −→ X for which {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in some separable Orlicz space Lϕ (μ). We also provide a characterization of the Pettis integral of Dunford integrable functions by mean of weak compactness in separable Orlicz spaces and give a necessary and sufficient condition for the uniform integrability of {xf : x ∈ BX }, whenever f : Ω −→ X ∗ is Gel’fand integrable. Mathematics Subject Classification (2000). Primary 46G10; Secondary 28B05. Keywords. Vector measures, Pettis integral, Orlicz spaces.

1. Introduction and preliminaries We shall use the following notations: throughout, X will denote a Banach space with dual X ∗ and BX its closed unit ball; while the closed unit ball of X ∗ is denoted by BX ∗ . Regarding the integrals considered herein we shall follow ([3]), along with most definitions. Let us recall a result of De La Vallée Poussin’s ([9], Theorem 2, p. 3). Theorem 1.1. Let (Ω, Σ, μ) be a finite measure space. A subset A ⊂ L1 (μ) is uniformly integrable if and only if there is an N -function ϕ such that A is bounded in the Orlicz space Lϕ (μ). As to N -functions,their properties and various classes thereof, we refer to ([4] , [9]). An N -function ϕ is submultiplicative if, for all s, t > 0, we have that ϕ (st) ≤ ϕ (s) ϕ (t). Throughout, by (Ω, Σ, μ), we mean a finite measure space. Research supported partially by C.D.C.H. Universidad Central de Venezuela grant 03-00-60402005.

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D. Barcenas and C.E. Finol

Recall that a subset A of L1 (μ) is said to be uniformly integrable if it is bounded in L1 (μ) and for each ε > 0, there is δ > 0 such that μ(E) < δ ⇒ |f |dμ < ε E

uniformly in A. A weakly measurable function f : Ω → X is called Dunford integrable if for each x∗ ∈ X ∗ the measurable scalar function x∗ f is integrable. If f is Dunford integrable, then it can be proved ([8]) that for each measurable set E ∈ Σ, there ∗ is x∗∗ E ∈ X such that ∗∗ ∗ xE (x ) = x∗ f dμ, ∀x∗ ∈ X ∗ . E

The x∗∗ is called the Dunford integral of f over E and it is denoted functional ∗∗ by D E f dμ. If xE ∈ X ⊂ X ∗∗ ∀E ∈ Σ, we say function that the Dunford integral f is Pettis integrable. In this case we write P E f dμ instead of D E f dμ. Another integral we deal with in this paper is the so-called the Gel’fand integral. A function f : Ω → X ∗ is called Gel’fand integrable if for each x ∈ X, xf is integrable. Proceeding as in the case of Dunford integrable functions ([8]), ∗ if f is Gel’fand integrable, then for each measurable set E ∈ Σ, there is x∗∗ E ∈X ∗ ∗ such that x (x) = E xf dμ. The functional xE is called the Gel’fand integral of f over E. This fact is denoted by G E f dμ. Another characterization of uniformly integrability, obtained by Dunford and Pettis ([6], Theorem II, p. 39), reads as follows: Theorem 1.2. A subset A ⊂ L1 (μ) is uniformly integrable if and only if it is relatively weakly compact in L1 (μ). Since a strongly measurable function f : Ω → X is Pettis integrable if and only if it is Dunford integrable and the set {x∗ f : x∗ ∈ BX ∗ } is uniformly integrable in L1 (μ). J.Uhl([12]) has obtained the following consequence of De La Vallée Poussin’s theorem. Theorem 1.3. Let f : Ω → X be a strongly measurable function. Then, f is Pettis integrable with respect to μ, if and only if there is an N -function φ such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lφ (μ). Yet another improvement of the De La Vallée Poussin’s theorem is the following: Theorem 1.4 (([2], Theorem 2.5)). A subset A of L1 (μ) is uniformly integrable,if and and only if, there is a submultiplicative N -function ϕ such that A is relatively weakly compact in Lϕ (μ). The starting point for preparing this research has been the above Uhl’s and Alexopoulos’ theorems, together with a better understanding of vector integration ([3], [10], [11]).

On Vector Measures, Uniform Integrability and Orlicz Spaces

53

A weakly measurable function f : Ω → X is said to be determined by a weakly compact generated (W CG) subspace of X, if there is a weakly compact generated subspace D of X such that one of the following conditions is met. • If x∗ |D = 0, then x∗ f = 0 μ a.e. • For each x∗ ∈ X ∗ , there exists a sequence {ϕn }∞ n=1 of D-valued simple functions such that x∗ f = lim x∗ ϕn μ a.e. (see [11, Definition 2.1]). Regarding to the Pettis integrability of Dunford integrable functions, the following holds: Theorem 1.5. ([11]) A Dunford integrable function f : Ω → X is Pettis integrable if and only if D f dμ is a countably additive vector measure and f is weakly compact Ω

generated determined.

2. The results In this section we put together the main results of our paper. We start with a characterization of the countable additivity of the Dunford integral. Our characterization is similar to that of Pettis integral for strongly measurable functions, obtained by Uhl (Theorem 1.3). Theorem 2.1. Let (Ω, Σ, μ) be a finite measure space and f : Ω → X be a Dunford integrable function. Then the following statements are equivalent. (i) The Dunford integral of f is countably additive; that is, the set function D f dμ : Σ → X ∗∗ , defined by D f dμ (E) = D f dμ, E

is countable additive. (ii) There is a submultiplicative N -function ϕ such that the set {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in the Orlicz space Lϕ (μ). (iii) There is a submultiplicative N -function ϕ such that {x∗ f ∈ BX ∗ } is bounded in Lϕ (μ) and {x∗ f : x∗ ∈ BX ∗ } does not contain any basic sequence equivalent to the unit vector basis of l1 . Proof. Let f : Ω → X be a Dunford integrable function; put ν(E) = D f dμ (E ∈ Σ). E

It is easy to see that ν is a vector measure and according to ([11]), ν is a countably additive if and only if the operator: T : X ∗ → L1 (μ) defined by T (x∗ ) = x∗ f is weakly compact. This is equivalent to say that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in L1 (μ). By the Dunford Pettis characterization of relative weak compactness in L1 (μ), it is equivalent to the uniform integrability of {x∗ f : x∗ ∈ BX ∗ } in L1 (μ) and by Theorem 1.4, it is equivalent to the existence

54


of a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in the Orlicz space Lϕ (μ). Therefore (i) and (ii) are equivalent statements. (iii)⇒(ii). Since submultiplicative functions obviously satisfy the Δ2 condition, for all t ≥ 0, then the corresponding Orlicz space is weakly sequentially complete. Therefore, if {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ), then, according to Rosenthal l1 theorem ([5], Theorem 2.e.5, p. 99), either {x∗ f : x∗ ∈ BX ∗ } contains a weakly Cauchy sequence or {x∗ f : x∗ ∈ BX ∗ } contains a subsequence equivalent to the unit vector basis of l1 . Thus, if (iii) holds, then every subset of the bounded set {x∗ f : x∗ ∈ BX ∗ } contains a weakly Cauchy sequence and because, Lϕ (μ) is weakly sequentially complete, then every subset of {x∗ f : x∗ ∈ BX ∗ } contains a weakly convergent sequence. By Eberlein Smulyan’s theorem, {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact and consequently (iii)⇒(ii). (ii)⇒(iii). If (ii) holds, then {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ) because it is relatively weakly compact in Lϕ (μ). Hence, applying again the relative weak compactness of {x∗ f : x∗ ∈ BX ∗ }, by Eberlein Smulyan’s theorem, each sequence contains a weakly convergent subsequence and by Rosenthal’s l1 theorem, {x∗ f : x∗ ∈ BX ∗ } does not contain any isomorphic copy of l1 . An alternative proof can be obtained by using the Bessaga Pelczyinski principle ([1], p. 14). From the above theorem ad Theorem 1.5, we have the following characterization of Pettis integrability of Dunford integrable functions. Corollary 2.2. Let f : Ω → X be a Dunford integrable function. Then the following statements are equivalent: (i) f is Pettis integrable. (ii) f is weakly compact generated determined and there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). (iii) f is weakly compact generated determined and there is a submultiplicative N -function such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lϕ (μ) and does not contain any subsequence isomorphic to the unit vector basis of l1 . We restate Uhl’s theorem as follows: Theorem 2.3. A strongly measurable function f : Ω → X is Pettis integrable if and only if there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). Proof. If f : Ω → X is strongly measurable, then its range is essentially separable and, consequently, weakly compact generated determined. On the other hand, being f Pettis integrable, then it is Dunford integrable with ν(E) = D E f dμ, a countably additive vector measure. Hence {x∗ f : x∗ ∈ BX ∗ } is uniformly integrable and consequently there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ).


55

For the converse, we suppose that f is strongly measurable and that there is a submultiplicative N -function ϕ such that {x∗ f : x∗ ∈ BX ∗ } is relatively weakly compact in Lϕ (μ). Since f is strongly measurable it has range weakly compactly generated determined. Since Lϕ (μ) ⊂ L1 (μ) ([4], [9]), we have that f is Dunford integrable with the Dunford integral of f countably additive. Therefore, f is Pettis integrable. We have the following version of Corollary 5.2 from ([7]). Corollary 2.4. Let f : Ω −→ X be a Dunford integrable function. If there is p > 1 such that {x∗ f : x∗ ∈ BX ∗ } is bounded in Lp (μ) then the Dunford integral of f is countably additive. Proof. Bounded subset of Lp (μ) (p > 1) are uniformly integrable in L1 (μ) and, consequently, the Dunford integral of f is countably additive. Regarding to the Gel’fand integral we have the following result. Theorem 2.5. Let X be a Banach space. The following statements are equivalent. (a) X does not contain a complemented copy of l1 . (b) A function f : Ω → X ∗ is Gel’fand integrable if and only if there is a submultiplicative N -function ϕ such that {xf : x ∈ BX } is relatively weakly compact in Lϕ (μ). Proof. If X does not contain a complemented copy of l1 , then X ∗ does not contain any copy of l∞ and by a theorem of Diestel and Faires, ([3], Theorem 2, p. 20), the Gel’fand integrable function f defines a countably additive vector measure. Define T : X → L1 (μ) by T (x) = xf ; T is a bounded linear operator and we want to prove that T is weakly compact. If ν(E) = G f dμ, then ν is a countably E

additive vector measure for which ν(E)(x) = 0 for every x ∈ X whenever μ(E) = 0. By ([3], Theorem I.2.1), given ε > 0, there is δ > 0 such that μ(E) < δ ⇒ ν(E) < ε, which implies

μ(E) < δ ⇒ sup

x∈BX

|xf | dμ < 4ε.

(2.1)

E

On the other hand, by the Bartle, Dunford and Schwartz theorem ([3], Corollary 6, p. 14), ν(Σ) is relatively weakly compact, which implies that xf dμ : E ∈ Σ, x ∈ BX E

is bounded in C. Consequently,

|xf |dμ < ∞.

sup x∈BX

Ω

(2.2)

56


From (2.1) and (2.2) we conclude that {xf : x ∈ BX } is uniformly integrable and {xf : x ∈ BX } is relatively weakly compact in Lϕ (μ) for some submultiplicative N -function ϕ. Conversely, if X contains a complemented copy of l1 , then X ∗ contains a copy of l∞ ; proceeding as in ([3], Example II.3.3, p. 53), we may construct a c0 -valued Dunford integrable function whose Dunford integral is not countably additive. More precisely, if the Dunford integral of the function f : [0, 1] → c0 is not countably additive, Y is a closed subspace of X ∗ and T : c0 → Y is an isomorphism, then, the function g := T f is defined on [0, 1], with values in X ∗ , and it is easy to see that its Gel’fand integral is not countably additive. This function is also Gel’fand integrable when it is considered as taking values in l∞ , and the values of both integrals coincide. Since the Dunford integral of f is not countably additive, then {xf : x ∈ Bl1 } is not uniformly integrable in L1 [0, 1] and, according to Theorem 1.4, there is not any submultiplicative function ϕ with {xf : x ∈ Bl1 } relatively weakly compact in Lϕ (μ). It is natural to ask whether a Gel’fand integrable function f whose Gel’fand integral is countably additive is Pettis integrable. The answer to this question is negative as the following example shows: Example. Take X = C[0, 1]. Then X contains a copy of every separable Banach space and, consequently, X contains a copy of l1 which is uncomplemented in C [0, 1] , since ,otherwise, B [0, 1] = C ∗ [0, 1] would contain a copy of l∞ . Since C [0, 1] contains a copy of l1 , then its dual, B[0, 1], does not have the weak Radon-Nikodym property ([8] Theorem 12.1) and, therefore, there is a countably additive vector measure ν : Σ −→ B[0, 1] (Σ the Borel sets in [0, 1]) such that ν does not have a weak Radon-Nikodym derivative; on the other hand, by ([8]) Theorem 11.1 there is g : [0, 1] −→ B[0, 1], Gel’fand integrable, such that ν(E) = G gdμ. E

Plainly, g is not weakly equivalent to any Pettis integrable function. In fact; if there is f such that f is Pettis integrable and xf = xg for all x ∈ X, then x∗ (P gdu) = x∗ ( f du) ∀x∗ ∈ X ∗∗ and E ∈ Σ E

E

and so x∗ ν(E) = x∗ (P

f du) =

E

x∗ f dμ.

E

This implies, by Hahn-Banach’s theorem, that ν has a weak Radon-Nikodym derivative; a contradiction.


57

Acknowledgment We wish to thank the referee for his suggestions which considerably improved the paper.

References [1] F. Albiac and N.J. Kalton,Topics in Banach Spaces Theory, Springer, 2006. [2] J. Alexopoulos, De La Vallée Poussin’s theorem and weakly compact sets in Orlicz spaces, QM 17 (1994), 231–238. [3] J. Diestel and J.J. Uhl, Vector Measures, Math. Survey 15, AMS. sre. Providence, RI (1977). [4] M.A. Krasnoselskii and V.B. Rutickii, Convex functions and Orlicz spaces, Noordhoff, Groningen 1961. [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Sequence Spaces, 1977. [6] P. Meyer, Probabilités et Potentiels, Publications de L’Institut de Math. de L’Université de Strasburg. Act. Sci. et Industrielles, Hermann, Paris 1966. [7] K. Musial, Topics in the theory of Pettis integration, Rend. Istit. mat. Univ. Trieste, 23 (1992), 177–262. [8] K. Musial, Pettis integral, Handbook of Measure Theory 531–568 (edited by E. Pap) North Holland, Amsterdam 2002. [9] M.M. Rao and Z. Ren, The Theory of Orlicz spaces, Marcel Dekker, New York, 1991. [10] S. Schwabik and Y. Guoju, Topics in Banach spaces integration, series in Real Analysis, vol. 10, World Scientific, Singapore (2005). [11] G. Stefansson, Pettis Integrability, Trans. A.M.S. 330 1 (1992), 401–418. [12] J.J. Uhl, A characterization of strongly measurable Pettis integrable functions, Proc. A.M.S., 34 2 (1972), 425–42. Diomedes Barcenas Departamento de Matem´ atica Universidad de Los Andes Mérida, 5101, Venezuela e-mail: [email protected] Carlos E. Finol Escuela de Matem´ atica Universidad Central de Venezuela Caracas, Venezuela e-mail: [email protected]


The Bohr Radius of a Banach Space Oscar Blasco Abstract. Let 1 ≤ p, q < ∞ and let X be a complex Banach space. For each ∞ n ∞ f (z) = n=0 xn z with f H (D,X) ≤ 1 we define Rp,q (f, X) = sup{r ≥ 0 : ∞ p x0 + ( n=1 xn r n )q ≤ 1} and denote the Bohr radius of X by Rp,q (X) = inf{Rp,q (f, X) : f H ∞ (D,X) ≤ 1}. The aim of this note is to study for which spaces X = Ls (μ) or X = s one has Rp,q (X) > 0. Mathematics Subject Classification (2000). Primary 46E40; Secondary 30B10. Keywords. Bohr radius, vector-valued analytic functions.

1. Introduction and preliminaries In 1914 H. Bohr [3] showed that ∞

1 |an |( )n ≤ f ∞ , 3 n=0

(1.1)

∞ where f (z) = n=0 an z n a bounded analytic function on the open unit disc. The value 1/3 is sharp. A bit later some other proofs of such inequality were given (see [9, 10]). Also several authors have found some extensions (see [4, 5, 8, 11]). Another basic inequality was discovered in [9, Corollary 2.7] much later and will play a special role for us, namely ∞ 1 2 |a0 | + |an |( )n ≤ 1, (1.2) 2 n=1 whenever f H ∞ ≤ 1 and the value 1/2 is sharp in this case. Later on some multidimensional analogues of Bohr’s inequality in which the disc D is replaced by a domain Ω ⊂ Cm were considered (see [1, 2]) and several applications of this multidimensional Bohr radius and connections concerning local Banach space theory have been recently achieved (see [7, 6]). Partially supported by Proyecto MTM2008-04594/MTM.

60

O. Blasco

Our point of view will be to keep D as domain for the functions but allow them to take values in a complex (possibly infinite-dimensional) Banach space. Throughout this paper X stands for a complex Banach space and H p (D, X), usual, for 1 ≤ p ≤ ∞, denotes the Hardy spaces of X-valued holomorphic functions from the unit disc. n Definition 1.1. Given f (z) = ∞ n=0 xn z with f H ∞ (D,X) ≤ 1 we denote $ % ∞ n R(f, X) = sup r ≥ 0 : xn r ≤ 1 . (1.3) n=0

Let us now define the Bohr’s radius of X by R(X) = inf{R(f, X) : f H ∞ (D,X) ≤ 1}. That is to say

$

R(X) = sup r ≥ 0 :

∞

(1.4) %

xn r ≤ f H ∞ (D,X) n

.

n=0

Since C is embedded into any complex Banach space we have, due to (1.1) that R(X) ≤ 13 for any Banach space. However the notion is not very useful even in the finite-dimensional case for dimension greater than one. Let us denote Cm p m the space Cm endowed with the norm the wp = ( i=1 |wi |p )1/p , or w∞ = supm i=1 |wi |. Theorem 1.2. Let m ≥ 2 and 1 ≤ p ≤ ∞. Then R(Cm p ) = 0. Proof. It suffices to do the case m = 2. In the case p = ∞ one can easily find f with f ∞ = 1 and R(f, Cm ∞ ) = 0, (take for instance f (z) = e1 + e2 z). This shows that R(C2∞ ) = 0. Assume 1 0, a value 0 < γ < 1 such that 1 − (1 − γ)1/p < εγ 1/p .

(1.5)

Now define f (z) = ((1 − γ)1/p , γ 1/p z) = (1 − γ)1/p e1 + γ 1/p e2 z. Clearly sup|z| 1. m This shows that R(f, Cm p ) ≤ ε. Hence R(Cp ) = 0. Assume now p = 1. As above for each ε > 0 we can find 0 < γ < 1 satisfying & √ 1 − 1 − γ < ε γ. (1.6)

and define

√ √ & γ 1−γ 1 & √ √ (1, 1) + (1, −1)z = ( 1 − γ + γz, 1 − γ − γz). f (z) = 2 2 2

The Bohr Radius of a Banach Space Observe that f (z)1

= ≤

61

& 1 & √ √ | 1 − γ + γz| + | 1 − γ − γz| 2 1/2 & √ √ 1 & √ | 1 − γ + γz|2 + | 1 − γ − γz|2 = 1. 2

On the other hand, from(1.6), x0 1 + εx1 1 =

& √ 1 − γ + ε γ > 1.

m This shows that R(f, Cm 1 ) ≤ ε. Hence R(C1 ) = 0.

However, following the observation in [9] and extending inequality (1.2), we are going to define another a modified Bohr radius which needs not be zero even for infinite-dimensional Banach spaces. Definition ∞1.3. Let 1 ≤ p, q < ∞ and let X be a complex Banach space. Given f (z) = n=0 xn z n with f H ∞ (D,X) ≤ 1 we denote

q $ ∞ % p n xn r ≤1 . (1.7) Rp,q (f, X) = sup r ≥ 0 : x0 + n=1

We now define Rp,q (X) = inf{Rp,q (f, X) : f H ∞ (D,X) ≤ 1}.

(1.8)

Of course R1,1 (X) = R(X) and we have the following chain of inclusions: Rp1 ,q1 (X) ≤ Rp2 ,q2 (X),

p1 ≤ p2 ,

q1 ≤ q2 .

(1.9)

To compute the precise value of Rp,q (Cm 2 ) is difficult in general, even for m = 1. In [9, Cor. 2.7] it was shown that R2,1 (C) = 12 . Let us adapt the same argument to cover the cases 1 ≤ p ≤ 2. p Proposition 1.4. If 1 ≤ p ≤ 2 then Rp,1 (C) = 2+p . ∞ Proof. Let f (z) = n=0 an z n belong to the unit ball of H ∞ (D, C). We recall the estimate, first observed by Wiener [3],

|an | ≤ 1 − |a0 |2

(1.10)

(see also [9] for a proof). From (1.10) one concludes that |a0 |p +

∞ n=1

|an |rn ≤ |a0 |p + (1 − |a0 |2 )

r 1−r

(1.11)

r Since 1 ≤ p ≤ 2 we estimate (1.11) by |a0 |p + 2p (1 − |a0 |p ) 1−r . Now |a0 |p + p p 2 r p p (1 − |a0 | ) 1−r ≤ 1 if and only if r ≤ 2+p . This gives that Rp,1 (f, C) ≥ 2+p . p Hence Rp,1 (C) ≥ 2+p .

62

O. Blasco z−a 1−az

For the converse we use Moebius transformations φa (z) = 2 ∞ n n Since φa (z) = −a + 1−a n=1 a z one obtains a ap +

for 0 < a < 1.

∞ 1 − a2 n n 1 − a2 ra . a r = ap + a n=1 a 1 − ra

This shows that Rp,1 (φa , C) =

1−ap 1−a p

(1 + a) + a( 1−a 1−a )

Taking limits as a → 1 one gets Rp,1 (C) ≤

.

p 2+p .

2. The Bohr radius Rp,q (X) for Lp -spaces Using the same example as in Theorem 1.2 one gets the following: Proposition 2.1. Rp,q (Cm ∞ ) = 0 for any m ≥ 2 and 1 ≤ p, q < ∞. Theorem 2.2. Let m ≥ 2. Then Rp,p (Cm 2 ) > 0 if and only if p ≥ 2. Proof. Assume p ≥ 2. From (1.9) it suffices to see that R1,2 (Cm 2 ) > 0. Now given f in the unit ball of H ∞ (D, Cm ) one has, in particular, that f 2H 2 (D,Cm ) = 2 2 ∞ 2 n=0 xn 2 ≤ 1. Therefore

2

∞

∞ ∞ n 2 2n x0 + xn r ≤ x0 + xn r n=1

n=1

n=1

r2 ≤ x0 + 2(1 − x0 ) 1 − r2 2 r . ≤ max 1, 2 1 − r2 2

r √1 , one obtains that for R1,2 (Cm ) ≥ √1 . Now, since 2 1−r 2 = 1 for r = 2 3 3 Conversely, assume now that 1 ≤ p < 2. Arguing as in Theorem 1.2, one has that for each ε > 0 we can find 0 < γ < 1 such that

(1 − γ)p/2 + εp γ p/2 > 1. (2.1) √ √ Now selecting f (z) = 1 − γe1 + γe2 z and using (2.1) we get Rp,p (f, Cm 2 ) ≤ ε. This implies that Rp,p (Cm 2 ) = 0. Let us now study the situation for Lp -spaces in the infinite-dimensional case. Theorem 2.3. Let 1 ≤ p, q, s < ∞ and let (Ω, Σ, μ) be a measure space such that there exists a sequence of pairwise disjoint sets with 0 < μ(An ) < ∞. Then Rp,q (Ls (μ)) = 0 whenever 1 ≤ q < s.

The Bohr Radius of a Banach Space Proof. Let 0 < β < 1 and a =

1−β 2−β .

63

Set

a0 = β 1/s μ(A0 )−1/s

and an = an/s μ(An )−1/s

for n ≥ 1. Now define φn = an χAn and Fβ (z) =

∞

φn z n .

n=0

Clearly Fβ belongs to the unit ball of H ∞ (D, H), because ∞ ∞ Fβ (z)sLs (μ) = |z|ns |an |s dμ ≤ |an |s μ(An ) = 1. n=0

An

On the other hand

q ∞ p φn Ls (μ) rn φ0 Ls (μ) +

n=0

=

ap0 μ(A0 )p/s

n=1

= β

p/s

+

= β p/s +

+

q

∞

an μ(An )1/s rn

n=1 ∞

(a

1/s

q n

r)

n=1

a1/s r 1 − a1/s r

q .

1/s

a r q Now β p/s + ( 1−a 1/s r ) ≤ 1 if and only if

r≤

a−1/s (1 − β p/s )1/q (1 − β p/s )1/q (2 − β)1/s = . 1 + (1 − β p/s )1/q (1 − β)1/s 1 + (1 − β p/s )1/q

Since 1/q > 1/s, taking limits as β goes to 1 one gets that Rp,q (Ls (μ)) = 0.

Corollary 2.4. Let 1 ≤ p, s < ∞ and 1 ≤ q < s. Then Rp,q (s ) = Rp,q (Ls (R)) = 0. A look to the proof in Theorem 2.2 shows that actually for X = L2 (μ) one gets R1,2 (L2 (μ)) ≥ √13 . Let us give some lower estimates of Rp,q (Ls (μ)) for some values of q ≥ s. As usual p stands for the conjugate exponent satisfying 1/p + 1/p = 1. Theorem 2.5. Let 1 < s < ∞, q = max{s, s } and 1 ≤ p ≤ q. Then Rp,q (Ls (μ)) ≥

(q q /q

p1/q . + pq /q )1/q

(2.2)

∞ n belong to the unit ball of Proof. Let X = Ls (μ) and let f (z) = n=0 xn z ∞ H (D, X). It follows easily from complex interpolation (considering X1 to be ∞ L1 (μ) or L∞ (μ) and X2 to be L2 (μ)) that ( n=0 xn q )1/q ≤ f H q (D,Ls (μ)) . In ∞ q q particular we have n=1 xn ≤ 1 − x0 , n ≥ 1.

64

O. Blasco Therefore

x0 + p

∞

q xn r

n

≤ x0 + (1 − x0 ) p

n=1

q

rq 1 − rq

q/q .

Hence, for p ≤ q, we estimate

q ∞ q rq p n xn r ≤ x0 p + (1 − x0 p ) x0 + p (1 − rq )q/q n=1 rq q . ≤ max 1, p (1 − rq )q/q Note that

q rq p (1−r q )q/q

= 1 gives the value r =

1/q (p q)

q /q )1/q (1+( p q)

.

References [1] L. Aizenberg Multidimensional analogues of Bohr’s theorem on power series Proc. Amer. Math. Soc. 128 (1999), 1147–1155. [2] L. Aizenberg, A. Aytuna and P. Djakov Generalization of a theorem of Bohr for basis in spaces of holomorphic functions in several variables J. Math. Anal. Appl. 258 (2001), 429–447. [3] H. Bohr A theorem concerning power series Proc. London Math. Soc. (2)13 (1914), 1–15. [4] E. Bombieri Sopra un teorema di H. Bohr e G. Ricci sulle funzioni maggioranti delle serie di potenze Bull. Un. Mat. Ital. (3)17 (1962), 276–282. [5] E. Bombieri and J. Bourgain A remark on Bohr’s inequality Inter. Math. Res. Notices 80 (2004), 4307–4329. [6] A. Defant, D. Garc´ıa and M. Maestre Bohr’s power series theorem and local Banach space theory J. reine angew. Math. 557 (2003), 173–197. [7] A. Defant, C. Prengel, Christopher Harald Bohr meets Stefan Banach. Methods in Banach space theory, London Math. Soc. Lecture Note Ser., 337, Cambridge Univ. Press, Cambridge, (2006), 317–339, [8] P.B. Djakov and M.S. Ramanujan A remark on Bohr’s theorem and its generalizations J. Anal. 8 (2000), 65–77. [9] V. Paulsen, G. Popescu, D. Singh On Bohr’s inequality Proc. London Math. Soc. 85 (2002), 493–512. ¨ [10] S. Sidon Uber einen Satz von Herrn Bohr Math. Z. 26 (1927), 731–732. [11] M. Tomic Sur un théorème de H. Bohr Math. Scand. 11 (1962), 103–106. Oscar Blasco Departamento de An´ alisis Matem´ atico Universidad de Valencia E-46100 Burjassot Valencia, Spain e-mail: [email protected]


Spaces of Operator-valued Functions Measurable with Respect to the Strong Operator Topology Oscar Blasco and Jan van Neerven Abstract. Let X and Y be Banach spaces and (Ω, Σ, μ) a finite measure space. In this note we introduce the space Lp [μ; L (X, Y )] consisting of all (equivalence classes of) functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x is strongly μ-measurable for all x ∈ X and ω → Φ(ω)f (ω) belongs to L1 (μ; Y ) for all f ∈ Lp (μ; X), 1/p + 1/p = 1. We show that functions in Lp [μ; L (X, Y )] define operator-valued measures with bounded p-variation and use these spaces to obtain an isometric characterization of the space of all L (X, Y )-valued multipliers acting boundedly from Lp (μ; X) into Lq (μ; Y ), 1 q < p < ∞. Mathematics Subject Classification (2000). 28B05, 46G10. Keywords. Operator-valued functions, operator-valued multipliers, vector measures.

1. Introduction Let (Ω, Σ, μ) be a finite measure space and let X and Y be Banach spaces over K = R or C. In his talk at the 3rd meeting on Vector Measures, Integration and Applications (Eichst¨ att, 2008), Jan Fourie presented some applications of the following extension of an elementary observation due to Bu and Lin [2, Lemma 1.1]. Proposition 1.1. Let Φ : Ω → L (X, Y ) be a strongly μ-measurable function. For all ε > 0 there exists strongly μ-measurable function f ε : Ω → X such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)f ε (ω) + ε. The first named author is partially supported by the Spanish project MTM2008-04594/MTM. The second named author is supported by VICI subsidy 639.033.604 of the Netherlands Organisation for Scientific Research (NWO).

66

O. Blasco and J. van Neerven

Recall that a function φ : Ω → Z, where Z is a Banach space, is said to be strongly μ-measurable if there exists a sequence of Σ-measurable simple functions φn : Ω → Z such that for μ-almost all ω ∈ Ω one has limn→∞ φn (ω) = φ(ω) in Z. In Proposition 1.1, the strong μ-measurability assumption on Φ refers to the norm of L (X, Y ) as a Banach space. The next two examples show that the conclusion of Proposition 1.1 often holds if we impose merely strong μ-measurability of the orbits of Φ. Example 1. Consider X = ∞ (Z), let T be the unit circle, and define Φ : T → ∞ (Z) = L (1 (Z), K)

by Φ(t) := (eint )n∈Z . For all x ∈ 1 (Z) the function t → Φ(t)x = n∈Z xn eint is continuous, but the function t → Φ(t) fails to be strongly measurable. Taking for f the constant function with value u0 ∈ 1 (Z), defined by u0 (0) = 1 and u0 (n) = 0 for n = 0, we have Φ(t) = |Φ(t)f (t)| = | u0 , Φ(t)| = 1 ∀t ∈ T. Example 2. Consider X = C([0, 1]) and define Φ : [0, 1] → M ([0, 1]) = L (C([0, 1]), K) by Φ(t) := δt . For all x ∈ X the function t → Φ(t)x = x(t) is continuous, but the function t → Φ(t) fails to be strongly measurable. If f : [0, 1] → X is a strongly measurable function such that (f (t))(t) = 1 for all t ∈ [0, 1] (e.g., take f (t) ≡ 1), we have Φ(t) = | f (t), Φ(t)| = 1

∀t ∈ [0, 1].

Thus it is natural to ask whether strong μ-measurability of Φ can be weakened to strong μ-measurability of the orbits ω → Φ(ω)x for all x ∈ X, or even to μmeasurability of the functions ω → Φ(ω)x. Although in general the answer is negative even when dim Y = 1 (Example 5), various positive results can be formulated under additional assumptions on X or Φ (Propositions 2.2, 2.4, and their corollaries). One of the applications of Proposition 1.1 was the study of multipliers between spaces of vector-valued integrable functions. In [5], for 1 p, q < ∞, Mult(Lp (μ; X), Lq (μ; Y )) is defined to be the space of all strongly μ-measurable functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X). It is shown (see [5, Proposition 3.4]) that for 1 q < p < ∞ and 1/r = 1/q − 1/p one has a natural isometric isomorphism Mult(Lp (μ; X), Lq (μ; Y )) Lr (μ; L (X, Y )). We observe (Proposition 3.1) that the strong μ-measurability of Φ as function with values in L (X, Y ) is not really needed to define bounded operators from Lp (μ; X) into Lq (μ; Y ); it is possible to weaken the measurability assumptions on the multiplier functions by only requiring strong μ-measurability of its orbits. This will motivate the introduction of an intermediate space between Lp (μ; L (X, Y )) and the space Lps (μ; L (X, Y )) of functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x belongs to Lp (μ; Y ) for all x ∈ X. This is done by selecting the functions in

Spaces of Operator-valued Functions

67

Lps (μ; L (X, Y )) for which ω → Φ(ω)f (ω) belongs to L1 (μ; Y ) for all f ∈ Lp (μ; X), 1/p+1/p = 1. We shall denote this space by Lp [μ; L (X, Y )]. We shall see that, for 1 p < ∞, functions in this space define L (X, Y )-valued measures of bounded p-variation (Theorems 3.5 and 3.8), and prove that one has a natural isometric isomorphism Mult[Lp (μ; X), Lq (μ; Y )] Lr [μ; L (X, Y )], where 1/r = 1/q − 1/p and Mult[Lp (μ; X), Lq (μ; Y )] is defined to be the linear space of all functions Φ : Ω → L (X, Y ) such that ω → Φ(ω)x is strongly μmeasurable for all x ∈ X and ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X) (Theorem 3.6).

2. Strong μ-normability of operator-valued functions Let (Ω, Σ, μ) be a finite measure space and let X and Y be Banach spaces. Definition 2.1. Consider a function Φ : Ω → L (X, Y ). 1. Φ is called strongly μ-normable if for all ε > 0 there exists strongly μmeasurable function f ε : Ω → X such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)f ε (ω) + ε. 2. Φ is called weakly μ-normable if for all ε > 0 there exist strongly μ-measurable functions f ε : Ω → X and g ε : Ω → Y ∗ such that for μ-almost all ω ∈ Ω one has f ε (ω) 1, g ε (ω) 1, and Φ(ω) | Φ(ω)f ε (ω), g ε (ω)| + ε. Clearly, every weakly μ-normable function is strongly μ-normable. In the case Y = K the notions of weak and strong μ-normability coincide and we shall simply speak of normable functions. It will be convenient to formulate our results on μ-normability in the following more general setting. Let S an arbitrary nonempty set. A function f : Ω → S is called a Σ-measurable elementary function ' if for n 1 there exist disjoint sets An ∈ Σ and elements sn ∈ S such that n1 An = Ω and f = n1 1An ⊗ sn . Since no addition is defined in S, this sum should be interpreted as shorthand notation to express that f ≡ sn on An . A function g : S → R is called bounded from above if sups∈S g(s) < ∞. The set of all such functions is denoted by BA (S). Proposition 2.2. Let Φ : Ω → BA (S) be such that for all s ∈ S the function ω → (Φ(ω))(s) is μ-measurable. If there is a countable subset C of S such that for all φ ∈ Φ(Ω) we have sup φ(s) = sup φ(s), s∈S

s∈C

then for all ε > 0 there exists a Σ-measurable elementary function f ε : Ω → S such that for μ-almost all ω ∈ Ω one has sup (Φ(ω))(s) (Φ(ω))(f ε (ω)) + ε. s∈S

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Proof. The function ω → sups∈C (Φ(ω))(s) is μ-measurable, as it is the pointwise supremum of a countable family of μ-measurable functions. Let (s(n) )n1 be an enumeration of C. For n 1 put An := ω ∈ Ω : sup Φ(ω)(s) (Φ(ω))(s(n) ) + ε . s∈S

These sets are μ-measurable, and therefore there exist sets An ∈ Σ such ' 'n that μ(An ΔAn ) = 0. Also, n1 An = Ω. Put B1 := A1 and Bn+1 := An+1 \ m=1 Bn ' for n 1. The sets Bn are Σ-measurable, disjoint. Since B0 := Ω \ n1 Bn is a μ-null set in Σ, the function 1Bn ⊗ s(n) , f ε := n0

where s(0) ∈ S is chosen arbitrarily, has the desired properties.

From this general point of view one obtains the following corollary. Corollary 2.3. Let X and Y be Banach spaces and consider a function Φ : Ω → L (X, Y ). 1. If X is separable and ω → Φ(ω)x is μ-measurable for all x ∈ X, then Φ is strongly μ-normable; 2. If X and Y are separable and ω → | Φ(ω)x, y ∗ | is μ-measurable for all x ∈ X and y ∗ ∈ Y ∗ , then Φ is weakly μ-normable. Proof. To prove part 2 we apply Proposition 2.2 to the set S = BX×Y ∗ (the unit ball of X × Y ∗ with respect to the norm (x, y ∗ ) = max{x, y ∗}) and the functions ω → | Φ(ω)x, y ∗ |, and note that Σ-measurable elementary functions with values in a Banach space are strongly μ-measurable. Since X is separable, for C we may take a set of the form {(xj , yk∗ ) : j, k 1}, where (xj )j1 is a dense sequence in BX and (yk∗ )k1 is a sequence in BY ∗ which is norming for Y . The proof of part 1 is similar. Proof of Proposition 1.1. By assumption, Φ can be approximated μ-almost everywhere by a sequence of simple functions with values in L (X, Y ). Each one of the countably many operators in the ranges of these functions is normed by some ( of X such separable subspace of X. This produces a separable closed subspace X that for μ-almost all ω ∈ Ω, Φ(ω)L (X,Y ) = Φ(ω)L (X,Y ( ). Now we may apply part 1 of Corollary 2.3.

Instead of a countability assumption on the set S we may also impose regularity assumptions on μ and Φ: Proposition 2.4. Let μ be a finite Radon measure on a topological space Ω. Let Φ : Ω → BA (S) be such that for all s ∈ S the function ω → (Φ(ω))(s) is lower


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semicontinuous. Then for all ε > 0 there exists a Borel measurable elementary function f ε : Ω → S such that for μ-almost all ω ∈ Ω one has sup (Φ(ω))(s) (Φ(ω))(f ε (ω)) + ε. s∈S

Proof. Let us first note that the function m(ω) := sup(Φ(ω))(s) s∈S

is lower semicontinuous, since it is the pointwise supremum of a family of lower semicontinuous functions. In particular, m is Borel measurable. Fix ε > 0. Using Zorn’s lemma, let (Ωi )i∈I be a maximal collection of disjoint Borel sets such that the following two properties are satisfied for all i ∈ I: (a) μ(Ωi ) > 0; (b) there exists si ∈ S such that m(ω) (Φ(ω))(si ) + ε for all ω ∈ Ωi . Clearly, (a) implies that the index set I is countable. We claim that ) μ Ω\ Ωi = 0. i∈I

The proof is then finished by taking f ε := i∈I 1Ωi ⊗ si and extending this definition to the remaining Borel μ-null set by assigning an arbitrary constant value on it; by (b) and the claim, this function satisfies the required inequality μ-almost everywhere. ' To prove the claim let Ω := Ω\ i∈I Ωi and suppose, for a contradiction, that μ(Ω ) > 0. By passing to a Borel subset of Ω we may assume that supω ∈Ω m(ω ) < ∞. Let M := ess supω ∈Ω m(ω ). The set A := {ω ∈ Ω : m(ω ) M − 13 ε} is Borel and satisfies μ(A) > 0. Since μ is a Radon measure we may select a compact set K in Ω such that K ⊆ A and μ(K) > 0. For any ω ∈ K we can find s ∈ S such that m(ω ) (Φ(ω ))(s ) + 13 ε. By lower semicontinuity, the set O := ω ∈ Ω : (Φ(ω ))(s ) < (Φ(ω))(s ) + 13 ε is open and contains ω . Choosing such an open set for every ω ∈ K, we obtain an open cover of K, which therefore has a finite subcover. At least one of the finitely many open sets of this subcover intersects K in a set of positive measure. Hence, there exist ω0 ∈ K and s0 ∈ S, as well as an open set O0 ⊆ Ω such that ω0 ∈ O0 , μ(K ∩ O0 ) > 0, m(ω0 ) (Φ(ω0 ))(s0 ) + 13 ε, and (Φ(ω0 ))(s0 ) < (Φ(ω))(s0 ) + 13 ε

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for all ω ∈ O0 . Hence, for μ-almost all ω ∈ K ∩ O0 , m(ω) − 13 ε M − 13 ε m(ω0 ) (Φ(ω0 ))(s0 ) + 13 ε < (Φ(ω))(s0 ) + 23 ε. It follows that the Borel set (K ∩ O0 ) \ N , where N is some Borel set satisfying μ(N ) = 0, may be added to the collection (Ωi )i∈I . This contradicts the maximality of this family. Corollary 2.5. Let μ be a finite Radon measure on a topological space Ω and let X and Y be Banach spaces. Consider a function Φ : Ω → L (X, Y ). 1. If ω → Φ(ω)x is lower semicontinuous for all x ∈ X, then Φ is strongly μ-normable. 2. If ω → | Φ(ω)x, y ∗ | is lower semicontinuous for all x ∈ X and y ∗ ∈ Y ∗ , then Φ is weakly μ-normable. Here are two further examples. Example 3. Consider Ω = (0, 1), X = L1 (0, 1), Y = K, and let Φ : (0, 1) → L∞ (0, 1) = L (L1 (0, 1), K) be defined by Φ(t) := 1(0,t) . For all x ∈ L1 (0, 1) the t function t → Φ(t)x = 0 x(s) ds is continuous. Corollary 2.5 asserts that Φ is normable. In fact, for f (t) := 1t 1(0,t) one even has Φ(t) = |Φ(t)f (t)| = 1

∀t ∈ (0, 1).

Example 4. Let X1 , X2 be Banach spaces and let T : X1 → X2 be a bounded linear operator with T = 1. Consider Ω = [0, 1], X = C([0, 1], X1 ), Y = X2 and let Φ : Ω → L (X, Y ) be defined by Φ(t) := Tt , where Tt (x) = T (x(t)) for x ∈ X. For all x ∈ X the function t → Tt x is continuous. Corollary 2.5 asserts that Φ is weakly (and hence strongly) normable. In fact, for each ε > 0 and t ∈ [0, 1] we can select xε ∈ BX1 and y ∗ε ∈ BX2∗ such that | T xε , y ∗ε | > 1 − ε. Defining f ε := 1 ⊗ xε and g ε := 1 ⊗ y ∗ε one has Φ(t) | Φ(t)f ε (t), g ε (t)| + ε

∀t ∈ [0, 1].

In the Examples 1, 2 and 3 the norming was exact. The next proposition formulates a simple sufficient (but by no means necessary) condition for this to be possible: Proposition 2.6. Let X and Y be Banach spaces and consider a function Φ : Ω → L (X, Y ). 1. Suppose that Φ : Ω → L (X, Y ) is strongly μ-normable. If X is reflexive, there exists a strongly μ-measurable function f : Ω → X such that for μ-almost all ω ∈ Ω one has f (ω) 1 and Φ(ω) = Φ(ω)f (ω). 2. Suppose that Φ : Ω → L (X, Y ) is weakly μ-normable. If X and Y are reflexive, there exist strongly μ-measurable functions f : Ω → X and g : Ω → Y ∗ such that for μ-almost all ω ∈ Ω one has f (ω) 1, g(ω) 1, and Φ(ω) = | Φ(ω)f (ω), g(ω)|.


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Proof. We shall prove part 1, the proof of part 2 being similar. For every n 1 choose a strongly μ-measurable function fn : Ω → X such that for μ-almost all ω ∈ Ω one has fn (ω) 1 and Φ(ω) Φ(ω)fn (ω) + n1 . 2 Since μ is finite, the sequence (fn )∞ n=1 is bounded in the reflexive space L (μ; X) and therefore it has a weakly convergent subsequence (fnk )∞ . Let f be its weak k=1 limit. By Mazur’s theorem there exist convex combinations gj in the linear span of 1 (fnk )∞ k=j such that gj − f < j . By passing to a subsequence we may assume that limj→∞ gj = f μ-almost surely. Clearly, for μ-almost all ω ∈ Ω one has gj (ω) 1 and Φ(ω) Φ(ω)gj (ω) + n1j .

The result follows from this by passing to the limit j → ∞.

The following example shows that the separability condition of Proposition 2.2 and the lower semicontinuity assumption of Proposition 2.4 and its corollaries cannot be omitted, even when X is a Hilbert space and Y = K. Example 5. Let Ω = (0, 1), X = l2 (0, 1), and Y = K. Recall that l2 (0, 1) is the Banach space of all functions φ : (0, 1) → R such that φ2 := sup |φ(t)|2 < ∞, U ∈U

t∈U

where U denotes the set of all finite subsets of (0, 1). Note that for all φ ∈ l2 (0, 1) the set of all t ∈ (0, 1) for which φ(t) = 0 is at most countable; this set will be referred to as the support of φ. Define Φ : (0, 1) → L (l2 (0, 1), K) by Φ(t)φ := φ(t). Clearly, Φ(t) = 1 for all t ∈ (0, 1). Also, Φ(t)φ = 0 for all t outside the countable support of φ and therefore this function is always measurable. Suppose now that a strongly measurable function f : (0, 1) → l2 (0, 1) exists such that 1 |Φ(t)f (t)| + 12 for almost all t ∈ (0, 1). Let N be a null set such that this inequality holds for all t ∈ (0, 1) \ N . For t ∈ (0, 1) \ N it follows that |(f (t))(t)| 12 . Let fn : (0, 1) → l2 (0, 1) be simple functions such that limn→∞ fn = f pointwise almost everywhere, say on (0, 1) \ N for some null set N . The range of each fn consists of finitely many elements of l2 (0, 1), each of which has countable support. Therefore there exists a countable set B ⊆ (0, 1) such that the support of f (t) is contained in B for all t ∈ (0, 1) \ N . For t ∈ (0, 1) \ (N ∪ N ), the inequality |(f (t))(t)| 12 implies that t ∈ B. Hence, (0, 1) \ (N ∪ N ) ⊆ B, a contradiction.

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3. Spaces of operator-valued functions Throughout this section, (Ω, Σ, μ) is a finite measure space and X and Y are Banach spaces. We introduce the linear spaces M (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φ is strongly μ-measurable }, Ms (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φx is strongly μ-measurable ∀x ∈ X}, Mw (μ; L (X, Y )) := {Φ : Ω → L (X, Y ) : Φx is weakly μ-measurable ∀x ∈ X}. Two functions Φ1 and Φ2 in M (μ; L (X, Y )) are identified when Φ1 = Φ2 μ-almost everywhere, two functions Φ1 and Φ2 in Ms (μ; L (X, Y )) are identified when Φ1 x = Φ2 x μ-almost everywhere for all x ∈ X, and Φ1 and Φ2 in Mw (μ; L (X, Y )) are identified when Φ1 x, y ∗ = Φ2 x, y ∗ μ-almost everywhere for all x ∈ X and y∗ ∈ Y ∗. As special cases, for X = K we put M (μ; X) := M (μ; L (K, X)) (which coincides with Ms (μ; L (K, X))) and Mw (μ; X) := Mw (μ; L (K, X)). The following easy fact will be useful below. Proposition 3.1. For Φ ∈ Ms (μ; L (X, Y )) and f ∈ M (μ; X), g(ω) := Φ(ω)f (ω) defines a function g ∈ M (μ; Y ). Proof. For simple functions f this is clear. The general case follows from this, using that μ-almost everywhere limits of strongly μ-measurable functions are strongly μ-measurable. For 1 p ∞ we consider the normed linear spaces Lp (μ; L (X, Y )) := Φ ∈ M (μ; L (X, Y )) : ΦLp(μ;L (X,Y )) < ∞ , Lps (μ; L (X, Y )) := Φ ∈ Ms (μ; L (X, Y )) : ΦLps (μ;L (X,Y )) < ∞ , Lpw (μ; L (X, Y )) := Φ ∈ Mw (μ; L (X, Y )) : ΦLpw (μ;L (X,Y )) < ∞ , where

1/p Φ(ω)p dμ(ω) , Ω 1/p := sup Φ(ω)xp dμ(ω) ,

ΦLp(μ;L (X,Y )) := ΦLps (μ;L (X,Y ))

x 1

ΦLpw (μ;L (X,Y )) := sup

Ω

sup

x 1 y ∗ 1

1/p | Φ(ω)x, y ∗ |p dμ(ω) ,

Ω

with the obvious modifications for p = ∞. As special cases we write Lp (μ; X) := Lp (μ; L (K, X)) = Lps (μ; L (K, X)) and Lpw (μ; X) := Lpw (μ; L (K, X)). Note that all these definitions agree with the usual ones.


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Let us recall some spaces of vector measures that are used in the sequel. The reader is referred to [3] and [4] for the concepts needed in this paper. Fix 1 p ∞ and let E be a Banach space. We denote by V p (μ; E) the Banach space of all vector measures F : Σ → E for which 1 (1A ⊗ F (A)) < ∞, F V p (μ;E) := sup μ(A) Lp (μ;E) π∈P(Ω) A∈π

where P(Ω) stands for the collection of all finite partitions of Ω into disjoint sets of strictly positive μ-measure. Similarly we denote by Vwp (μ; E) the Banach spaces of all vector measures F : Σ → E for which 1 F Vwp (μ;E) := sup (1A ⊗ F (A)) p < ∞. μ(A) Lw (μ;E) π∈P(Ω) A∈π

In both definitions of the norm we make the obvious modification for p = ∞. Note that F V 1 (μ;E) and F Vw1 (μ;E) equal the variation and semivariation of F with respect to μ, respectively. It is well known that for 1 p < ∞ and 1/p + 1/p = 1 one has a natural isometric isomorphism

(Lp (μ; E))∗ V p (μ; E ∗ ). We now concentrate on the case E = L (X, Y ). For each Φ ∈ L1 (μ; L (X, Y )) one may define a vector measure F : Σ → L (X, Y ) by F (A) := Φ dμ A

which satisfies F V 1 (μ;L (X,Y )) = ΦL1 (μ;L (X,Y )) . In the next proposition we extend this definition to functions Φ ∈ Lps (μ; L (X, Y )), 1 < p < ∞. The case p = 1 will be addressed in Remark 3.3 and Theorem 3.8. Proposition 3.2. Assume that Φ ∈ Lps (μ; L (X, Y )) for some 1 < p < ∞. Define F : Σ → L (X, Y ) by F (A)x := Φ(ω)x dμ(ω), x ∈ X. A

Then F is an L (X, Y )-valued vector measure and, for any q ∈ [1, p], one has F Vwq (μ;L (X,Y )) ΦLqs (μ;L (X,Y )) . Proof. Let us first prove that F is countably additive. Let (An )n1 be a sequence of ' pairwise disjoint sets in Σ and let A = n1 An . Put T := F (A) and Tn := F (An ).

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Then, N N Tn = sup T x − Tn x T − n=1

x =1

n=1

= sup ' x =1

sup

x =1

nN +1

An

1/p Φ(ω)xp dμ(ω) μ

Ω

ΦLp(μ;L (X,Y )) Hence T =

Φ(ω)x dμ(ω) )

1/p An

nN +1

μ

)

1/p An

.

nN +1

n1

Tn in L (X, Y ). Next,

| F (A), e∗ |q 1/q q−1 π∈P(Ω) e∗ =1 A∈π (μ(A)) # * F (A) +# # ∗ # = sup sup sup αA , e # # (μ(A))1/q π∈P(Ω) e∗ =1 (αA ) q =1 A∈π F (A) sup αA = sup (μ(A))1/q L (X,Y ) π∈P(Ω) (αA ) q =1 A∈π F (A) = sup sup sup αA x (μ(A))1/q π∈P(Ω) (αA ) q =1 x =1 A∈π 1A = sup Φ(ω)x dμ(ω) sup sup αA 1/q (μ(A)) π∈P(Ω) (αA ) q =1 x =1 Ω A∈π 1/q sup Φ(ω)xq dμ(ω)

F Vwq (μ;L (X,Y )) = sup

x =1

sup

Ω

= ΦLqs (μ;L (X,Y )) .

Remark 3.3. The same results holds for functions Φ ∈ L1s (μ; L (X, Y )) provided the family {ω → Φ(ω)x : x ∈ BX } is equi-integrable in L1 (μ; X). The next definition introduces a new class of Banach spaces intermediate between Lp (μ; L (X, Y )) and Lps (μ; L (X, Y )). Definition 3.4. For 1 p ∞ we consider the Banach space Lp [μ; L (X, Y )] := {Φ ∈ Ms (μ; L (X, Y )) : ΦLp[μ;L (X,Y )] < ∞}, where

ΦLp[μ;L (X,Y )] :=

Φ(ω)f (ω) dμ(ω).

sup f Lp (μ;X) =1

Ω


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It is clear that Lp (μ; L (X, Y )) → Lp [μ; L (X, Y )] → Lps (μ; L (X, Y )) with contractive inclusion mappings. Using these spaces we can prove the following improvement of Proposition 3.2. Theorem 3.5. Let 1 < p < ∞. Then Lp [μ; L (X, Y )] → V p (μ; L (X, Y )) and the inclusion mapping is contractive. Proof. Using the inclusion into Lp [μ; L (X, Y )] → Lps (μ; L (X, Y )), from Proposition 3.2 we see that F (A)x := A Φ(ω)x dμ(ω) defines a vector measure F : Σ → L (X, Y ). Now, if π ∈ P(Ω), then for ε > 0 and each A ∈ π there exist xA ∈ BX and ∗ yA ∈ BY ∗ so that #, -#p ε # p ∗ # . Φ(ω)xA dμ(ω), yA F (A) < # # + card(π) A Hence, F (A)p (μ(A))p−1 A∈π #, -#p 1 # ∗ # Φ(ω)xA dμ(ω), yA # # +ε p−1 (μ(A)) A A∈π , -p 1 ∗ Φ(ω)x dμ(ω), β y +ε sup A A A 1/p (βA ) p =1 A∈π (μ(A)) A , p βA xA ∗ Φ(ω) dμ(ω) 1A ⊗ , 1 ⊗ y +ε sup A A (μ(A))1/p A∈π (βA ) p =1 Ω A∈π p βA xA dμ(ω) 1A ⊗ +ε sup Φ(ω) (μ(A))1/p (βA ) p =1 Ω A∈π p Φ(ω)f (ω) dμ(ω) + ε sup f Lp (μ;X) =1

Ω

ΦpLp [μ;L (X,Y )]

+ ε.

Since ε > 0 was arbitrary, this gives the result.

For 1 p, q < ∞ we define Mult[Lp (μ; X), Lq (μ; Y )] to be the linear space of all Φ ∈ Ms (μ; L (X, Y )) such that ω → Φ(ω)f (ω) belongs to Lq (μ; Y ) for all f ∈ Lp (μ; X). By a closed graph argument the linear operator

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MΦ : f → Φf is bounded, and the space Mult[Lp (μ; X), Lq (μ; Y )] is a Banach space with respect to the norm ΦMult[Lp (μ;X),Lq (μ;Y )] := MΦ L (Lp (μ;X),Lq (μ;Y )) . We refer to [5] for further details and and some results on spaces of multipliers between different spaces of vector-valued functions, extending those proved in [1] for sequence spaces. Theorem 3.6. Let X and Y be Banach spaces and let 1 q < p < ∞. We have a natural isometric isomorphism Mult[Lp (μ; X), Lq (μ; Y )] Lr [μ; L (X, Y )], where 1/r = 1/q − 1/p. Proof. The case q = 1 corresponds to r = p and the result is just the definition of the space Lp [μ; L (X, Y )]. Assume 1 < q < p and Φ ∈ Lr [μ; L (X, Y )]. Let f ∈ Lp (μ; X). Then for any φ ∈ Lq (μ) we have that ω → f (ω)φ(ω) belongs to Lr (μ; X). Hence Φ(ω)f (ω)|φ(ω)| dμ(ω) ΦLr [μ;L (X,Y )] φLq (μ) f Lp(μ;X) . Ω

Taking the supremum over the unit ball of Lq (μ) the first inclusion is achieved. Conversely, let Φ ∈ Mult[Lp (μ; X), Lq (μ; Y )]. Let g ∈ Lr (μ; X), and choose ψ ∈ Lq (μ) and f ∈ Lp (μ; X) in such a way that g = ψf and gLr (μ;X) = ψLq (μ) f Lp(μ;X) . Now observe that Φ(ω)g(ω) = ψ(ω)Φ(ω)f (ω) ∈ L1 (μ; Y ) and Φ(ω)g(ω) dμ(ω) ψLq (μ) ΦMult[Lp (μ;X),Lq (μ;Y )] f Lp(μ;X) .

Ω

The next result establishes a link with the notion of strong μ-measurability. Proposition 3.7. Let X be a Banach space, 1 p ∞, and let Φ ∈ Lp [μ; L (X, Y )] be strongly μ-normable. Then ω → Φ(ω) belongs to Lp (μ) and 1/p Φ(ω)p dμ(ω) ΦLp [μ;L (X,Y )] . Ω

Proof. By assumption, for any ε > 0 there exists f ε ∈ M (μ; X) such that for μ-almost all ω ∈ Ω one has f ε (ω) 1 and Φ(ω) Φ(ω)(f ε (ω)) + ε. If εn ↓ 0, then for μ-almost all ω ∈ Ω Φ(ω) = lim Φ(ω)f εn (ω). n→∞

The strong μ-measurability of ω → Φ(ω)x for all x ∈ X implies the strong μmeasurability of the functions ω → Φ(ω)f εn (ω). It follows that ω → Φ(ω) is μ-measurable.


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Let φ ∈ Lp (μ) and consider ω → φ(ω)f ε (ω) ∈ Lp (μ; X). Then Φ(ω)|φ(ω)| dμ(ω) Φ(ω)(φ(ω)f ε (ω)) dμ(ω) + εφL1 (μ) Ω

Ω

ΦLp[μ;L (X,Y )] φLp (μ) + εφL1 (μ) .

Since ε > 0 was arbitrary, this gives the result.

By invoking Proposition 2.2 we shall now deduce some further results under the assumption that the space X is separable. The first should be compared the remarks preceding Proposition 3.2 (where functions Φ ∈ L1 (μ; L (X, Y )) are considered) and Remark 3.3 (where functions Φ ∈ L1s (μ; L (X, Y )) are considered). Theorem 3.8. Let X be a separable Banach space and let Φ ∈ L1 [μ; L (X, Y )] be given. Define F : Σ → L (X, Y ) by F (A)x := Φ(ω)x dμ(ω), x ∈ X. A

Then F is an L (X, Y )-valued vector measure and F V 1 (μ;L (X,Y )) ΦL1 [μ;L (X,Y )] . Proof. First we prove that F is countably ' additive. Let (An )n1 be a sequence of pairwise disjoint sets in Σ and let A = n1 An . Put T := F (A) and Tn := F (An ). Combining Proposition 2.2 and Proposition 3.7 one obtains that Φ ∈ L1 (μ). Hence, N N Tn = sup T x − Tn x T − n=1

x =1

n=1

= sup ' x =1

nN +1

Φ(ω)x dμ(ω)

An

Φ(ω) dμ(ω).

'

An

Hence T = n1 Tn in L (X, Y ). Next, using that F (A) A Φ(ω) dμ(ω), from Proposition 3.7 we conclude that F (A) ΦL1 [μ;L (X,Y )] . F V 1 (μ;L (X,Y )) = sup nN +1

π∈P(Ω) A∈π

Our final result extends the factorization result that was used in the proof of Theorem 3.6. Theorem 3.9. Let 1 p1 , p2 , p3 < ∞ satisfy 1/p1 = 1/p2 +1/p3 and let X be a separable Banach space. A function Φ ∈ Ms (μ; L (X, Y )) belongs to Lp1 [μ; L (X, Y )] if and only if Φ = ψΨ for suitable functions ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )]. In this situation we may choose ψ and Ψ in such a way that ΦLp1 [μ;L (X,Y )] = ψLp2 (μ) ΨLp3 [μ;L (X,Y )] .

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Proof. To prove the ‘if’ part let Φ ∈ Lp1 [μ; L (X, Y )]. Using Proposition 3.7 together with Proposition 2.2 one has that Φ ∈ Lp1 (μ). Put $ Φ(t) if Φ(t) = 0, Φ(t)p1 /p3 Φ(t) p1 /p2 ψ(t) := Φ(t) , Ψ(t) := 0 if Φ(t) = 0.

Clearly ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )]. Now for each g ∈ Lp3 (μ; X), invoking Proposition 3.1, one has that Ψg ∈ M (μ, Y ) and Ψ(t)g(t) Φ(t)p1 /p3 g(t). Hence the right-hand side defines a function in L1 (μ) and therefore Ψg ∈ L1 (μ, Y ). The above decomposition satisfies the required identity for the norms. To prove the ‘only if’ part let ψ ∈ Lp2 (μ) and Ψ ∈ Lp3 [μ; L (X, Y )] be given. For each f ∈ Lp1 (μ; X) we have ψf ∈ Lp3 (μ; X). Hence Ψ(ψf ) ∈ L1 (μ; Y ).

References [1] J.L. Arregui, O. Blasco, (p, q)-Summing sequences of operators. Quaest. Math. 26 (2003), no. 4, 441–452. [2] Q. Bu and P.-K. Lin, Radon-Nikodym property for the projective tensor product on K¨ othe function spaces. J. Math. Anal. Appl. 293 (2004), no. 1, 149–159. [3] J. Diestel, J.J. Uhl, Vector measures. Mathematical Surveys 15, Amer. Math. Soc., Providence (1977). [4] N. Dinculeanu Vector measures. Pergamon Press, New York (1967). [5] J.H. Fourie and I.M. Schoeman, Operator-valued integral multiplier functions. Quaest. Math. 29 (2006), no. 4, 407–426. Oscar Blasco University of Valencia Departamento de An´ alisis Matem´ atico E-46100 Burjassot, Valencia, Spain e-mail: [email protected] Jan van Neerven Delft University of Technology Delft Institute of Applied Mathematics P.O. Box 5031 NL-2600 GA Delft, The Netherlands e-mail: [email protected]


Defining Limits by Means of Integrals Antonio Boccuto and Domenico Candeloro Abstract. A particular notion of limit is introduced, for Riesz space-valued functions. The definition depends on certain ideals of subsets of the domain. It is shown that, according with our definition, every bounded function with values in a Dedekind complete Riesz space admits limit with respect to any maximal ideal. Mathematics Subject Classification (2000). Primary 28B15; Secondary 46G10. Keywords. Riesz space, ideal convergence, integral.

1. Introduction In this paper a definition of convergence with respect to ideals is introduced. To every ideal I of subsets of an abstract non-empty set T a dual filter is associated in a natural way: it is the family of the complements of all elements of I. If the dual filter is an ultrafilter, then the involved ideal is maximal, and vice versa. It is shown that every bounded map, taking values in any Dedekind complete Riesz space, admits a “limit” with respect to any maximal ideal. This result can be achieved in the real-valued case by simply integrating with respect to a suitable “ultrafilter measure”, since all bounded functions are integrable. In the general case, the goal is obtained by using the powerful tools of the Chojnacki integral (see [3]) and the representation of Riesz spaces as ideals of suitable spaces of continuous functions (Maeda-Ogasawara-Vulikh theorem, [7, 8, 9]). Applications are given in finding extensions of finitely additive measures.

2. Preliminaries We begin with introducing the following basic notions (see also [4, 5, 6]). Definition 2.1. Let I denote any fixed ideal of subsets of an abstract set T . We say that I is admissible if it contains all finite subsets of T . In particular, the ideal consisting precisely of the finite subsets of T will be denoted by If in , and of course

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A. Boccuto and D. Candeloro

is the minimal admissible ideal. So, any admissible ideal must contain If in . On the opposite side, I is called maximal if no proper ideal in T strictly contains I. Since the family of all complements of elements from an ideal I forms a filter F , called the dual filter of I, we can say that I is admissible if and only if, for every element t0 ∈ T , the set T \ {t0 } belongs to the dual filter of I: filters of this kind are also called free. On the other hand, I is maximal if and only if its dual filter U is an ultrafilter: any subset of T either belongs to I or to U. Thanks to the Axiom of Choice, it is well known that any filter is contained in some ultrafilter (and therefore any ideal is contained in a maximal ideal). We will deal with two-valued additive set functions P : P(T ) → {0, 1} with P (T ) = 1. Such maps are also called ultrafilter measures: indeed, the family of all sets U such that P (U ) = 1 is an ultrafilter in T and consequently the family of all P -null sets is a maximal ideal. Assume now that I is any admissible ideal in an abstract set T , and let f : T → R be any real-valued mapping. We say that f is I-convergent to an element x ∈ R, if for every ε > 0 the set {t ∈ T : |f (t) − x| > ε} belongs to I. When this is the case, we also write x = I − limt∈T f (t). It is well known that the I-limit is unique (if it exists), and enjoys all linearity and monotonicity properties of usual limits, see [5]. However we mainly will be concerned with ideal convergence in the case of maximal ideals. Indeed, we have (see also [1]): Proposition 2.2. Let P : P(T ) → {0, 1} be an ultrafilter measure, and f : T → R any bounded function. Then, if I denotes the ideal of P -null sets, the function f has limit with respect to I, and we have I − lim f (t) = f (t) dP. t∈T

T

Conversely, given any maximal ideal I of subsets of T , any bounded mapping f : T → R has limit w.r.t. I, and this limit is nothing but the integral of f with respect to a suitable ultrafilter measure. Proof. Let us denote by U the dual filter of I. Then it is easy to see that sup inf f (t) = inf sup f (t)

U ∈U t∈U

U ∈U t∈U

and that the common value is the integral T f (t) dP : we shall denote by J this quantity. Now, fix arbitrarily ε > 0, and set Aε := {t ∈ T : |f (t)−J| > ε}. We claim that Aε belongs to I: indeed, if Aε ∈ U we should have either inf U∈U supt∈U ≤ J −ε or supU ∈U inf t∈U ≥ J + ε, and both these cases are impossible. Hence the first part is proved. Conversely, given any maximal ideal I in T and any bounded function f : T → R, the I-limit of f is nothing but the integral w.r.t. the (unique) ultrafilter measure whose null sets are precisely the elements of I.

Defining Limits by Means of Integrals

81

In the sequel, we shall extend our notion of limit to the case of Riesz spacevalued mappings, and show that any bounded function taking values in every Dedekind complete Riesz space has limit (at least in a weak sense) when I is maximal.

3. I-convergence in Riesz spaces From now on we shall assume that R is a Dedekind complete Riesz space. Our purpose here is to show that, if f : T → R is any bounded map, then, for every maximal ideal I in T , f has I-limit in R, at least in a weak sense. We start with the following definition. Definition 3.1. Let R be any Dedekind complete Riesz space, and f : T → R be any map. Given an ideal I of subsets of T , we say that f has I-limit in R if there exist an element l in R and a decreasing net (rλ )λ in R, with inf λ rλ = 0 and such that for each λ the set {t ∈ T : |f (t) − l| ≤ rλ } is the complement of some element N ∈ I. In other terms, we have I − lim f (t) = l if and only if inf sup f (t) = l = sup inf f (t),

U ∈U t∈U

U ∈U t∈U

where U is the dual filter of I. We shall say that a Dedekind complete Riesz space R has the strong limit property if, for any abstract set T , any bounded mapping f : T → R, and any maximal ideal I in T , there exists in R the I-limit of f . We first observe that the space R has the strong limit property, thanks to 2.2. From this, it clearly follows that the space RD , for any discrete space D, has the strong limit property, when endowed with the natural order and algebraic structure. We shall now give a further example. Let us assume that an abstract discrete space D is fixed, and denote by C(βD) the space of all continuous mappings ψ : βD → R, where βD denotes the ˇ Stone-Cech compactification of D. The space C(βD) has a natural ordering, and is stable under arbitrary suprema and infima of bounded subsets. Given any bounded map ψ : D → R, we denote by ψ the unique continuous extension of ψ to βD. Moreover, given a bounded family (ψh )h∈H of mappings from D to R, the notation suph ψh always means the pointwise supremum; in case the. mappings ψh are elements from C(βD), the lattice supremum will be denoted by h∈H ψh . Similar notations will be adopted for infima.

82


Now, it is not difficult to check that, given any uniformly bounded family (ψh )h∈H of functions defined on D and taking values in R, we always have

/ 0 ψ h (d) = sup ψh (d), ψ h (d) = inf ψh (d), h∈H

h∈H

h∈H

h∈H

for all d ∈ D. Fix any abstract set T , together with an ultrafilter U of subsets of T , and choose any bounded mapping f : T → C(βD). We shall state the following result, whose proof now is straightforward. Lemma 3.2. Let us set: J1 :=

0 /

f (t),

/ 0

J2 :=

U ∈U t∈U

f (t),

U∈U t∈U

where the involved suprema and infima are taken in C(βD). Then we have J1 = J2 . The assertion of Lemma 3.2 is precisely that C(βD) has the strong limit property. We now turn to the concept of weak limit. To this aim, we need some definitions, concerning embeddings of a Dedekind complete Riesz space R. Definition 3.3. We say that R is embedded in another Dedekind complete Riesz space R∗ if there exist a one-to-one Riesz homomorphism (called embedding) j : R → R∗ such that j(R) is a Dedekind complete subspace of R∗ , and an onto homomorphism π : R∗ → R (called projection) such that π ◦ j is the identity map on R. When this happens, we also say that R∗ is an extension of R. In general, embeddings reduce the gap existing between the two quantities 0 / / 0 f (t), and f (t), F ∈F t∈F

F ∈F t∈F

whenever f : T → R is a bounded map, and F any filter in T : indeed, we have Theorem 3.4. Assume that f : T → R is a bounded map, and F is any filter in T . If R∗ is any extension of R, with embedding j and projection π, then we always have

/ 0 / 0 f (t) ≤ π j(f (t)) F ∈F t∈F

≤π

0 / F ∈F t∈F

F ∈F t∈F

j(f (t))

≤

0 /

f (t).

F ∈F t∈F

The proof is straightforward. Let us turn now to the definition of maximal extension.


83

Definition 3.5. Let R be any Dedekind complete Riesz space, and R∗ any extension of R. We say that the extension of R into R∗ is maximal if, for every abstract set T , every ultrafilter U in T , and any bounded mapping f : T → R, it holds:

0 / / 0 π j(f (t)) = π j(f (t)) . U ∈U t∈U

U∈U t∈U

If this happens, then the common value will be called weak limit of f with respect to the dual ideal I of U. In particular, if R has an extension R∗ with the strong limit property, then R is obviously a maximal extension. We shall see later that this is the case, for example, if R has a strong unit, i.e., a strictly positive element u such that for every x ∈ R there is a positive real number λ > 0 for which |x| ≤ λu. More generally, given a Riesz space R, we say that a bounded mapping f : T → R has weak limit with respect to some ideal I in T , if the range of f is contained in a Dedekind complete space R0 with a maximal extension R0∗ , and

/ 0 0 / j(f (t)) = π j(f (t)) , π ∗

F ∈F t∈F

F ∈F t∈F

where F is the dual filter of I. In this case, the weak limit, i.e., the common value, is an element l of R0 , and we write l := (w) − I − lim f (t). Thanks to 3.4, we can see that, in case f : T → R is a bounded map, and I is any ideal in T , the existence of the strong I-limit implies that any weak limit coincides with it. Moreover, it is not difficult to check that the weak limit, when existing, enjoys the usual linearity properties, provided it is performed always with respect to the same maximal extension. We now turn to show that a weak I-limit always exists, as soon as f : T → R is a bounded mapping and I is a maximal ideal in T . To this aim, we recall the following version of the Maeda-Ogasawara-Vulikh Theorem (see [7]; [9], Theorems V.3.1, p. 131 and V.4.2, p. 138). Theorem 3.6. Every Dedekind complete Riesz space R is algebraically and lattice ( Ω : f is continuous, and isomorphic to an order dense ideal of C∞ (Ω) = {f ∈ R {ω ∈ Ω : |f (ω)| = +∞} is nowhere dense in Ω}, where Ω is a suitable compact Hausdorff extremely disconnected topological space. Moreover, if R has a strong order unit, then it is algebraically and lattice isomorphic to C(Ω) = {f ∈ RΩ : f is continuous}. We also recall the following well-known result, which in turn is related to the Maeda-Ogasawara-Vulikh representation theorem (see [3, 8]).

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Theorem 3.7. Let Ω be any Hausdorff, compact, extremely disconnected space, and denote by D the support set of Ω, endowed with the discrete topology. Then, Ω ˇ (with its original topology) is a subspace of the Stone-Cech compactification βD and there exists a continuous onto mapping r : βD → Ω, (retract) such that r|Ω is the identity map. Thanks to the previous theorems, we can deduce that, if R is a Dedekind complete Riesz space endowed with a strong order unit, then R admits a maximal extension. Indeed, we can view R as the space C(Ω), for a suitable Stone space Ω, and C(Ω) is embedded in C(βD), according with Theorem 3.7: indeed, for any mapping φ ∈ C(Ω) we can define j(φ) : βD → R as follows: j(φ)(ξ) = φ(r(ξ)), for all ξ ∈ βD, and also we can define a projection π : C(βD) → C(Ω) as: π(ψ) := ψ|Ω for all ψ ∈ C(βD). As already observed in Lemma 3.2, the space C(βD) has the strong limit property, hence the claim is proved. We now turn to the general case of a bounded map f , taking values in a Dedekind complete Riesz space R. Let us fix a Dedekind complete Riesz space R, and fix any abstract set T , together with a maximal ideal I of subsets of T , and its dual ultrafilter U. Theorem 3.8. Let f : T → R be any bounded mapping. Then there exists a weak I-limit of f , in the sense of Definition 3.5. Proof. We note that, if f : T → R is bounded and e is the least upper bound of the range of |f |, then the range of f is contained in the vector lattice V [e] := {b ∈ R : there exists λ > 0 with |b| ≤ λ e}. Since R is Dedekind complete, V [e] is too, and hence V [e] is a Dedekind complete Riesz space with a strong order unit. So, for the previous remark, V [e] has a maximal extension, and this shows that f has weak limit in V [e], hence in R. More precisely, as soon as f : T → R is a bounded map, and e is any positive element of R dominating |f |, we can choose the space C(βD) as an extension of V [e], and define f0 (t)(ξ) = f (t)(r(ξ)) for all t ∈ T and ξ ∈ βD; then, denoting by l the strong limit of f0 , the weak limit of f is π(l), i.e., l|Ω , where Ω is the Stone space such that V [e] is isomorphic to C(Ω).

4. Applications In this section we shall prove that I-limits can be used to extend measures. Definition 4.1. Given an abstract non-empty set T , and fixed a non-trivial algebra A of subsets of T , let us denote by D the family of all finite tagged partitions of T into non-empty subsets from A: this means that every element Π ∈ D is a finite collection of pairwise disjoint elements from A, say A1 , . . . , Ak , (which we shall call intervals), whose union is T , with attached a corresponding finite number of elements t1 , . . . , tk from T , such that ti ∈ Ai for all i = 1, . . . , k.


85

Fixed two such partitions, Π1 and Π2 , we say that Π2 refines Π1 if any interval from Π2 is a subset of some interval from Π1 (without any condition on the attached points). Fixed any partition Π ∈ D, we denote by D(Π) the family of all partitions which refine Π. Now, if we consider D as an abstract set, the families D(Π) form, as Π runs in D, a filter basis in D: this means that, if we collect all the families (of subsets of D), which contain some family of the type D(Π), they form a filter in D. This filter is called the refinement filter of D, based on A. Assume now that a non-empty abstract set T is fixed, with an algebra A of subsets of T . Let us suppose that a bounded finitely additive measure m : A → R is given. Then we have: Proposition 4.2. There exists at least one finitely additive measure m ( : P(T ) → R such that m ( |A = m. Proof. For any Π ∈ D, say Π = {(A1 , t1 ), (A2 , t2 ), . . . , (Ak , tk )}, define Σ(f, Π) =

k

f (tj )m(Aj ),

j=1

for all bounded maps f : T → R. Now, let U be any ultrafilter on D containing the refinement filter. Denoting by e any strong order unit in R such that the range of m is contained in V [e], it is clear that Σ(f, ·) is a bounded map from D to V [e], and hence has weak I-limit, with respect to the dual ideal I of U: let us put J (f ) := (w) − I − lim Σ(f, Π), Π∈D

for every bounded map f : T → R, and with respect to the same maximal extension of V [e]. We claim that the mapping m(A) ( := J (1A ), defined on all subsets A of T , is a finitely additive extension of m. First of all, it is easy to show that J is a linear mapping. So, it remains to prove that J (1A ) = m(A) whenever A ∈ A. Indeed, let us fix A ∈ A, and set f = 1A . Let now ΠA be the partition whose intervals are A and T \ A. For any Π ∈ D(ΠA ) we have clearly Σ(f, Π) = m(A), from which it is easy to deduce that J (f ) = m(A), i.e., the claim. This concludes the proof. A similar argument leads to prove the following result, whose meaning is that, when a finitely additive measure is defined in the whole of P(T ), then every bounded function on T admits integral. Corollary 4.3. Let P : P(T ) → R be any positive finitely additive measure, and let f : T → R be any bounded map. Now, consider an ultrafilter U containing the refinement filter like in the previous proof, with A = P(T ). Denoted by I the dual ideal of U, for each bounded map f : T → R let us define f dP := (w) − I − lim Σ(f, Π), T

where Σ(f, Π) has the usual meaning.

86


Then, f → T f dP is a linear monotonic functional, which coincides with the elementary integral, whenever f has finite range. Further applications concern the existence of invariant measures: for example, we can see that, given any non-empty set T and any map τ : T → T , it is always possible to construct a finitely additive τ -invariant measure μ : P(T ) → R. More precisely, we have the following result. Theorem 4.4. Let T be any abstract non-empty set, and τ : T → T be a mapping. Assume that m : A → R is any τ -invariant bounded finitely additive measure, defined in some algebra A of subsets of T . (This means that, for all A ∈ A, we have τ −1 (A) ∈ A and m(A) = m(τ −1 (A)).) Then there exists a bounded finitely additive measure μ : P(T ) → R, extending m and still invariant with respect to τ , i.e., μ(A) = μ(τ −1 (A)), for all A ⊂ T . Proof. Thanks to 4.2, we can find a bounded finitely additive extension m0 of m to the whole of P(T ). Then define a sequence (mn )n of measures, by induction: mn (A) = mn−1 (τ −1 (A)), for all A ⊂ T, n ≥ 1. Clearly, all measures mn coincide with m in A. Then set: μn := n1 ni=1 mi , for all n ≥ 1. If e denotes any upper bound for the range of |m0 |, then it is clear that each measure μn has the same upper bound, and that |μn (A) − μn (τ −1 (A))| ≤ 2e n for all A ⊂ T . Now choose any maximal admissible ideal I in the set N, and any maximal extension of V [e], in order to construct a (pointwise) weak I-limit μ0 of the sequence (μn )n . Then μ0 is the required extension: indeed, by construction it is additive and extends m, so the only property we have to prove is invariance; but for all sets A ∈ A we have |μ(A) − μ(τ −1 (A))| = | (w) − I − lim μn (A) − (w) − I − lim μn (τ −1 (A)) | 2e = 0, ≤ lim n n which concludes the proof.

References [1] E. Barone, A. Giannone, R. Scozzafava, On some aspects of the theory and applications of finitely additive probability measures. Pubbl. Istit. Mat. Appl. Fac. Ingr. Univ. Stud. Roma, Quaderno No. 16 (1980), 43–53. [2] A. Boccuto, D. Candeloro, Sandwich theorems, extension principle and amenability. Atti Sem. Mat. Fis. Univ. Modena 42 (1994), 257–271. [3] W. Chojnacki, Sur un théorème de Day, un théorème de Mazur-Orlicz et une généralisation de quelques théorèmes de Silverman. Colloq. Math. 50 (1986), 257– 262.


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[4] J.S. Connor, Two-valued measures and summability. Analysis 10 (1990), 373–385. ˇ at, W. Wilczy´ [5] P. Kostyrko, T. Sal´ nski, I-convergence. Real Anal. Exch. 26 (2000/ 2001), 669–685. [6] B.K. Lahiri, P. Das, I and I ∗ -convergence in topological spaces. Math. Bohemica 130 (2005), 153–160. [7] F. Maeda, T. Ogasawara, Representation of Vector Lattices. J. Sci. Hiroshima Univ. Ser. A 12 (1942), 17–35 (Japanese). [8] J. Rainwater, A note on projective resolutions. Proc. Am. Math. Soc. 10 (1959), 734–735. [9] B.Z. Vulikh, Introduction to the theory of partially ordered spaces. Wolters-Noordhoff Sci. Publ., Groningen, 1967. Antonio Boccuto and Domenico Candeloro Dipartimento di Matematica e Informatica via Vanvitelli, 1 I-06123 Perugia, Italy e-mail: [email protected] [email protected]


A First Return Examination of Vector-valued Integrals Donatella Bongiorno Abstract. We prove that for each Bochner integrable function f there exists a trajectory yielding the Bochner integral of f , and that on infinite-dimensional Banach spaces there exist Pettis integrable functions f such that no trajectory yields the Pettis integral of f . Mathematics Subject Classification (2000). Primary 28B05, 48G10; Secondary 26A42. Keywords. Trajectory, first return integral, Bochner integral, Pettis integral, McShane integral.

1. Introduction Let X be a Banach space. It is well known that the Bochner integral of a function f : [0, 1] → X can be obtained as a limit of suitable Riemann sums, in particular it is McShane integrable (see [7, Theorem 5.1.5]). Unfortunately the partitions used in the those theories are not completely free, as in the Riemann integral. In view to cover this gap, U.B. Darji and M.J. Evans [1] proved that if f : [0, 1]n → R is Lebesgue integrable then it is possible to find a dense sequence t (in the sequel called trajectory) such that for each ε > 0 there exists a positive constant δ with 1 f (r(t, J)) |J| − f < ε, (1) 0 x∈J∈P

for each partition P of [0, 1] with mesh(P) = sup{|J| : J ∈ P} < δ. Here r(t, J) denotes the first point of t belonging to J. In this paper we investigate the possible extension of Darji-Evans theorem to vector-valued functions. For simplicity we restrict our attention to the case of functions of one real variable.

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It is well known that there are three possible distinct extensions of the Lebesgue integral, in case the range space is an infinite-dimensional Banach space; namely the Bochner integral, the Pettis integral and the McShane integral. Definition. Let X be a Banach space and let f : [0, 1] → X be Bochner (resp. Pettis or McShane) integrable with integral A. We say that a trajectory t yields the integral A if for each ε > 0 there is a constant δ > 0 such that f (r(t, J))|J| − A < ε, (2) J∈P

for each partition P of [0, 1] with mesh(P) < δ. According with this definition the mentioned Darji-Evans theorem in [0, 1] can be reformulated as follows. Theorem. If f : [0, 1] → R is Lebesgue integrable, then there exists a trajectory t yielding the Lebesgue integral of f . In this paper we will prove that this theorem holds also for Bochner integrable functions (Theorem 1) and cannot holds for Pettis and McShane integrable functions (Theorems 2 and 3, respectively). We also prove that, in the case that the dual of the range space is w∗ -separable, a class of Pettis integrable functions (not Bochner integrable) for which the Darji-Evans theorem holds is the class of scalarly null functions (Theorem 4).

2. Preliminaries The set of all real (resp. natural) numbers is denoted by R (resp. N). The Lebesgue measure of a subset E of R is denoted by |E|. We write a.e. for almost everywhere. The word measurable is always refereed to the Lebesgue measure. A sequence t ≡ {tn } of distinct points of [0, 1], dense in [0, 1], is called trajectory. Given a trajectory t and an interval J ⊂ [0, 1] we denote by r(t, J) the first element of t that belongs to J. We call partition of [0, 1]' any finite collection of non-overlapping compact n intervals J1 , . . . , Jn such that i=1 Ji = [0, 1]. Given a partition P = {J1 , . . . , Jn } we set mesh(P)=supi |Ji |. Throughout this paper we denote by X a fixed Banach space, by · its norm, by B(x, r) the ball with center x and radius r, and by X ∗ the topological dual of X. We recall that a function f : [0, 1] → X is said to be strongly measurable if there exists a sequence of simple functions {fn } such that fn (x) − f (x) → 0, a.e. in [0, 1]. A strongly measurable function f : [0, 1] → X is said to be Bochner integrable if there exists a sequence of simple functions {fn } such that 1 lim fn (x) − f (x) dx = 0. n

0

A First Return Examination of Vector-valued Integrals

91

1 1 In this case limn 0 fn dx is called the Bochner integral of f on [0, 1], where 0 fn dx is the Bochner integral of the simple function fn defined in the obvious way. It is known that a strongly measurable function f : [0, 1] → X is Bochner 1 integrable if and only if 0 f dx < ∞ (see [2, Theorem 2, §2, Chapter II]). A function f : [0, 1] → X is said to be weakly measurable if for each x∗ ∈ X ∗ the scalar function x∗ f is measurable. A weakly measurable function f : [0, 1] → X is said to be Pettis integrable if for each measurable set E ⊂ [0, 1] there exists xE ∈ X such that for all x∗ ∈ X ∗ the scalar function x∗ f is Lebesgue integrable and ∗ x (xE ) = x∗ f dx. E

In this case xE is called the Pettis integral of f on E. Each Bochner integrable function is McShane integrable and each McShane integrable function is Pettis integrable with the same value of the integrals (see [7, Theorems 5.1.2 and 6.1.2]). Pettis and McShane integrals coincide if X is super reflexive and if X = c0 (Γ), where Γ is a generic set (see [3]). A function f : [0, 1] → X is said to be scalarly negligible if x∗ f = 0 a.e. in [0, 1]. Each scalarly negligible function is Pettis integrable with Pettis integral equal to zero.

3. Bochner integrable functions Theorem 1. If f : [0, 1] → X is Bochner integrable, then there exists a trajectory t in [0, 1] yielding the Bochner integral of f . Within the proof of this theorem we need the following variant of the Lusin theorem: (RLT ) If F ⊂ [0, 1] is measurable, f : [0, 1] → X is strongly measurable and ε > 0 then there is a closed nowhere dense set E ⊂ F such that |F \ E| < ε and f |E is continuous. Proof of (RLT). By the Lusin theorem for strongly measurable functions (see [4, Theorem 3, Chapter III, §15, No. 8]) there exists a closed set S ⊂ F such that |F \ S| < ε/2 and f |S is continuous. If the interior of S is empty we take E = S. Otherwise we define S1 ⊂ S, removing a countable dense subset in the interior of S, and we use the regularity of the Lebesgue measure to find a closed set E ⊂ S1 such that |S1 \ E| < ε/2. So E is a nowhere closed subset of F with |F \ E| ≤ |F \ S| + |S \ S1 | < ε and f |E is continuous. Proof of Theorem 1. In this proof we use some ideas from [5]. For each n ∈ N let Fn = {t ∈ [0, 1] : f (t) ≤ n}. We start by finding inductively a sequence {En } of closed nowhere dense sets such that En ⊂ En+1 , En ⊂ Fn , |Fn \ En | < 1/2n , and f |En is continuous, for each n ∈ N. To find a nowhere dense closed set E1 ⊂ F1 such that |F1 \ E1 | < 1/2, and f |E1 is continuous, we apply (RLT ) to the strongly measurable function f , to the

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set F1 , and to the constant 1/2. Then, assumed that E1 ⊂ E2 ⊂ · · · ⊂ En−1 have been defined, we proceed as follows. We apply (RLT ) to the strongly measurable function f , to the set Fn \ En−1 , and to the constant 1/2n . So we find a nowhere dense closed set A ⊂ (Fn \ En−1 ) such that |(Fn \ En−1 ) \ A| < 1/2n , and f |A is continuous. We define En = En−1 ∪ A. Then En is nowhere dense and closed, En−1 ⊂ En , En ⊂ Fn−1 ∪ A ⊂ Fn , |Fn \ En | = |(Fn \ En−1 ) \ A| < 1/2n, and f |En is continuous. ' Let E = n En . Since En is closed and nowhere dense, then [0, 1] \ En is open and dense. Therefore, by the Baire category theorem, [0, 1] \ E is dense in [0, 1]. Let {rn } ⊂ [0, 1] \ E be dense in [0, 1]. We define inductively a trajectory t = {tn } as follows. By the definition of {rn } it follows r1 ∈ [0, 1] \ E1 . Let I be the connected component of [0, 1] \ E1 containing r1 . We define t1 = inf I. Then, assumed that t2 , . . . , tn−1 have been defined, by the definition of {rn } it follows that rn ∈ [0, 1] \ Em , for m = n, n + 1, . . . . Since the set {t1 , t2 , . . . , tn−1 } is finite, there exists a first index kn such that rn belongs to some connected component J of [0, 1] \ Ekn and one of the extreme points of J is not in {t1 , t2 , . . . , tn−1 }. We define tn as this extreme point. Remark that, by the density of {rn }, t coincide with the class of all extreme points of the connected components of the open sets [0, 1] \ E1 , . . . , [0, 1] \ En , . . . . Remark also that, by the definition of Fn , limn |Fn | = 1. Then, since |En | = |FN | − |FN \ En | ≥ |Fn | −

1 , 2n

we have limn |En | = 1. Therefore, if T ⊂ [0, 1] is an interval, there exists nT ∈ N such that T ∩ EnT = ∅. Moreover, since EnT is nowhere dense, it is T ⊂ EnT . Let I be a connected component of [0, 1] \ EnT such that I ∩ T = ∅. Then T contains at least one extreme point of I. As remarked before, this extreme point belongs to t; consequently t is dense in [0, 1], therefore t is a trajectory. Now we remark that: p) r(t, J) ∈ En implies J ∩ En = ∅, for each interval J ⊂ [0, 1] and for each n ∈ N. In fact, by the definition of t, the condition r(t, J) ∈ En implies r(t, J) ∈ Em for some m > n; i.e., r(t, J) is one of the end points of a connected component I of [0, 1] \ Em . Moreover, since En ⊂ Em , there exists a connected component (a, b) of [0, 1] \ En such that I ⊂ (a, b). Using again the particular construction of t, it follows that the end points of (a, b) belong to t and both precede r(t, J). Then a, b ∈ J. So, by r(t, J) ∈ I ⊂ (a, b), we have J ⊂ (a, b). Consequently J ∩ En = ∅. Finally we prove that t yields the Bochner integral of f . Given ε > 0, let N be such that ∞ n+1 (f + 1) ≤ ε, and ≤ ε. (3) 2n [0,1]\EN N


93

Moreover let g be the extension of f |EN to [0, 1] such that g is linear on each connected component of [0, 1] \ EN . It is easy to prove that g is continuous on [0, 1] and that g(x) ≤ N for each x ∈ [0, 1]. Then, g being Riemann integrable, there exists a positive constant δ such that 1 g(r(t, Ji ))|Ji | − g < ε, (4) 0 i

for each usual partition P = {Ji } of [0, 1] with mesh(P) < δ. We set fP (t) = f (r(t, Ji )), if t ∈ Ji , gP (t) = g(r(t, Ji )), if t ∈ Ji . Therefore

1

fP = 0

f (r(t, Ji ))|Ji |,

i 1

gP = 0

g(r(t, Ji ))|Ji |.

i

' By E \ EN = n≥N (En+1 \ En ) and by |[0, 1] \ E| = 0 it follows that, for almost all t ∈ [0, 1] \ EN there exists n ≥ N such that t ∈ En+1 \ En . Let J ∈ P such that t ∈ J. Then, by p) we infer r(t, J) ∈ En+1 . So fP (t) = f (r(t, J)) ≤ n + 1 a.e. in [0, 1]. Thus, since En+1 \ En ⊂ Fn+1 \ En = (Fn+1 \ Fn ) ∪ (Fn \ En ), by (3) we have fP ≤ [0,1]\EN

≤

∞ N ∞

(n + 1)|En+1 \ En | (n + 1)(|Fn+1 \ Fn | + |Fn \ En |)

(5)

N

(f + 1) +

≤ [0,1]\EN

≤

∞ n+1 N

2n

2ε.

Moreover, since f > N on [0, 1] \ FN , by (3) it follows gP ≤ N · |[0, 1] \ EN | [0,1]\EN

≤ ≤ ≤

N · (|[0, 1] \ FN | + |FN \ EN |) N f + N 2 [0,1]\FN 2ε.

(6)

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Therefore, if P = {Ji } is a partition of [0, 1] with mesh(P) < δ, by |FN \ EN | < 1/2N and by (3), (4), (5) and (6) we infer: 1 f (r(t, Ji )) |Ji | − f 0 i 1 ≤ (f (r(t, Ji )) − g(r(t, Ji ))) |Ji | + g(r(t, Ji )) |Ji | − g 0 i i 1 + f − g 0 1

fP − gP + ε + f − g [0,1]\EN 0 fP + gP + ε ≤ [0,1]\EN [0,1]\EN + f + g + g [0,1]\EN [0,1]\FN FN \EN N ≤ 5ε + f + f + N 2 [0,1]\EN [0,1]\FN ≤ 8ε.

≤

This completes the proof.

4. Pettis integrable functions Theorem 2. There exists a strongly measurable function f : [0, 1] → l2 (N), Pettis integrable, such that no trajectory in [0, 1] yields the Pettis integral of f . Proof. Let {en } be the canonical basis of l2 (N). For n ≥ 2 let {an } be a decreasing sequence of positive numbers such that an − an+1 = 1/n2 and limn an = 0. We define ∞ n en χEn (x), f (x) = n=2

where En = (an+1 , an ]. The function f is strongly measurable and Pettis integrable with integral ∞ f= n|E ∩ En | en , xE = E

n=2

for each measurable set E ⊂ [0, 1]. In fact, for each x∗ = (x1 , x2 , . . . , xn , . . . ) ∈ l2∗ (N) we have ∞ x∗ f = nxn |E ∩ En | = x∗ (xE ). E

n=2


95

Remark that f is not Bochner integrable, since 1 ∞ ∞ 1 = +∞. f = n|En | = n 0 2 2 Now we show that no trajectory in [0, 1] yields the Pettis integral of f . Let t be an arbitrary trajectory in [0, 1], and let 0 < ε < 1/2. ∞ Given δ > 0, let n0 ∈ N be such that an0 < δ and n0 n1 en < 1 − 2ε. First of all we remark that an0 > 1/n0 .

(7)

In fact an0

=

(an0 − an0 +1 ) + (an0 +1 − an0 +2 ) + · · · + (an0 +k − an0 +k+1 ) + · · ·

=

|En0 | + |En0 +1 | + · · · + |En0 +k | + · · · 1 1 1 + + ···+ + ··· n20 (n0 + 1)2 (n0 + k)2 1 1 1 + + ···+ + ··· n0 (n0 + 1) (n0 + 1)(n0 + 2) (n0 + k)(n0 + k + 1) 1 1 1 1 + − + ··· − n0 n0 + 1 n0 + 1 n0 + 2 1 . n0

= > = =

Moreover remark that, by the definition of f , it follows f (r(t, [0, an0 ])) = n en , for some n ≥ n0 . Then, by (7), we have f (r(t, [0, an0 ]) an0 ≥ n0 an0 > 1.

(8)

Now let P be a partition with mesh(P) < δ such that i) [0, an0 ] ∈ P; ii) [an0 , an0 −1 ], . . . , and [a3 , a2 ] are union of elements of P; iii) if J ∈ P with inf J = an+1 , n = 2, . . . , n0 − 1, then |J| < ε/(2n + 1) n0 . For n = 2, . . . , n0 − 1 we denote by Jn ∈ P the interval with inf Jn = an+1 . Then by definition of f and by iii) it follows 1 f (r(t, J)) |J| − en n J⊂[an+1 ,an ] = f (r(t, Jn )) |Jn | + f (r(t, J)) |J| − n|En | en Jn =J⊂[an+1 ,an ]

= f (r(t, Jn )) |Jn | + n|En \ Jn | en − n|En |en = f (r(t, Jn )) |Jn | − n|Jn |en .

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Hence

1 en n

f (r(t, J)) |J| −

J⊂[an+1 ,an ]

= 0, if r(t, Jn ) = an+1 = ((n + 1)en+1 − nen ) |Jn |, if r(t, Jn ) = an+1 .

Consequently 1 ε < (2n + 1)|Jn | < f (r(t, J)) |J| − en . n n0 J⊂[an+1 ,an ]

(9)

Thus, by (7), (8) and (9) we infer ∞ 1 en f (r(t, J)) |J| − n n=1 J∈P ∞ 0 −1 1 n 1 ≥ f (r(t, [0, an0 ])) an0 − − en f (r(t, J)) |J| − en n n n=n0 n=2 J⊂[an+1 ,an ] ≥ 1 − ε − (1 − 2ε) = ε .

This completes the proof.

Theorem 3. There exists a strongly measurable function f : [0, 1] → l (N), McShane integrable, such that no trajectory in [0, 1] yields the McShane integral of f . 2

Proof. Let f : [0, 1] → l2 (N) be defined as in Theorem 2. It was proved that f is strongly measurable and Pettis integrable on [0, 1]. Then, by a theorem proved by R.A. Gordon in [6] (see also [7, Theorem 6.2.1]) f is McShane integrable on [0, 1] with the same value of the integrals. Therefore no trajectory yields the McShane integral of f , by Theorem 2. Theorem 4. If X ∗ is w∗ -separable and f : [0, 1] → X is scalarly null, then there exists a trajectory t in [0, 1] yielding the Pettis integral of f . ' Proof. Let {x∗n } be w∗ -dense in X ∗ , and let E = n En , where En = {t ∈ [0, 1] : x∗n f (t) = 0}. By hypothesis |E| = 0. Then x∗n f (t) = 0 for each n ∈ N and each t ∈ [0, 1]\E. Now, for x∗ ∈ X ∗ and t ∈ [0, 1]\E there exists a sequence {nk } of indices such that x∗nk f (t) → x∗ f (t). Consequently x∗ f (t) = 0 and f (t) = sup x∗ =1 |x∗ f (t)| = 0, for each t ∈ [0, 1] \ E. Let t = {tn } be a trajectory in [0, 1] such that tn ∈ E, for each n. Then, for each δ > 0 and each partition P with mesh(P) < δ, we have J∈P f (r(t, J))|J| = 0. This completes the proof, since the Pettis integral of f is zero. Acknowledgment The author is indebted to the referee for various helpful comments that helped improve the paper.


97

References [1] U.B. Darji and M.J. Evans, A first return examination of the Lebesgue integral, Real Anal. Exchange, 27 (2001/2002), 573–581. [2] J. Diestel and J.J. Uhl,Vector measures, Math. Surveys, 15 (1977). [3] L. Di Piazza and D. Preiss, When do McShane and Pettis integrals coincide?, Illinois J. Math., 47 (2003), 1177–1187. [4] N. Dinculeanu, Vector measures, Pergamon Press, Oxford, 1967. [5] D. Fremlin, Notes on first-return integration, Preprint available at http://www.essex.ac.uk/maths/staff/fremlin/n07k04.ps [6] R.A. Gordon, The McShane integral of Banach-valued functions, Illinois J. of Math., 34 (1990), 557–567. [7] S. Schwabik and Y. Guoju, Topics in Banach space integration. Series in Real Analysis, 10. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. Donatella Bongiorno Dipartimento di Metodi e Modelli Matematici Viale delle Scienze I-90100 Palermo, Italy e-mail: [email protected]


A Note on Bi-orthomorphisms Gerard Buskes, Robert Page, Jr. and Rusen Yilmaz Abstract. We show that the space of bi-orthomorphisms forms a vector lattice.The space of orthomorphisms on a semiprime f -algebra is a vector sublattice of the space of bi-orthomorphisms and an ideal in the case that the f -algebra is Dedekind complete. Mathematics Subject Classification (2000). 46A40, 46B42. Keywords. Separately order bounded, bilinear map of order bounded variation, disjointness preserving bilinear operator, bi-orthomorphism.

1. Introduction Let A be a vector lattice. The authors of [13] were the first to study what, in this paper, we will call bi-orthomorphisms, i.e., bilinear maps A × A → A that are orthomorphisms in each variable separately. Their paper is based on a certain calculus for order bounded bilinear maps with values in a Dedekind complete vector lattice (Theorem 2.3 of [13]). The bilinear map φ : R2 × R2 → R defined by φ((x1 , x2 ), (y1 , y2 )) = (x1 − x2 )(y1 − y2 ) provides a counterexample to the Kantorovich-like formulas for the positive part and absolute value of bilinear maps in Theorem 2.3 of [13], though a calculus of order bounded variation like in [5] does yield the right formulas. One of the goals of [13] is to understand the space of quasimultipliers in relation to the space of bi-orthomorphisms, in case that A is an f -algebra. In that direction the main results in [13] (e.g., Theorem 4.7) are correct, because the lattice calculations for bi-orthomorphisms that are needed for these results are, in fact, more elementary than the Kantorovich-like formulas (and do not need Dedekind completeness of the range space) as we will show in this paper. We prove in this note that the space Orth(A, A) of bi-orthomorphisms is a vector lattice, which contains Orth(A) as a vector sublattice in case that A is a semiprime f -algebra. We prove that Orth(A) is an ideal in the vector lattice Orth(A, A) in case A is a semiprime Dedekind complete f -algebra. Our approach to the vector ¨ I˙ TAK Grant CODE The third named author gratefully acknowledges support from the T UB 2221.

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lattice structure of Orth(A, A) starts with (the more general) bilinear maps that are disjointness preserving and order bounded in each variable separately. In that direction we slightly finesse Theorem 3.4 in [8], resulting in a more concise proof of a theorem that follows from combining Theorems 1.12 and 2.11 in the Ph.D. thesis by Page (see [9]). For general information about vector lattices we refer to [8]. We remark that bilinear maps that are separately band preserving have recently been studied in the paper [5]. For a general survey on bilinear maps on products of vector lattices we refer to [4].

2. Preliminaries Let A and B be vector lattices. An operator T : A → B is said to be disjointness preserving if a ⊥ b implies T a ⊥ T b for all a ∈ A and b ∈ B. An operator T : A → A is called band preserving if a ⊥ b in A implies T a ⊥ b, where a ⊥ b means |a| ∧ |b| = 0. Clearly, every band preserving operator is disjointness preserving. A band preserving operator which is also order bounded is said to be an orthomorphism. The set of all orthomorphisms on A is denoted by Orth(A). If A is an Archimedean vector lattice, then Orth(A) is an Archimedean f -algebra under multiplication by composition, with the identity operator I on A as a multiplicative identity. It is well known that every order bounded operator T : A → B between two Archimedean vector lattices which preserves disjointness has a modulus |T | and |T |(|x|) = |T (|x|)| = |T (x)| holds for all x ∈ A. From here on, let A, B, and C be Archimedean vector lattices. A bilinear operator T : A × B → C is said to be of order bounded variation if, for all (x, y) ∈ A+ × B + , the set N,M n,m

|T (an , bm )| : an ∈ A+ , bm ∈ B + and

N n=1

an = x,

M

bm = y (N, M ∈ N)

m=1

is order bounded in C. T is said to be order bounded if for all (x, y) ∈ A+ × B + we have that {T (a, b) : 0 ≤ a ≤ y, 0 ≤ b ≤ y} is order bounded. We denote by Lbv (A, B; C) the set of all bilinear operators of order bounded variation and by Lb (A, B; C) the vector space of all order bounded bilinear maps. T is positive if for all x ∈ A+ and y ∈ B + we have T (x, y) ∈ C + . We denote by Lr (A, B; C) the span of the positive bilinear maps in Lb (A, B; C). Each element of Lr (A, B; C) is called regular. The vector spaces Lbv (A, B; C) and Lb (A, B; C) are ordered vector spaces under the ordering defined by T1 ≤ T2 if T2 − T1 is positive. The inclusions Lr (A, B; C) ⊂ Lbv (A, B; C) ⊂ Lb (A, B; C) hold. The converse inclusions need not hold, even if C = R, as was shown in [10]. However, if C is Dedekind complete, then Lr (A, B; C) = Lbv (A, B; C) and this space is then a Dedekind complete vector lattice. In addition, if C is Dedekind

A Note on Bi-orthomorphisms

101

complete then for any T ∈ Lbv (A, B; C), the modulus |T | is determined by the following formula N M N,M |T |(x, y) = sup |T (an , bm )| : an ∈ A+ , bm ∈ B + , an = x, bm = y n,m

n

m

and |T (x, y)| ≤ |T |(|x|, |y|) for each x ∈ A and each y ∈ B (see [5, 7]). A bilinear map T : A × B → C is said to be separately disjointness preserving if +

+

(a1 , b1 ) ⊥ (a2 , b2 ) in A × B implies T (a1 , b) ⊥ T (a2 , b) and T (a, b1) ⊥ T (a, b2 ) for all a ∈ A and b ∈ B. A bilinear operator T : A × A → A is called separately band preserving if x ⊥ y in A

implies

T (x, z) ⊥ y and T (z, x) ⊥ y

for all z ∈ A. A bilinear operator T : A × B → C is called separately order bounded (respectively a Riesz bimorphism) if a → T (a, y) (a ∈ A)

and

b → T (x, b) (b ∈ B)

are order bounded (respectively a Riesz homomorphism). T is a Riesz bimorphism if and only if |T (a, b)| = T (|a| , |b|) for all a ∈ A and b ∈ B. Clearly, every separately band preserving bilinear operator is disjointness preserving. A separately band preserving bilinear operator which is also separately order bounded is called a bi-orthomorphism and the set of all bi-orthomorphisms of A × A into A is denoted by Orth(A, A). Finally we will use Fremlin’s Archimedean tensor product of two Archimedean vector lattices. In [6], Fremlin introduced for every two Archimedean vector lattices ¯ which is called the Archimedean E and F a new Archimedean vector lattice E ⊗F vector lattice tensor product of E and F , defined by the following universal prop¯ such that whenever G is a erty: there exists a Riesz bimorphism E × F → E ⊗F vector lattice and T is a Riesz bimorphism E × F → G then there exists a unique ¯ → G for which Riesz homomorphism T ⊗ : E ⊗F T (x, y) = T ⊗ (x⊗ y) (x ∈ E, y ∈ F ). ¯ has He also showed that the Archimedean vector lattice tensor product E ⊗F the following additional universal property. For every positive map T of E × F into any uniformly complete (hence Archimedean) vector lattice G there exists a ¯ → G such that T (x, y) = T ⊗ (x ⊗ y) for all unique positive linear map T ⊗ : E ⊗F x ∈ E, y ∈ F . It follows immediately that the map S → S ◦ ⊗ defines a bijective ¯ G) onto Lr (E, F ; G), where Lr (E ⊗F, ¯ G) denotes the positive map from Lr (E ⊗F, ¯ to G. partially ordered vector space of all linear regular maps from E ⊗F A slight extension of Fremlin’s universal property is the following result in [5]. Theorem 2.1. For every bilinear map T of E × F into any uniformly complete (hence Archimedean) vector lattice G which is of order bounded variation there ¯ → G such that T (x, y) = exists a unique order bounded linear map T ⊗ : E ⊗F

102


T ⊗ (x ⊗ y) (x ∈ E, y ∈ F ). The map T → T ⊗ is a linear order isomorphism ¯ G), where Lb (E ⊗F, ¯ G) denotes the partially between Lbv (E, F ; G) and Lb (E ⊗F, ¯ to G. ordered vector space of all order bounded linear maps from E ⊗F

3. Separately disjointness preserving operators The modulus of any separately disjointness preserving separately order bounded bilinear map exists as we will prove in this section. We need the following Lemma, the equivalent of which for linear maps is well known (see, e.g., Theorem 1.1 in [11]). Lemma 3.1. Let E, F and G be vector lattices. Let T : E × F → G be a bilinear map and assume that N M N,M sup |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y n,m

n

m

exists in G for all x ∈ E + and all y ∈ F + . Then the map N M N,M |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y (x, y) → sup n,m

n

m

(x ∈ E , y ∈ F ) extends to a linear map E × F → G which in the ordered vector space Lbv (E, F ; G) is the least upper bound of {T, −T }. +

+

Proof. Let Gδ be the Dedekind completion of G. We interpret the map T : E ×F → G as a map T : E × F → Gδ and accordingly we read the formula in the lemma in Gδ rather than in G. According to [5] there exists a linear map E × F → Gδ which extends N M N,M |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = x, ym = y (x, y) → sup n,m

n

m

and that linear map is the modulus |T | of T in Lbv (E, F ; G ). This |T | has all its values in G. Moreover, in the ordered vector space Lbv (E, F ; G) we have that |T | ≥ T and |T | ≥ −T . It is immediate that for any S ∈ Lbv (E, F ; G) which is an upper bound for {T, −T } we have S ≥ |T |. The lemma follows. δ

The next theorem generalizes Theorem 3.4 in [8] and combines in one proof (rather than two separate arguments) the information from Theorems 1.12 and 2.11 in Page’s Ph.D. thesis [9] (see our comments after the theorem). Since the Ph.D. thesis of Page is less easily accessible, and for completeness sake, we have included the result with the more concise proof. Theorem 3.2. Let E, F and G be vector lattices. Let T : E × F → G be a separately disjointness preserving bilinear map. Then the following are equivalent. (1) T is separately order bounded. (2) T is order bounded.


103

(3) T is of order bounded variation. (4) T is regular. Moreover, if T satisfies any of the above, then |T | exists and ||T |(e, f )| = |T (e, f )| = |T |(|e|, |f |) for all (e, f ) ∈ E × F. In particular, |T | is a Riesz bimorphism. Proof. Obviously (4)⇒(3)⇒(2)⇒(1). Thus it suffices to show (1)⇒(4). Assume that T : E × F → G is separately order bounded. Let e ∈ E + and f ∈ F + . Define the maps Tf : E → G by Tf (x) = T (x, f ) (x ∈ E) e T : F → G by e T (y) = T (e, y) (y ∈ F ). Since e T and Tf are order bounded operators which preserve disjointness by the hypotheses, |e T | and |Tf | exist and

|e T |(f ) = |e T (f )| = |T (e, f )| = |Tf (e)| = |Tf |(e). Now let x = (x1 , . . . , xm ) and y = (y1 , . . . , yn ) be partitions of e ∈ E + and f ∈ F + respectively (as defined in [5]). Then we have n m m n m n |T (xi , yj )| = |T (xi , yj )| = |xi T (yj )| i=1 j=1

i=1

=

= =

j=1

m n i=1 m i=1 m i=1

i=1

j=1

m n |xi T |( |xi T |(yj ) = yj )

j=1

|xi T |(f ) = |Tf (xi )| =

i=1 m i=1 m

|xi T (f )| =

j=1 m

|T (xi , f )|

i=1

m |Tf |(xi ) = |Tf | xi

i=1

i=1

= |Tf (e)| = |T (e, f )|. Thus, by the previous lemma, |T | exists in Lbv (E, F ; G) and |T | (e, f ) is given by the formula N M N,M sup |T (xn , ym )| : xn ∈ E + , ym ∈ F + , xn = e, ym = f . n,m

n

m

for e ∈ E and f ∈ F . Then T is regular in Lbv (E, F ; G), which proves (1)⇒(4). Moreover, |T |(e, f ) = |T (e, f )| for e ∈ E + and f ∈ F + . Using the latter and the fact that T is separately disjointness preserving, we obtain that for arbitrary e ∈ E and f ∈ G the elements in {|T | (e+ , f + ), |T | (e+ , f − ), |T | (e− , f + ), |T | (e− , f − )} are pairwise disjoint hence +

+

||T | (e, f )| = |T | (e+ , f + ) + |T | (e+ , f − ) + |T | (e− , f + ) + |T | (e− , f − ) = |T | (|e| , |f |). Then |T | is a Riesz bimorphism. This proves the theorem.

104


Some remarks are in order. First of all we comment that, in general, separately order bounded bilinear maps are not necessarily order bounded, as was shown by Swartz in [12]. Secondly, in Theorem 3.4 of [8] Kusraev and Tabuev proved (2) ⇔ (4) (as well as the fact that |T | is a Riesz bimorphism). Page proved (3) ⇒ (4) in Theorem 2.11 of his Ph.D. thesis [9] (as well as the fact that |T | is a Riesz bimorphism) and he proves (1) ⇒ (3) in Theorem 1.12. The proof by Kusraev and Tabuev is similar to our proof above of (1) ⇒ (4) while our proof above combines the two proofs by Page. We also, as an example, make explicit a comment from the introduction, which is one of our reasons for writing this note. Example 3.3. For the bilinear map φ : R2 × R2 → R defined by φ((x1 , x2 ), (y1 , y2 )) = (x1 − x2 )(y1 − y2 ) we have that φ is of order bounded variation. Hence |φ| exists and its values are calculated via the formula in Section 1. Consequently, φ also is order bounded but |φ| cannot be calculated via a Kantorovic formula for the absolute value as formulated in Theorem 2.3 of [13]. We collect a number of corollaries to Theorem 3.2. As a first easy application, we add a statement to the equivalences in Theorem 3.2 which is the analogue for bilinear maps of the first part of Theorem 3.3 in [2]. Corollary 3.4. Let E, F and G be vector lattices. Let T : E × F → G be a separately disjointness preserving bilinear map. Then each of (1)–(4) in Theorem 3.2 is equivalent to | (x1 , x2 ) | ≤ | (y1 , y2 ) | ⇒ |T (x1 , x2 ) | ≤ |T (y1 , y2 ) | for all (x1 , x2 ) , (y1 , y2 ) ∈ E × F.

()

Proof. Notice that | (x1 , x2 ) | ≤ | (y1 , y2 ) | if and only if |x1 | ≤ |y1 | and |x2 | ≤ |y2 |. If T is separately order bounded then (using the notation of the proof of Theorem 3.2 in this paper) by Theorem 3.3 in [2] |T (x1 , x2 ) | = |x1 T (x2 )| ≤ |x1 T (y2 )|, while by the same reasoning |x1 T (y2 )| = |T (x1 , y2 ) | = |Ty2 (x1 ) | ≤ |Ty2 (y1 ) | = |T (y1 , y2 ) |. Hence (1) in 3.2 implies the statement () above. Conversely, the statement () obviously implies order boundedness of T . The proof of the next corollary is left to the reader. In case T is order bounded it is also contained in Theorem 3.4 of [8]. Corollary 3.5. If T is separately disjointness preserving and separately order bounded, then |T |, T + and T − are Riesz bimorphisms and |T | = T + − T − . Our next corollary deals with bi-orthomorphisms. Corollary 3.6. Let A be a vector lattice. If T ∈ Orth(A, A), then |T |, T + and T − are also in Orth(A, A).


105

Proof. Assume that x ⊥ y in A and z ∈ A. If T ∈ Orth(A, A), then |T | exists by the above theorem and ||T |(x, z)| ∧ |y| = |T (x, z)| ∧ |y| = 0

and

||T |(z, x)| ∧ |y| = |T (z, x)| ∧ |y| = 0.

holds, and so |T | ∈ Orth(A, A). It follows that T + and T − are also in Orth(A, A). Corollary 3.7. If A is an Archimedean vector lattice, then Orth(A, A) is a vector lattice. The next lemma for order bounded separately disjointness preserving operators is Theorem 3.6 of [8]. Lemma 3.8. Let E, F and G be Archimedean vector lattices. If T : E × F → G is separately disjointness preserving and separately order bounded, then there exists ¯ → G such that an order bounded disjointness preserving operator T ⊗ : E ⊗F T (x, y) = T ⊗ (x ⊗ y) (x ∈ E, y ∈ F ) and |T |⊗ = |T ⊗ |. Proof. Let Gδ be the uniform completion of G. By Theorem 2.1, the map T → T ⊗ ¯ ; Gδ ) is a linear order isomorphism. Since T + ∧T − = from Lbv (E, F ; Gδ ) to Lb (E ⊗F + ⊗ 0 we have that (T ) ∧ (T − )⊗ = 0 and T ⊗ = (T + )⊗ − (T − )⊗ . Also (T + )⊗ and (T − )⊗ are Riesz homomorphisms by the definition of tensor product. But (T + )⊗ and (T − )⊗ take their values in G, and then so does T ⊗ . Hence T ⊗ is order bounded disjointness preserving and |T |⊗ = (T + + T − )⊗ = (T + )⊗ + (T − )⊗ = (T ⊗ )+ + (T ⊗ )− = |T ⊗ |.

An analogue of a known Radon-Nikodym Theorem for disjointness preserving linear maps is next. It generalizes Proposition 3.7 (3) of [8] (where G is Dedekind complete and T is order bounded). Theorem 3.9. Let E, F and G be Archimedean vector lattices. If G is uniformly complete and S : E × F → G is a positive bilinear map and T : E × F → G is a separately disjointness preserving map which is separately order bounded and S ≤ |T | then there exists π ∈ Orth(I(T (E × F ))) (where I(T (E × F )) is the ideal generated by T (E × F ) in G) such that S = π ◦ |T |. If G is Dedekind complete then π can be selected in Orth(G). The latter theorem follows immediately from the following polar decomposition theorem for separately order bounded separately disjointness preserving bilinear operators. Theorem 3.10. Let E, F and G be vector lattices. If G is uniformly complete and T : E ×F → G is a separately order bounded and separately disjointness preserving operator, then there exists a bijective π ∈ Orth(I(T (E × F ))) such that T = π ◦ |T | (where I(T (E × F )) is the ideal generated by T (E × F ) in G). If G is Dedekind complete then π can be selected in Orth(G).

106


¯ → G is order bounded and disjointness preservProof. The map T ⊗ : E ⊗F ing. By the polar decomposition theorem ([3, Theorem 7]), there exists π ∈ ¯ ))) such that T ⊗ = π ◦ |T ⊗ |. First note that Orth(I(T ⊗ (E ⊗F T ⊗ ◦ ⊗ = (π ◦ |T ⊗ |) ◦ ⊗ = π ◦ |T |⊗ ◦ ⊗ = π ◦ |T |. Next we observe that (π ◦ |T |)⊗ (x ⊗ y) = (π ◦ |T |)(x, y) = (T ⊗ ◦ ⊗)(x, y) = T ⊗ (⊗(x, y)) = T ⊗ (x ⊗ y) for all x ∈ E and x ∈ F , and so T ⊗ = (π ◦ |T |)⊗ . Then T = π ◦ |T | because the map T → T ⊗ is injective. ¯ )) = I(T (E × F )). Indeed, certainly T (E × F ) ⊂ Note also that I(T ⊗ (E ⊗F ⊗ ¯ ¯ , and therefore T (E ⊗F ). On the other hand, ⊗(E × F ) is majorizing in E ⊗F ⊗ ⊗ ¯ ¯ I (T (E ⊗F )) ⊂ I (T (E × F )). Then I (T (E ⊗F )) = I (T (E × F )) and π ∈ Orth (I (T (E × F ))). As a consequence we are able to locate Orth(A) as an ideal in Orth(A, A)) when A is a Dedekind complete semiprime f -algebra. Proposition 3.11. (i) If A is a semiprime f -algebra then Orth(A) is a vector sublattice of Orth(A, A). (ii) If A is a semiprime Dedekind complete f -algebra then Orth(A) is an order ideal in Orth(A, A). Proof. We first prove (i). Let A be a semiprime f -algebra. The map ϕ : Orth(A) → Orth(A, A) defined by (ϕ(π))(f, g) = π(f )g (π ∈ Orth(A) and f, g ∈ A) is an injective Riesz homomorphism. This proves (i). Now let A be a semiprime Dedekind complete f -algebra. Let T : A × A → A be a bilinear map of order bounded variation and let π in Orth(A) be such that |T | ≤ |ϕ(π)|. By 3.9, there exists π ∈ Orth(A) such that T + = π ◦ |ϕ(π)|. Now π ◦ |ϕ(π)|(f, g) = π (|π| (f )g) = π (|π| (f ))g. Hence T + ∈ ϕ(Orth(A)). Similarly T − ∈ ϕ(Orth(A)) and consequently T ∈ ϕ(Orth(A)). This proves (ii). Example 3.12. In general, Orth(A) = Orth(A, A). Indeed, take A = C[0, 1] and define an f -algebra multiplication by (f ∗ g)(x) = x · (f · g)(x) (f, g ∈ A and x ∈ [0, 1]). Then the ordinary multiplication (f, g) → f ·g is in Orth(A, A) but not in Orth(A). We ask the following question. Question 3.13. When is Orth(A, A) an f -algebra? When exactly Orth(A, A) is an f -algebra is indeed unclear. However, the results above lead to the following improvement of Theorem 4.7 in [13], to which we refer for the definition of a minimal ultra-approximate identity. The proof follows the same lines as [13] with a small addition at the end of the proof of Theorem 4.6 in [13] where it needs to be observed that band preserving operators on a Banach lattice are indeed orthomorphisms. Proposition 3.14. Let A be an f -algebra which is also a Banach algebra with a minimal ultra-approximate identity. Then Orth(A, A) is a Banach f -algebra.


107

References [1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press, 1985. [2] Y.A. Abramovich, E.L. Arensen, and A.K. Kitover, Banach C(K)-modules and Operators Preserving Disjointness, Pitman Research Notes in Mathematics Series, 277, Longman Scientific & Technical, Harlow, 1992. [3] K. Boulabiar and G. Buskes, Polar decomposition of order bounded disjoint preserving operators, Proc. Amer. Math. Soc. 132 (2005), no. 3, 799–806. [4] Q. Bu, G. Buskes, and A.G. Kusraev, Bilinear maps on products of vector lattices: A survey, Positivity, Trends in Mathematics, Positivity, 97–126, Trends Math., Birkh¨ auser, Basel, 2007. [5] G. Buskes and A. van Rooij, Bounded variation and tensor products of Banach lattices, Positivity 7 (2003), 47–59. [6] D.H. Fremlin, Tensor products of Archimedean vector lattices, Amer. J. Math. 94 (1972), 778–798. [7] A.G. Kusraev, When all separately band preserving bilinear operators are symmetric, Vladikavkaz Mat. Zh. 9, no. 2 (2007), 22–25. [8] A.G. Kusraev and S.N. Tabuev, On disjointness preserving bilinear operators, Vladikavkaz Math. Zh. 6, no. 1 (2004), 58–70. [9] R. Page, On bilinear maps of order bounded variation, Thesis University of Mississippi, 2005. [10] A.L. Peressini and D.R. Sherbert, Ordered topological tensor products, Proc. London Math. Soc. 19 (1969), 177–190. [11] A. van Rooij, When do the regular operators between two Riesz spaces form a Riesz space?, Report 8410, Catholic University Nijmegen, (1984). [12] C. Swartz, Bilinear mappings between lattices, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 33(81) (1989), no. 2, 147–152. [13] R. Yilmaz and K. Rowlands, On orthomorphisms, quasi-orthomorphisms and quasimultipliers, J. Math. Anal. Appl. 313 (2006), 120–131. [14] A.C. Zaanen, Riesz Spaces II, North-Holland (1983). Gerard Buskes Department of Mathematics, University of Mississippi University, MS 38677, USA e-mail: [email protected] Robert Page, Jr. Department of Mathematics, Framingham State College Framingham, MA 01701, USA Rusen Yilmaz Department of Mathematics, Faculty of Arts and Science Rize University 53100 Rize, Turkey e-mail: [email protected]


Compactness of Multiplication Operators on Spaces of Integrable Functions with Respect to a Vector Measure Ricardo del Campo, Antonio Fernández, Irene Ferrando, Fernando Mayoral and Francisco Naranjo Abstract. We study properties of compactness of multiplication operators between spaces of p-power integrable scalar functions with respect to a vector measure m. Mathematics Subject Classification (2000). Primary 46G10, 46E30 Secondary 47B65, 47H07. Keywords. Vector measure, integrable function, multiplication operator, weakly compact operator.

1. Introduction The aim of the present paper is to study compactness and weak compactness properties of multiplication operators between spaces of p-power integrable functions with respect to a vector measure. In the following, m : Σ −→ X will be a countably additive vector measure defined on a σ-algebra Σ of subsets of a nonempty set Ω with values in a real Banach space X. We denote by X its dual space and by B(X) the unit ball of X. The properties of the Banach lattices Lp (m) and Lpw (m), for p ≥ 1, can be found in ([4] and [6]). In particular, neither Lp (m) nor Lpw (m) have to be reflexive spaces even if p > 1. For a given function g ∈ L0 (m), we can always consider the multiplication operator Mg : f ∈ L0 (m) −→ Mg (f ) := gf ∈ L0 (m) . This research has been partially supported by La Junta de Andaluc´ıa. The authors acknowledge the support of the Ministerio de Educaci´ on y Ciencia of Spain and FEDER, under projet MTM2006-11690-C02.

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In paper [2] we proved that if g ∈ Lq (m), then the multiplication operators Mg : Lp (m) −→ L1 (m) and Mg : Lpw (m) −→ L1 (m) are well defined, and if g ∈ Lqw (m), then the multiplication operators Mg : Lpw (m) −→ L1w (m) and Mg : Lp (m) −→ L1 (m) are also well defined. Moreover, we characterize the continuity of these operators. Furthermore, for g ∈ L∞ (m) the following multiplication operators Mg : Lp (m) −→ Lp (m) and Mg : Lpw (m) −→ Lpw (m) are well defined. Although the multiplication operator from Lpw (m) to Lp (m) is not always defined, we found, in the cited paper [2], conditions under which the multiplication from Lpw (m) to Lp (m) is continuous. We can notice that there exists an asymmetric behavior of multiplication operators depending on its domain when the continuity is considered. In the following section we will prove that this asymmetric behavior disappears when compactness and weak compactness are considered. In the sequel, if E and F are Banach lattices we denote by B (E, F ) the Banach space of all linear and continuous operators from E into F , and by K (E, F ) and W (E, F ) the ideals of compact and weakly compact operators, respectively. 1 1 Throughout the paper p, q > 1 will be conjugated exponents, that is, + = 1. p q

2. Compactness and weak compactness Next results will make evident that the behavior of multiplication operators must be the same when we deal with compactness-type conditions instead of continuity. Lemma 2.1. Let p, r ≥ 1 two real numbers and let g ∈ L0 (m) a function for which Mg ∈ W (Lp (m), Lr (m)). Then Mg ∈ B (Lpw (m), Lr (m)) and Mg (B(Lpw (m))) ⊆ Mg (Lp (m)) In particular, Mg ∈

Lr (m)

.

W (Lpw (m), Lr (m)).

Proof. First we are going to prove that Mg (Lpw (m)) ⊆ Lr (m). Let f ∈ Lpw (m). For each n = 1, 2, . . . , consider the set An := {w ∈ Ω : |f (w)| ≤ n}, and put fn := f χAn . Since fn ∈ Lp (m), and |fn | ≤ |f | we have fn Lp (m) ≤ f Lpw (m) ,

n = 1, 2, . . .

Now, from Mg ∈ W (L (m), L (m)), it follows that (gfn )n has a weakly convergent subsequence (gfnk )k in Lr (m). By [3, Corollary 2.2], there exists (ϕk )k such that ϕk ∈ co gfnj : j ≥ k , k = 1, 2, . . . p

r

and (ϕk )k converges to some h in Lr (m). Since (Ank )k ↑ Ω, it follows that ϕk χAnk = gf χAnk for all k = 1, 2, . . . and thus (ϕk )k → gf pointwise m-a.e. This gives gf = h m–a.e. and Mg (f ) = gf ∈ Lr (m). Next, we are going to prove that Mg (B(Lpw (m))) ⊆ Mg (B(Lp (m)))

Lr (m)

.

Compactness of Multiplication Operators

111

Note that, gf − gfn Lr (m) = gf χΩ\An Lr (m) → 0 by the Dominated Convergence Theorem, and so, the convergence of (gfn )n to gf is, in fact, in the norm of Lr (m). Thus, given ε > 0 and f ∈ B(Lpw (m)), there exists n ∈ N such that gf − gfn Lr (m) < ε. Hence, Mg (f ) = gf = gfn + (gf − gfn ) ∈ Mg (B(Lp (m))) + εB(Lr (m)), which proves that Mg (B(Lpw (m))) ⊆ Mg (B(Lp (m)))

Lr (m)

.

With the help of the previous lemma we can prove the following proposition. Proposition 2.2. Let 1 < p < ∞, g ∈ L0 (m), r = 1 or r = p and assume that A ∈ {W, K}. Then the following conditions are equivalent: 1) 2) 3) 4)

Mg Mg Mg Mg

∈ A (Lpw (m), Lr (m)). ∈ A (Lpw (m), Lrw (m)). ∈ A (Lp (m), Lrw (m)). ∈ A (Lp (m), Lr (m)).

Proof. The implications 1) =⇒ 2) and 2) =⇒ 3) are evident. 3) =⇒ 4) If Mg ∈ A (Lp (m), Lrw (m)) then Mg ∈ B (Lp (m), Lrw (m)) and from [2, Theorem 4] we obtain that Mg ∈ B (Lp (m), Lr (m)). In particular, the inclusion Mg (Lp (m)) ⊆ Lr (m) holds, and hence Mg ∈ A (Lp (m), Lr (m)). 4) =⇒ 1) Apply Lemma 2.1. Next we are going to characterize the weak compactness of the multiplication operator Mg in terms of the function g. Theorem 2.3. Let g ∈ L0 (m). The following conditions are equivalent: 1) g ∈ Lq (m). 2) Mg ∈ W Lp (m), L1 (m) . 3) Mg ∈ B Lpw (m), L1 (m) . Proof. 1) =⇒ 2) Taking into account [5, Definition 3.6.1, Proposition 3.6.5] it is sufficient to prove that hn L1 (m) → 0 for every disjoint sequence (hn )n ⊆ Mg (B (Lp (m))). Consider the disjoint measurable sets An := {w ∈ Ω : hn (w) = 0}, for n = 1, 2, . . . . Thus, hn = Mg (fn ) = gfn = gfn χAn = gχAn fn for some sequence (fn )n ⊆ B (Lp (m)). From H¨ older’s inequality we deduce that hn L1 (m)

=

Mg (fn )L1 (m) = gχAn fn L1 (m)

=

gχAn Lq (m) fn Lp (m) ≤ gχAn Lq (m) ,

but gχAn Lq (m) → 0 since (gχAn )n is an order bounded disjoint sequence in Lq (m) and the space Lq (m) is order continuous. 2) =⇒ 3) It is a direct application of Lemma 2.1. Finally 3) =⇒ 1) is obtained from [2, Theorem 4].

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Next, we will consider, for a fixed set G ∈ Σ, the measurable space (G, ΣG ), where ΣG := {A ∈ Σ : A ⊆ G}, and the vector measure mG : A ∈ ΣG −→ mG (A) := m(A) ∈ X. We denote, respectively, by EG and RG the extension and restriction maps from L0 (mG ) to L0 (m). It is easy to establish that EG (RG (f )) = χG f , for all f ∈ L0 (m). 1 0 , for all Theorem 2.4. Let g ∈ L (m) and denote by Gn := w ∈ Ω : |g (w) | ≥ n n = 1, 2, . . . . The following assertions are equivalent: 1) g ∈ L∞ (m) and Lp (mGn ) is reflexive for all n = 1, 2, . . . . 2) Mg ∈ W (Lpw (m), Lp (m)). 3) Mg ∈ B (Lpw (m), Lp (m)). Proof. 1) =⇒ 2) Since g ∈ L∞ (m), by [2, Theorem 8] we have that Mg ∈ B (Lpw (m), Lpw (m)). Consider now the sequence MgχGn n of multiplication operators defined in Lpw (m). For each n = 1, 2, . . . we claim that MgχGn belongs to W (Lpw (m), Lp (m)). Indeed, note that the composition Mg

RG

EG

n n → Lpw (mGn ) = Lp (mGn ) −−−− → Lp (m) Lpw (m) −−−−→ Lpw (m) −−−−

is a weakly compact operator, since Lp (mGn ) is reflexive and, particularly, Lpw (mGn ) = Lp (mGn ). But, for each f ∈ Lpw (m), EGn RGn Mg (f ) = MgχGn (f ). Let us consider now the set G := {w ∈ Ω : g(w) = 0} and denote Cn := G \ Gn for all n = 1, 2, . . . Thus we have Mg − MgχG = MgχCn B(Lp (m),Lp (m)) p p n B(Lw (m),Lw (m)) w w = sup gχCn f Lpw (m) : f Lpw (m) ≤ 1 1 → 0. n Therefore Mg is the uniform limit in B (Lpw (m), Lpw (m)) of the operators MgχGn which belong to W (Lpw (m), Lp (m)). Since Lp (m) is closed in Lpw (m) we conclude that Mg ∈ W (Lpw (m) , Lp (m)). The implication 2) =⇒ 3) is evident. 3) =⇒ 1) We apply [2, Theorem 10], having in mind that Lp (m) is reflexive if and only if Lp (m) = Lpw (m), see [4, Corollary 3.10]. ≤

gχCn L∞ (m) ≤

Having in mind that the spaces Lp (m) are ideal spaces and that for a measurable function g the multiplication operator Mg is the same as the superposition operator F generated by the super-measurable function f (s, x(s)) = g(s)x(s), we obtain as a particular case of [1, Theorem 2.5], the following characterization for the operator Mg . Theorem 2.5. Let g ∈ L0 (m), then the following conditions are equivalent: 1) g ∈ Lq (m) and gχB = 0, where B is the non-atomic part of the vector measure m. 2) Mg ∈ K Lp (m), L1 (m) .

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1 , for all Theorem 2.6. Let g ∈ L (m) and denote by Gn := w ∈ Ω : |g (w) | ≥ n n = 1, 2, . . . . The following assertions are equivalent: 1) g ∈ L∞ (m) and dim (Lp (mGn )) < ∞, for all n = 1, 2, . . . . 2) Mg ∈ K (Lp (m), Lp (m)). 0

References [1] J. Appell and P.P. Zabrejko, Nonlinear superposition operators, Cambridge University Press, Cambridge, 1990. [2] R. del Campo, A. Fern´ andez, I. Ferrando, F. Mayoral and F. Naranjo, Multiplication operators on spaces on integrable functions with respect to a vector measure, J. Math. Anal. Appl. 343 (2008), 514–524. [3] J. Diestel, W.M. Ruess and W. Schachermayer, On weak compactness in L1 (μ, X), Proc. Amer. Math. Soc. 118 (1993), 447–453. [4] A. Fern´ andez, F. Mayoral, F. Naranjo, C. S´ aez and E.A. S´ anchez–Pérez, Spaces of p-integrable functions with respect to a vector measure, Positivity 10 (2006), 1–16. [5] P. Meyer–Nieberg, Banach Lattices, Springer-Verlag. Berlin. 1991. [6] S. Okada, W. Ricker and E.A. S´ anchez–Pérez, Optimal domain and Integral extension of Operators acting in Function Spaces, Operator Theory: Advances and Applications, vol. 180, Birkh¨ auser Verlag, Basel, 2008. Ricardo del Campo Dpto. Matem´ atica Aplicada I EUITA, Ctra. de Utrera Km. 1 E-41013 Sevilla, Spain e-mail: [email protected] Antonio Fern´ andez, Fernando Mayoral and Francisco Naranjo Dpto. Matem´ atica Apda. II Escuela Técnica Superior de Ingenieros E-41092 Sevilla, Spain e-mail: [email protected] [email protected] [email protected] Irene Ferrando I.M.P.A., Universidad Politécnica de Valencia E-46022 Valencia, Spain e-mail: [email protected]


Some Applications of Nonabsolute Integrals in the Theory of Differential Inclusions in Banach Spaces Kinga Cicho´ n and Mieczyslaw Cicho´ n Abstract. In this paper we present a brief historical review about multivalued integrals and its relations with differential inclusions. Then a new theorem about existence of solutions (in some weak sense) for differential inclusions in Banach spaces is proved (by using some properties of nonabsolute integrals). Mathematics Subject Classification (2000). Primary 28B20; Secondary 34A60. Keywords. Aumann integrals, Pettis integrals, Henstock-Kurzweil-Pettis integrals, selections, differential inclusions, nonlocal Cauchy problems.

1. Introduction When we try to solve some problems for differential equations, we deal with different kind of “solutions”. Most of them are related to some integral equations with an appropriate definition of an interval. For instance, the Henstock-Kurzweil integral was constructed as a solution to the Cauchy problem. It was claimed that to recover a function from its derivative, the Lebesgue integral is not sufficient, so it is not sufficiently convenient for solving differential equations (Kurzweil, 1957). Similar problem for weak derivatives was solved by introducing the Henstock-KurzweilPettis integral (1999). Now, various types of integrals are used for solving such problems. But when we try to extend these results for differential inclusions, we need some new definitions of the multivalued integrals. The problem lies in not sufficiently investigated properties of multivalued integrals. In this paper we present some brief historical review about multivalued integrals. Due to lack of systematic study of different kind of solutions for differential inclusions we concentrate on this topic. To do this we present some results about multivalued nonabsolute integrals. Both single and multivalued integrals are really useful in theories of differential equations and inclusions and allow to unify separately considered cases (cf. [14]). Applicability of such integrals will be clarified in the next sections.

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K. Cicho´ n and M. Cicho´ n

Then a new theorem about existence of solutions (in some weak sense) for differential inclusions in Banach spaces is proved (by checking some properties of nonabsolute integrals). For a brief survey about different kind of solutions for differential equations see [14]. Let E be a Banach space and E ∗ be its dual space. The closed unit ball of ∗ E is denoted by B(E ∗ ). By Br we denote a closed ball {x ∈ E : x ≤ r}. In the whole paper I stands for a compact interval [0, α]. By cwk(X) we denote the family of all nonempty convex weakly compact subsets of X. For every bounded and convex set C the support function of C is denoted by s(·, C) and defined on E ∗ by s(x∗ , C) = supx∈C x∗ , x, for each x∗ ∈ E ∗ . A multifunction G : E → 2E with nonempty, closed values is called weakly sequentially upper hemi-continuous (w-seq uhc) iff for each x∗ ∈ E ∗ s(x∗ , G(·)) : E → R is sequentially upper semicontinuous from (E, w) into R. For a bounded subset A ⊂ E, we define the measure of weak noncompactness ω(A) (in the sense of DeBlasi [15]): ω(A) is the infimum of all ε > 0 s.t. there exists a weakly compact set K in E with A ⊆ K + ε · B1 . By ωC we will denote the measure of weak noncompactness in the space C(I, E). Let F : [a, b] → E and A ⊂ [a, b]. The function f : A → E is a pseudoderivative of F on A if for each x∗ in E ∗ the real-valued function x∗ F is differentiable almost everywhere on A and (x∗ F ) = x∗ f a.e. on A (Pettis, 1938). Unfortunately, we are unable to present in this short paper all necessary definitions. We leave the reader to remind the classical definitions of integrals ([22]). For the definitions and results about different concepts of solutions we refer the readers to [14].

2. Multivalued integrals We present below a brief survey about nonabsolute multivalued integrals. Such integrals were defined, in general, to solve some problems for differential (or integral) inclusions. For theorems about applications in the theory of differential inclusions we refer to the papers cited in this section. In the next section we will present a new example of application for multivalued integrals. It is rather unknown, that the study of multivalued integrals was begun by Alexander Dinghas in 1956 [17] who adapted the definition of the Riemann integral to the multivalued context (the Riemann-Minkowski integral). The same idea of Riemann sums was rediscovered by Hukuhara in 1967 [23] and developed by Artstein and Burns [4] or Jarnik-Kurzweil [24] (without using the Hausdorff distance in the definition). But the theory of integration for multivalued mappings is intensively studied since Aumann’s work of 1965 which is based on another idea. The Aumann integral is well suited for applications to various mathematical fields, in particular to solve some problems for differential inclusions. In fact, the next two treatments of the multivalued integrals is still considered. Aumann’s idea is connected with using

Some Applications of Nonabsolute Integrals

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some selection theorems. In the last concept it was defined the multivalued integral as a usual integral in the space of subsets (by using the Radstr¨ om embedding theorem or other isometries for spaces of subsets) – cf. [26] or [10], for instance. As claimed above, the first one was begun by Aumann [5] in 1965 and since then this idea is intensively developed. We concentrate on this kind of integrals as it seems to be really fruitful when we solve some problems for differential inclusions. For the second one we have the Debreu integral which stands for the Bochner integral for multifunctions or the Castaing integral for multivalued Pettis integrals. Let us remark, that by using the Radstr¨ om embedding theorem or due to the Hörmander theorem it is possible to extend the other type of integrals for multifunctions with convex and compact values (the Debreu (Bochner) integral [16], the generalized McShane multivalued integral [8], the multivalued Birkhoff integral [9] or the (Debreu-)Pettis integral [10] or [9], for instance). The problem of integrability, properties of the integrals as well as comparison results between different kind of multivalued integrals are investigated as basic problems in many papers, including mentioned above. And now a few words about selected multivalued integrals. We denote by SF1 the set of all Bochner integrable selections of F, namely SF1 = {f ∈ L1 (E) : f (s) ∈ F (s) almost everywhere}. If F is a measurable multifunction and SF1 = ∅, then the Aumann integral (shortly (A)-integral) of F is given by (A) T F (s)ds = { T f (s)ds : f ∈ SF1 }. We will call a multifunction Aumann “integrable” if this set is not empty. One of the advantages for the Aumann integrals lies in the fact that the values of “integrable” multifunctions need be neither convex nor compact sets. Let us note, that the Debreu integral is also a multivalued extension of the Bochner integral and under classical assumptions these two mentioned integrals are equivalent. If we consider multivalued nonabsolute integrals then the Pettis integral is the oldest one. For instance, the results which are the most interesting for dealing with applications of multivalued integrals for differential inclusions can be found in the book of Castaing-Valadier [11] or in the paper of Tolstonogov [29]. The (Castaing-) Pettis integrability means the Lebesgue integrability of the function s(x∗ , F (·)) for multifunctions with convex compact [or: weakly compact] as well as “Pettis” integrability in the appropriate space of subsets (for each measurable subset A of the domain there exists a convex compact [weakly compact] set which realize the integral of s(x∗ , F (·)) over A). Cf. also [1]. A full theory for this topic (for many classes of values of F ), including more general definitions of the Pettis integral for multifunctions, can be found in [20]. The last paper deals also with Pettis integrability for multifunctions with possibly unbounded values and contains interesting examples of different kind of Pettis integrable multifunctions. Let us recall, that the Pettis integral for multifunctions was defined also via some isometries in [9] (the Debreu-type of integral) or via selection theorems (Valadier [30]). Then last concept (considered in [20]) is based on the idea of

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Aumann: (AP ) T F (s)ds = {(P ) T f (s)ds : f ∈ SFP e }, where SFP e denotes the set of all Pettis integrable selections of F provided that this set is not empty. As remarked above, an interesting discussion about all the types of Pettis integrals can be found for instance in [20]. The properties of multivalued Pettis integrals are intensively studied: [1], [10], [31], [2] or [20], for instance. Now, consider the multivalued integrals based on idea of Kurzweil and Henstock. If we deal with the integrals based on Riemann sums, we can treat such integrals as special cases of Henstock-Kurzweil integral. Thus, we can consider the integrals from [17], [23], [4], [24] or [8]. Although the application of such integrals for differential inclusions lies in a basis of the work of Jarnik and Kurzweil [24], the idea of using Aumann-type integrals is still better known. To facilitate some comparison of integrals, in the multivalued case, an Aumann-type integral will be also considered via Henstock-integrable selections. The really first definitions of multivalued Henstock-Kurzweil integrals can be found in [4] and [24]. For the Riemann-type definition and Debreu-type definition of the Henstock-Kurzweil multivalued integral see also [19]. A set-valued function F : [0, 1] → 2E is AumannHenstock-Kurzweil integrable if the collection of its HK-integrable selections SFHK is non-empty. In this case, the definition of Aumann-Henstock integrals of F is analogous to that of Aumann integral. As in a single-valued case it is possible to unify both concepts presented above by introducing a new integral. For multifunctions it can be found in [18]. If a multifunction has closed convex and bounded values we can define, similarly like in previous considerations, the multivalued Henstock-Kurzweil-Pettis integral by assuming scalar HK-integrability together with existence of a closed convex and bounded set IA in E such that s(x∗ , IA ) = (HK) A s(x∗ , F (t)) dt for each subinterval IA of I ([18], [19]). It is possible also to consider the integrability for different classes of subsets ([20] ). Some examples can be found in [18, Section 3].

3. Results Now, let us present a few definitions of solutions for the Cauchy problem (t ∈ I) and we show some dependencies between them: x (t) = f (t, x(t)) , x(0) = x0 ∗

(3.1)

∗

or a scalar problem considered for arbitrary x ∈ E : (x∗ x) (t) = x∗ f (t, x(t)) , x(0) = x0 .

(3.2)

For a more complete theory see [14]. Let us start with some assumptions. (1◦ ) Continuity of solutions: a) x is absolutely continuous (AC function), b) x is an ACG∗ function, c) x is weakly ACG∗ . (2◦ ) Differentiability hypotheses: a) x ∈ C 1 , b) x is weakly differentiable, c) x is differentiable a.e., d) x is pseudo-differentiable. A function x is a solution of (3.1) if it satisfies the initial condition together with: a) classical solution ⇔ (1◦ a) and (2◦ a), b) weak solution ⇔ (1◦ a) and


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(2◦ b), (Szep, 1971) c) Carathéodory solution ⇔ (1◦ a) and (2◦ c), d) pseudosolution ⇔ (1◦ a) and (2◦ d), (Knight, 1974) e) Kurzweil solution (K-solution) ⇔ (1◦ b) and (2◦ c), (Kurzweil, 1957) f) pseudo-K-solution ⇔ (1◦ c) and (2◦ d), (Cicho´ n et al., 1999) and the derivative of x taken in the sense of (2◦ ) satisfies the equation (3.1) (for each t in the cases (2◦ a) and (2◦ b), almost everywhere in the cases (2◦ c) and (2◦ e) or satisfies (3.2) in the cases (2◦ d) and (2◦ f )). Let us note that under the appropriate conditions of integrability of the righthand side each solution of the problem (3.1) is equivalent to the solution of the integral problem t x(t) = x0 + f (s, x(s))ds, (3.3) 0

where the integral is depending on the type of considered solutions of (3.1). Theorem 3.1. ([14]) If the right-hand side of (3.1) is integrable in the sense considered below, respectively, then each solution x of the problem (3.1) is equivalent to the solution y of the integral equation (3.3) in the following cases: (a) (b) (c) (d) (e) (f)

x x x x x x

– – – – – –

classical solution ⇔ the Riemann integral, weak solution ⇔ the weak Riemann integral, Carathéodory solution ⇔ the Bochner (Lebesgue) integral, pseudo-solution ⇔ the Pettis integral, K-solution ⇔ the Henstock-Kurzweil integral, pseudo-K-solution ⇔ the Henstock-Kurzweil-Pettis integral.

As a consequence of the above theorem we get the following fact ([14]): if we consider the following classes of solutions for the problem (3.1): (a) classical solutions, (b) Carathéodory solutions, (c) weak solutions, (d) pseudo-solutions, (e) Kurzweil solutions, (f) pseudo-K-solutions. Then (a) ⊂ (c) ⊂ (b) ⊂ (d) ⊂ (f ) and (b) ⊂ (e) ⊂ (f ). Moreover, all these inclusions are proper. Let us turn to the differential inclusions. Consider the following problem: x (t) ∈ F (t, x(t)),

x(0) = g(x),

t ∈ [0, α] = I .

(3.4)

In contrast to the single-valued case only in a limited number of papers it was considered the other solutions than Carathéodory ones. In the beginning, the Bochner integrability was replaced by the Pettis integrability, but due to the other assumptions as a solution was still considered a Carathéodory (cf. Castaing, Valadier [11] Theorem VI-7). But in the paper of Tolstonogov both type of solutions were checked: Carathéodory or weak one ([29], cf. also Maruyama [25]). The next result was obtained by Arino, Gautier and Penot [3] for weakly-weakly usc multifunctions F with almost weakly relatively compact images. In the last paper the existence of pseudo-solutions was proved, but under the assumptions from this paper they are in fact Carathéodory solutions, too. Let us note that in all mentioned papers it was considered the local problem, i.e., g(x) = x0 . In the last years, thanks to the progress of the theory for multivalued integrals, this theory was rediscovered. Let us mention the papers of Godet-Thobie and Satco [21], Satco [27],

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[28], Azzam-Laouir, Castaing and Thibault [7] or Azzam-Laouir and Boutana [6] (results for the different kind of multivalued problems). Let us begin by proving the following important lemma (cf. [12], [6] or [27]): Lemma 3.2. Assume that E ∗ is separable. Let v ∈ C(I, E) and F : I × E → 2E is such that: (i) F (·, x) – has a weakly measurable selection for each x ∈ E, (ii) F (t, ·) – w-seq. uhc for each t ∈ I, (iii) F (t, x) are nonempty, closed and convex, (iv) F (t, x) ⊂ G(t) a.e., G has nonempty convex and weakly compact values and is Pettis uniformly integrable on I. Then there exists at least one Pettis integrable selection z0 of F (·, v(·)). Proof. Take a sequence of simple functions vn , such that vn → v uniformly on I. Thus by (i) there exists a weakly measurable selection zk s.t. zk (·) ∈ F (·, vk (·)). By our assumption on E the selection zk is, in fact, measurable. Put H(t) = conv{zk (t) : k ≥ 1}. Since zk (·) is measurable, {zk (·) : k ≥ 1} is measurable and hence H(·) is measurable. Moreover H(t) ⊂ convF (t, V (t)), where V (t) = {vk (t) : k ≥ 1}. But (vk ) is a convergent sequence, so V (t) is relatively compact. By Lemma 2 from [12] and Mazur’s lemma we have that the values convF (t, V (t)) are weakly compact. Our multifunction H is weakly measurable, H(t) ⊂ G(t) a.e. therefore by our assumption (iv) and due to Lemma 2.2. from [10] H is Pettis integrable. Since Pe SH = ∅, it is sequentially compact for the topology of pointwise convergence on L∞ (I, E)⊗E ∗ (cf. [2] Proposition 3.4 and [27] for the definition of Pettis uniformly integrable multifunctions). Thus we are able to extract a subsequence (znk ) of Pe (zn ) which is convergent in σ(PE1 , L∞ (I, E) ⊗ E ∗ ) topology to some z0 ∈ SH , i.e., Pe 1 znk −→ z0 ∈ SH . Here PE denotes the space of Pettis integrable functions from I to E. Denote by D a dense sequence for the Mackey topology in the unit ball in E ∗ . For each fixed x∗ ∈ D, Lemma 12 from [27] applied to the sequence x∗ znk gives us the existence of the “Mazur” sequence vk ∈ conv{x∗ znm : m ≥ k} such that vk is a.e. pointwisely convergent to some measurable x∗ z0 . The convergence theorem (Lemma 1 from [12]) can be applied in our situation (with some necessary changes in a suitable part of the proof: Lemma III.33 together with Corollary I.15 from [11] instead of the Separation Theorem). Thus x∗ z0 (t) ≤ s(x∗ , F (t, v(t))) a.e. and finally z0 (t) ∈ F (t, v(t)). Let us present an existence result for the problem (3.4). We extend some previous results by using an assumption expressed in terms of the measure of weak noncompactness instead of the strong one. This allows to cover the case of weakly compact mappings (or the sum of Lipschitz and weakly compact mappings). Moreover, this results remain true also for standard Cauchy problem, i.e., when g(x) = x0 as well as for classical functions g which are useful to describe some phenomena k by using nonlocal (nonstandard) conditions. For instance: g(x) = n=1 cn · x(tn ) T for some 0 ≤ t1 ≤ · · · ≤ tk ≤ T or g(x) = T10 · 0 0 x(s)ds for T0 < T .


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Theorem 3.3. Assume that E has separable dual E ∗ . Let F : I × E → 2E with nonempty convex and weakly compact values satisfies (a) F (t, ·) is weakly sequentially upper hemi-continuous for each t ∈ I, (b) for each continuous function x : I → E the multifunction F (·, x(·)) has Pettis-integrable selection, (c) F (t, x) ⊂ G(t) a.e., for some Pettis (uniformly) integrable multifunction G with values in cwk(E), (d) there exists a constant k < 1 such that for each bounded subset B ⊂ E and J ⊂ I we have ω(F (J × B)) ≤ k · ω(B), (e) the function g : E → E is convex, bounded say by N1 , i.e., g(x) ≤ N1 for each x, weakly-weakly sequentially continuous and ω(g(B)) ≤ d · ωC (B) for each bounded subset B of E and for some d such that d + k < 1. Then there exists at least one pseudo-solution of the Cauchy problem (3.4) on some J ⊂ I. Proof. By the hypothesis on the space E we get the uniqueness of the pseudoderivative accurate to a set of measure zero. From the definition of the pseudosolution it follows that (as in earlier papers about nonlocal problems) a pseudosolution of our problem (3.4) is, at the same time, a solution of the integral inclusion t x(t) ∈ g(x) + (P ) 0 F (s, x(s))ds. Then, we are interesting in finding a fixed point of R : C(I, E) → 2C(I,E) : t R(x)(t) = g(x) + (P ) 0 F (s, x(s))ds. Let W = {f ∈ PE1 : f (t) ∈ G(t) a.e. on I} and U = {xf ∈ C(I, E) : xf (t) = t g(xf ) + (P ) 0 f (s)ds, t ∈ I, f ∈ W }. For f ∈ W and x∗ ∈ E ∗ we have x∗ f ≤ s(x∗ , G). Then, by our assumptions, W is Pettis uniformly integrable (cf. [21]). Thus for arbitrary xf ∈ U and t, τ ∈ I there exists an appropriate f ∈ W and t τ ∗ ∗ x f (s)ds − x∗ f (s)ds) xf (t) − xf (τ ) = sup x (xf (t) − xf (τ )) = sup ( x∗ ∈B ∗

= sup

x∗ ∈B ∗

τ

t

x∗ f (s)ds ≤ sup

x∗ ∈B ∗ t

x∗ ∈B ∗

0

0

s(x∗ , G(s))ds.

τ

By uniform Pettis integrability of G it follows that U is an equicontinuous subset of C(I, E). The property of the multivalued Pettis integral gives us the convexity of U . Then U is nonempty, closed, convex, bounded and equicontinuous in C(I, E). As the set U is strongly equicontinuous then for each M > 0 there exists α ∈ I t such that for each t ∈ [0, α] and f ∈ SFP e we have (P ) 0 f (s)ds ≤ M. We fix M > 0. We have the following estimation: t R(x)(t) ≤ g(x) + (P ) 0 F (s, x(s))ds ≤ N1 + M. Denote by N = N1 + M and so by BN the ball {x ∈ C(I, E) : x ≤ N }. e Lemma 3.2 ensures us, that for each x ∈ C(I, E) SFP (·,x(·)) is nonempty, then R(x) = ∅ (for each x ∈ BN ). By the properties of multivalued Pettis integrals,

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the values of R are weakly compact and convex (because F has such values). Let e e ⊂ W for each x ∈ V . Indeed, for g ∈ SFP (·,x(·)) V = U ∩BN . It is clear that SFP (·,x(·)) g(t) ∈ F (t, x(t)) ⊂ G(t) a.e. In the next part of the proof we will consider V as a domain of R. Obviously R : V → 2V . Now, we are in a position to show, that R has a weakly-weakly sequentially closed graph. Let (xn , yn ) ∈ GrR, (xn , yn ) → (x, y) weakly in C(I, E) × C(I, E). From our assumptions it follows that g(xn ) tends weakly to g(x). Moreover, yn is of the following form t e yn (t) = g(xn ) + (P ) 0 fn (s)ds , fn ∈ SFP (·,x , t ∈ I. n (·)) But ω({fn (t) : n ≥ 1}) ≤ ω({F (t, xn (t)) : n ≥ 1} ≤ k · ω{xn (t) : n ≥ 1}) a.e. Since xn is weakly convergent in C(I, E), {xn (t) : n ≥ 1} is relatively weakly compact in E. Hence ω({xn (t) : n ≥ 1}) = 0 and finally ω({fn (t) : n ≥ 1}) = 0 a.e. on I. By redefining a new multifunction H on the set of measure zero: H(t) = conv{fn (t) : n ≥ 1} we can say that H(t) are nonempty closed convex and weakly compact. Pe As SH is nonempty, convex and is sequentially compact for the topology of pointwise convergence on L∞ ⊗ E ∗ ([2] Proposition 3.4), then we extract a subsequence (fnk ) of (fn ) such that (fnk ) converges σ(PE1 , L∞ ⊗ E ∗ ) to a function Pe f ∈ SH in such a way that (xnk ) converges weakly to a continuous function x. Fix an arbitrary x∗ ∈ E ∗ . By weak sequential hemi-continuity of F (t, ·) and weak convergence of xnk (t) in E we obtain weak convergence of (s(x∗ , F (t, xnk (t))). Remember, that C ∈ cwk(E) iff s(·, C) is τ (E ∗ , E)-continuous on E ∗ . Denote by D a dense sequence for the Mackey topology in the ball in E ∗ . Since (fnk ) unit 1 ∞ ∗ Pe ∗ converges σ(PE , L ⊗ E ) to f ∈ SH , we have A x fnk (s)ds → A x∗ f (s)ds t for each measurable A ∈ I and x∗ ∈ E ∗ . Thus ynk = g(xnk ) + (P ) 0 fnk (s)ds t and for each x∗ ∈ D we obtain that x∗ ynk tends to x∗ g(x) + 0 x∗ f (s)ds, i.e., t t e y = g(x) + (P ) 0 f (s)ds. Whence y(t) = g(x) + (P ) 0 f (s)ds , f ∈ SFP (·,x(·)) and (x, y) ∈ GrR. Take an arbitrary bounded subset B of V . For any selection f of F (·, B(·)) t we have (P ) 0 f (s)ds ∈ t · convf (I) ⊂ t · convF (I × B(I)). Thus R(B)(t) ⊂ g(B) + t · convF (I × B(I)). By using the properties of ωC we are able to prove that R is a contraction with respect to the measure of weak noncompactness: ω(R(B)(t)) ≤ ω(g(B))+ ω(F (I × B(I)) ≤ d·ωC (B)+ k ·ω(B(I)) ≤ (d+ k)·ωC (B). property. Thus ωC R(B) ≤ (d + k)ωC (B). As d + k < 1 we obtain the desired ∞ Define a sequence of sets: K0 = V , Kn+1 = convR(Kn ), and a set K = n=0 Kn . All the sets Kn are nonempty equicontinuous closed and convex. Moreover, it can be proved (by induction) that it is a decreasing sequence of sets. t We have ω(R(Kn (t)) ≤ ω(g(Kn )) + ω((P ) 0 F (s, Kn (s))ds). By the mean value theorem for the Pettis integral: t (P ) t−τ F (s, Kn (s)ds ∈ τ · conv{F (s, Kn (s)) : s ∈ [t − τ, t]}. Thus ω(R(Kn (t))) ≤ d·ωC (Kn )+ t·ω(convF ([0, t]× Kn ([0, t]))) ≤ d·ωC (Kn ) +t · k · ω(Kn ([0, t])). For sufficiently small t we have ω(R(Kn (t)) < ωC (Kn ) since ωC (R(Kn )) = supt∈I ω(R(Kn (t))). Finally ωC (R(Kn )) < ωC (Kn ). As ω(Kn (t)) =


123

ω(convR(Kn−1 (t))) = ω(Kn−1 (t)) we obtain ωC (R(Kn )) = ωC (Kn−1 ) and consequently ωC (Kn−1 ) < ωC (Kn ). The sequence (ωC (Kn )) is decreasing and bounded below by zero, so is convergent. From the above consideration it follows also that ωC (Kn ) ≤ (d + k)n ωC (K0 ) and therefore the limit must be zero. The CantorKuratowski intersection lemma for the weak measure of noncompactness ([15]) ensures us that K is weakly compact (cf. [12]). By the properties of multivalued mappings we obtain that R is weakly-weakly upper semi-continuous on K. Then we have weakly-weakly upper semi-continuous multifunction R : K → cwk(K) (R(K) ⊂ K, cf. Lemma 2 in [12]), so the fixed point theorem of Kakutani type for weak topology ([3]) applies to the map R and we get a fixed point z of R. Of course, z is a pseudo-solution of problem (3.4).

References [1] A. Amrani, Lemme de Fatou pour l’int´ egrale de Pettis, Publ. Math. 42 (1998), 67–79. [2] A. Amrani, Ch. Castaing, Weak compactness in Pettis integration, Bull. Polish Acad. Sci. Math. 45 (1997), 139–150. [3] O. Arino, S. Gautier and J.P. Penot, A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations, Funkc. Ekvac. 27 (1984), 273–279. [4] Z. Artstein, J. Burns, Integration of compact set-valued functions, Pacific J. Math. 58 (1975), 297-3-7. [5] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1965), 1–12. [6] D. Azzam-Laouir, I. Boutana, Application of Pettis integration to differential inclusions with three-point boundary conditions in Banach spaces, Electron. J. Differential Equations 173 (2007), 1–8. [7] D. Azzam-Laouir, C. Castaing and L. Thibault, Three boundary value problems for second-order differential inclusions in Banach spaces. Well-posedness in optimization and related topics, Control Cybernet. 31 (2002), 659–693. [8] A. Boccuto, A.R. Sambucini, A McShane integral for multifunction, J. Concr. Appl. Math. 2 (2004), 307–325. [9] B. Cascales, J. Rodr´ıgues, Birkhoff integral for multi-valued functions, J. Math. Anal. Appl. 297 (2004), 540–560. [10] B. Cascales, V. Kadets and J. Rodr´ıgues, The Pettis integral for multi-valued functions via single-valued ones, J. Math. Anal. Appl. 332 (2007), 1–10. [11] Ch. Castaing, M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer, Berlin, 1977. [12] M. Cicho´ n, Differential inclusions and abstract control problems, Bull. Austral. Math. Soc. 53 (1996), 109–122. [13] M. Cicho´ n, Convergence theorems for the Henstock-Kurzweil-Pettis integral, Acta Math. Hungar. 92 (2001), 75–82. [14] M. Cicho´ n, On solutions of differential equations in Banach spaces, Nonlin. Anal. Th. Meth. Appl. 60 (2005), 651–667.

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[15] F. DeBlasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie 21 (1977), 259–262. [16] G. Debreu, Integration of correspondences, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability 1965/66, Berkeley, 1967, pp. 351–372. [17] A. Dinghas, Zum Minkowskischen Integralbegriff abgeschlossener Mengen, Math. Zeit. 66 (1956), 173–188. [18] L. Di Piazza, K. Musial, Set-valued Kurzweil-Henstock-Pettis integral, Set-Valued Anal. 13 (2005), 167–179. [19] L. Di Piazza, K. Musial, A decomposition theorem for compact-valued Henstock integral, Monatsh. Math. 148 (2006), 119–126. [20] K. El Amri, Ch. Hess, On the Pettis integral of closed valued multifunctions, SetValued Anal. 8 (2000), 329–360. [21] C. Godet-Thobie, B. Satco, Decomposability and uniform integrability in Pettis integration, Quaest. Math. 29 92006), 39–58. [22] R.A. Gordon, The Integrals of Lebesgue, Denjoy, Perron and Henstock, Providence, Rhode Island, 1994. [23] M. Hukuhara, Intégration des applications measurable dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223. [24] J. Jarnik, J. Kurzweil, Integral of multivalued mappings and its connection with ˇ differential relations, Casopis pro Peˇstov´ ani Matematiky 108 (1983), 8–28. [25] T. Maruyama, A generalization of the weak convergence theorem in Sobolev spaces with application to differential inclusions in a Banach space, Proc. Japan Acad. Ser. A Math Sci. 77 (2001), 5–10. [26] A.R. Sambucini, A survey on multivalued integration, Atti Sem. Mat. Fis. Univ. Modena 50 (2002), 53–63. [27] B. Satco, Volterra integral inclusions via Henstock-Kurzweil-Pettis integral, Discuss. Math. Differ. Incl. Control Optim. 26 (2006), 87–101. [28] B. Satco, Second-order three boundary value problem in Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral, J. Math. Anal. Appl. 332 (2007), 919–933. [29] A.A. Tolstonogov, On comparison theorems for differential inclusions in locally convex spaces. I. Existence of solutions, Differ. Urav. 17 (1981), 651–659 (in Russian). [30] M. Valadier, On the Strassen theorem, in: Lect. Notes in Econ. Math. Syst. 102, 203-215, ed. J.-P. Aubin, Springer, 1974. [31] H. Ziat, Convergence theorems for Pettis integrable multifunctions, Bull. Polish Acad. Sci. Math. 45 (1997), 123–137. Kinga Cicho´ n Institute of Mathematics, Faculty of Electrical Engineering Poznan University of Technology, Piotrowo 3a, 60-965 Pozna´ n, Poland e-mail: [email protected] Mieczyslaw Cicho´ n Faculty of Mathematics and Computer Science, Adam Mickiewicz University Umultowska 87, 61-614 Pozna´ n, Poland e-mail: [email protected]


Equations Involving the Mean of Almost Periodic Measures Silvia-Otilia Corduneanu Abstract. We use the theory of Fourier-Bohr series for almost periodic measures to looking for a complex-valued function f which is almost periodic on R and satisfies equation f (x) = My [f (x − y)μ(y)] + ν ∗ h(x),

x ∈ R.

(E)

In this context h is an almost periodic function on R, μ is a positive almost periodic measure on R and ν is a bounded measure also on R. With a suitable choice of the measures μ and ν equation (E) becomes +∞ t 1 f (x) = lim f (x − y)g(y) dy + ϕ(y)h(x − y) dy, x ∈ R, t→∞ 2t −t −∞ where g is an almost periodic function on R and ϕ belongs to L1 (R). Mathematics Subject Classification (2000). Primary 42A05, 42A10, 42A16, 42A38, 43A25, 43A40, 43A60; Secondary 39B32. Keywords. Almost periodic function, almost periodic measure, functional equation, Fourier series, Fourier-Bohr coefficients.

1. Introduction Let AP (R) be the space of all almost periodic complex-valued functions defined on R, ap(R) the space of all almost periodic complex-valued measures on R and ap+ (R) the subspace of ap(R) containing positive measures. We denote by M (μ) or Mx [μ(x)] the mean of an almost periodic measure μ and by M (f ) or Mx [f (x)] the mean of an almost periodic function f . L.N. Argabright and J. Gil de Lamadrid proved that the measure f μ which is defined by an almost periodic function f as density and an almost periodic measure μ as base, is also an almost periodic measure (see [2], [9]). If f ∈ AP (R) and μ ∈ ap+ (R), the mean My [f (x − y)μ(y)] having the parameter x ∈ R is an almost periodic function as function of variable

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x ∈ R (see [4]). In this paper we solve equation f (x) = My [f (x − y)μ(y)] + ν ∗ h(x),

x ∈ R,

(1.1)

where h ∈ AP (R), μ ∈ ap+ (R) and ν is a bounded complex-valued measure on R. A solution is an almost periodic function f which satisfies the equation. Let λ be the Lebesgue measure on R, g ∈ AP (R), ϕ ∈ L1 (R). In the case of μ = gλ and ν = ϕλ, equation (1.1) becomes +∞ 1 t f (x) = lim f (x − y)g(y) dy + ϕ(y)h(x − y) dy, x ∈ R. (1.2) t→∞ 2t −t −∞ For solving equation (1.1) we use the theory of Fourier-Bohr series for almost 1 periodic measures and that of Fourier series for almost periodic functions. Let R be the dual group of the group R. Consider f ∈ AP (R) and μ ∈ R. For every 1 the Fourier coefficient of f corresponding to the character γ is denoted γ ∈ R, by cγ (f ) and is defined by cγ (f ) = M (γf ). On the other hand the Fourier-Bohr coefficient of μ corresponding to the character γ is denoted by cγ (μ) and is defined 1 | M (γf ) = 0} is at most a by cγ (μ) = M (γμ). It is proved that the set {γ ∈ R countable set (see Theorem 1.15 from [3]). Hence, we can denote the previous set 1 | n ∈ N} and we define the Fourier series of f as being by {γn ∈ R ∞

cγn (f )γn .

n=1

We calculate the mean of the almost periodic function Φ : R → C defined by Φ(x) = My [f (x − y)μ(y)],

x∈R

and we obtain the equality M (Φ) = M (f )M (μ) which induces the another one cγ (Φ) = cγ (f )cγ (μ),

1 γ ∈ R.

(1.3)

Taking into account (1.3) and the property that two almost periodic functions coincide if they have the same Fourier coefficients we solve equations (1.1) and (1.2).

2. Preliminaries Consider C(R) the set of all bounded continuous complex-valued functions on R and denote by · the supremum norm defined on C(R). Let λ be the Lebesgue measure on R. The space of Lebesgue measurable functions f on R, with R |f (x)| dλ(x) < ∞ will be denoted by L1 (R). We use mF (R) to denote the space of all bounded 1 to denote the dual group of the group R. Consider [R] 1 the measures on R and R 1 space of all trigonometric polynomials on R. If ϕ ∈ L (R), the Fourier transform of ϕ, denoted by ϕ, 1 is given by 1 ϕ(x)γ(x) dλ(x), γ ∈ R, ϕ(γ) 1 = R

Equations Involving the Mean of Almost Periodic Measures

127

and if ν ∈ mF (R) the Fourier-Stieltjes transform of ν, denoted by ν1, is given by 1 ν1(γ) = γ(x) dν(x), γ ∈ R. R

For f ∈ C(R) and a ∈ R, the translate of f by a is the function fa (x) = f (x + a) for all x ∈ R. In [2], [3], [7], [8], [9], there are defined the almost periodic functions. Definition 2.1. A function f ∈ C(R) is called an almost periodic function on R, if the family of translates of f , {fa | a ∈ R} is relatively compact in the sense of uniform convergence on R. The set AP (R) of all almost periodic functions on R is a Banach algebra with 1 There exists respect to the supremum norm, closed to conjugation and contains R. a unique positive linear functional M : AP (R) → C such that M (fa ) = M (f ), for all a ∈ R, f ∈ AP (R) and M (1) = 1. We denote by 1 the constant function which is 1 for all x ∈ R. If f ∈ AP (R) we define the mean of f as being the 1 and we call above complex number M (f ), we put cγ (f ) = M (γf ) for all γ ∈ R 1 cγ (f ), the Fourier coefficient of f corresponding to γ ∈ R. Next, we recall the definition of the Fourier series of an almost periodic function. If f ∈ AP (R) the 1 | M (γf ) = 0} is at most a countable set (see Theorem 1.15 from [3]) set {γ ∈ R 1 | n ∈ N}. The Fourier series of f is and we denote it by {γn ∈ R ∞

cγn (f )γn .

n=1

If ϕ ∈ L1 (R) and h ∈ AP (R), their convolution, ϕ ∗ h, belongs to AP (R). We recall that ϕ ∗ h(x) = h(x − y)ϕ(y) dλ(y), x ∈ R. R

If ν ∈ mF (R) and h ∈ AP (R) we have that their convolution, ν ∗ h, belongs to AP (R). We remind the reader that ν ∗ h(x) = h(x − y) dν(y), x ∈ R. R

We denote by m(R) the space of complex Radon measures on R and by mB (R) the subspace of m(R) containing the translations-bounded measures (see [1]). Let K(R) be the subset of C(R) containing functions which have a compact support. We say that μ ∈ mB (R) is an almost periodic measure, and we denote it by μ ∈ ap(R), if for every ϕ ∈ K(R), ϕ ∗ μ ∈ AP (R) (see [2], [9]). Consider ϕ ∈ K(R) such that λ(ϕ) = 1. We can define the mean of an almost periodic measure μ ∈ ap(R) as being the number M (μ) ∈ C defined by M (μ) = M (ϕ ∗ μ). L.N. Argabright and J. Gil de Lamadrid proved that the measure f μ which is defined by an almost periodic function f as density and an almost periodic measure μ as base, is also an almost periodic measure (see [2], [9]). Denote by ap+ (R) the subspace of ap(R) containing positive measures. If f ∈ AP (R) and μ ∈ ap+ (R), the mean My [f (x − y)μ(y)] having the parameter x ∈ R is an almost periodic function as function of

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1 and we variable x ∈ R (see [4]). If μ ∈ ap(R) we put cγ (μ) = M (γμ) for all γ ∈ R 1 call cγ (μ), the Fourier-Bohr coefficient of μ corresponding to γ ∈ R.

3. Properties of the almost periodic functions Theorem 3.1. Consider f ∈ AP (R) and A ⊂ R a compact set. The function χ : R → C defined by χ(y) = f (x − y) dλ(x), y ∈ R, A

is an almost periodic function on R. 1 We obtain Proof. We first suppose that f = γ ∈ R. 1A (γ). γ(x)γ(y) dλ(x) = γ(y) 1A (x)γ(x) dλ(x) = γ(y)1 χ(y) = R

A

1 then χ ∈ AP (R). It immediately follows, that for f ∈ [R], 1 Therefore, if f = γ ∈ R, the function χ is an almost periodic function on R. Suppose that f ∈ AP (R) and is arbitrary. There exists a sequence (fn )n such that 1 ∧ (fn − f → 0). ((∀n ∈ N)(fn ∈ [R])) If we denote by

fn (x − y) dλ(x),

χn (y) =

y ∈ R, n ∈ N

A

we obtain that (∀n ∈ N)(χn ∈ AP (R)). For every n ∈ N and y ∈ R we have |χn (y) − χ(y)| ≤ |fn (x − y) − f (x − y)| dλ(x) ≤ λ(A)fn − f . A

It follows that χn → χ, uniformly, hence χ ∈ AP (R).

Theorem 3.2. Consider μ ∈ ap+ (R). For every f ∈ AP (R) the following equality is true 2 t 3 t My [f (x − y)μ(y)] dx = My f (x − y) dx μ(y) . (3.1) −t

−t

1 We obtain Proof. We first suppose that f = γ ∈ R. t My [γ(x)γ(y)μ(y)] dx = cγ (μ) −t

and

2 My

t

−t

3 γ(x)γ(y) dx μ(y) = cγ (μ)

t

γ(x) dx −t t

γ(x) dx. −t

1 it follows that the equality (3.1) So, because the equality (3.1) is valid for f ∈ R 1 is valid for f ∈ [R]. Consider f ∈ AP (R). There exists a sequence (fn )n such that 1 and fn → f in the sense of uniform convergence. Consider (∀n ∈ N)(fn ∈ [R])

Equations Involving the Mean of Almost Periodic Measures

129

t > 0. For every n ∈ N we have the following inequalities 2 t 3 t My [f (x − y)μ(y)] dx − My f (x − y) dx μ(y) −t

≤

−t

My [f (x − y)μ(y)] dx − 2

+ My ≤

−t t

t

t

−t

−t

My [fn (x − y)μ(y)] dx

2 3 fn (x − y) dx μ(y) − My

t

−t

3 f (x − y) dx μ(y)

t

−t

+ My

My [(f (x − y) − fn (x − y))μ(y)] dx 2

t

−t

3 fn (x − y) − f (x − y) dx μ(y)

≤ 4tM (μ)fn − f . Therefore the equality (3.1) holds for every f ∈ AP (R).

Theorem 3.3. Consider μ ∈ ap+ (R) and f ∈ AP (R). If Φ : R → C is the almost periodic function defined by Φ(x) = My [f (x − y)μ(y)],

x∈R

then the following equality is true M (Φ) = M (f )M (μ). Proof. We have that 1 t→∞ 2t

(3.2)

t

M (Φ) = lim

−t

My [f (x − y)μ(y)] dx,

so, equality (3.2) is equivalent to 1 t lim My {[f (x − y) − f (x)]μ(y)} dx = 0. t→∞ 2t −t

(3.3)

From Theorem 3.2 we have that 1 t My {[f (x − y) − f (x)]μ(y)} dx 2t −t 2 t 3 1 = My (f (x − y) − f (x)) dx μ(y) . 2t −t According to Theorem 1.12 from [3] we obtain that 1 t [f (x − y) − f (x)] dx = 0, lim t→∞ 2t −t uniformly with respect to y ∈ R. Hence equality (3.3) follows.

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Corollary 3.4. Consider μ ∈ ap+ (R) and f ∈ AP (R). If Φ : R → C is the almost periodic function defined by Φ(x) = My [f (x − y)μ(y)],

x∈R

then the following equality is true cγ (Φ) = cγ (f )cγ (μ),

1 γ ∈ R.

(3.4)

1 We observe that Proof. Consider γ ∈ R. γΦ(x) = My [(γf )(x − y)γμ(y)],

x ∈ R.

Hence, using Theorem 3.3, it results that cγ (Φ) = M (γΦ) = M (γf )M (γμ) = cγ (f )cγ (μ).

4. Equations with almost periodic measures and functions We solve the following equation (4.1) f (x) = My [f (x − y)μ(y)] + ν ∗ h(x), x ∈ R, where h ∈ AP (R), μ ∈ ap+ (R) and ν ∈ mF (R). A solution of (4.1) is an almost periodic function f which satisfies the equation. Theorem 4.1. Consider a function h ∈ AP (R) such that its Fourier series is ∞ cγn (h)γn . n=1

Let μ ∈ ap+ (R) such that there exists δ > 0 with the property cγn (μ) − 1 > δ, and ν ∈ mF (R) such that

∞

n∈N

|1 ν (γn )|2 < ∞.

n=1

Then equation (4.1) has a solution f ∈ AP (R). 1 Let us suppose that f ∈ AP (R) is a solution of equation Proof. Consider γ ∈ R. (4.1). Then the function Φ : R → C defined by Φ(x) = My [f (x − y)μ(y)],

x∈R

is an almost periodic function and cγ (Φ) = cγ (f )cγ (μ). Taking into account that cγ (ν ∗ h) = ν1(γ)cγ (h), one obtains cγ (f ) = cγ (f )cγ (μ) + ν1(γ)cγ (h). We return to looking for a solution of (4.1). This previous observations suggest us to consider the series ∞ ν1(γn )cγn (h) γn . (4.2) 1 − cγn (μ) n=1

Equations Involving the Mean of Almost Periodic Measures It is obvious that ν1(γn )cγn (h) 1 γn (x) ≤ | ν1(γn ) | | cγn (h) |, 1 − cγn (μ) δ

131

x ∈ R.

On the other hand we have the Parseval equality 2

M (|h| ) =

∞

| cγn (h) |2 ,

n=1

hence the Cauchy inequality ∞

| ν1(γn ) | | cγn (h) | ≤

n=1

∞

12 | ν1(γn ) |

2

n=1

∞

12 | cγn (h) |

2

,

n=1

gets us that the series (4.2) is uniform convergent on R. We denote by f the sum of this series. It is obvious that f is an almost periodic function. Based on the property that two almost periodic functions coincide if they have the same Fourier series we conclude that f is a solution of equation (4.1). Next we discuss the equation ∞ 1 t f (x) = lim f (x − y)g(y) dy + h(x − y)ϕ(y) dy, t→∞ 2t −t −∞

x ∈ R.

(4.3)

In this context g ∈ AP (R), h ∈ AP (R) and ϕ ∈ L1 (R). A solution of (4.3) is an almost periodic function f which satisfies the equation. Corollary 4.2. Consider a function h ∈ AP (R) such that its Fourier series is ∞

cγn (h)γn .

n=1

Let g ∈ AP (R) a positive function such that there exists δ > 0 with the property cγn (g) − 1 > δ, and ϕ ∈ L1 (R) such that

∞

n∈N

|ϕ(γ 1 n )|2 < ∞.

n=1

Then equation (4.3) has a solution f ∈ AP (R). Proof. Consider the measures μ = gλ and ν = ϕλ. The Lebesgue measure λ is almost periodic on R, therefore, the measure μ is a positive almost periodic on R. On the other hand the measure ν is bounded. With this choice equation (4.1) becomes (4.3). It is easy to see that cγ (μ) = cγ (g) and ν1(γ) = ϕ(γ), 1

1 γ∈R

Therefore we can apply Theorem 4.1 and we find a solution f ∈ AP (R) for (4.3).

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Notation 4.3. Consider h ∈ AP (R) having the Fourier series ∞

cγn (h)γn .

n=1

For every n ∈ N there exists an ∈ R such that γn (x) = eian x ,

x ∈ R.

We denote Exp(h) = {an ∈ R | n ∈ N}. Application 4.4. Consider a function h ∈ AP (R) with the Fourier series ∞

cγn (h)γn ,

n=1

which satisfies the properties that Exp(h) \ Z is a finite set and 0 ∈ / Exp(h). Then the equation 2π 1 t f (x) = lim f (x − y) dy + h(x − y) dy, x ∈ R (4.4) t→∞ 2t −t 0 has a solution f ∈ AP (R). Proof. We observe that 2π h(x − y) dy = 1[0,2π] ∗ h(x),

x ∈ R,

0

$

where

1, x ∈ [0, 2π] 0, x ∈ / [0, 2π].

1[0,2π] (x) =

Choosing g ≡ 1 on R we can see that equation (4.4) is a particular case of (4.3). The set Exp(h) = {an ∈ R | n ∈ N} can be represented as {an ∈ R | n ∈ N} = {bn ∈ Z | n ∈ N} ∪ {cm ∈ R \ Z | m = 1, 2, . . . , p}. The previous partition of {an ∈ R | n ∈ N} induces the following partition for 1 | n ∈ N} in the following way {γn ∈ R 1 | n ∈ N} = {γ 1 ∈ R 1 | n ∈ N} ∪ {γ 2 ∈ R 1 | m = 1, 2, . . . , p} {γn ∈ R n m where for every n ∈ N

γn1 (x) = eibn x , and for every m ∈ {1, 2, . . . , p} 2 γm (x) = eicm x ,

x∈R x ∈ R.

It is easy to see that for every n ∈ N, cγn (g) = 0 and 2π 1 1 e−ibn x dx = 0. [0,2π] (γn ) = 0

Equations Involving the Mean of Almost Periodic Measures Therefore

∞

2 |1 [0,2π] (γn )| =

n=1

p

133

2 2 |1 [0,2π] (γm )| < ∞,

m=1

hence we can apply Corollary 4.2 and we conclude that equation (4.4) has a solution f ∈ AP (R).

References [1] L.N. Argabright and J. Gil de Lamadrid, Fourier Analysis of Unbounded Measures on Locally Compact Abelian Groups, Mem. Amer. Math. Soc. 145 (1974). [2] L.N. Argabright and J. Gil de Lamadrid, Almost Periodic Measures, Mem. Amer. Math. Soc. 428 (1990). [3] C. Corduneanu, Almost Periodic Functions, Interscience Publishers, New York, London, Sydney, Toronto, 1968. [4] S.O. Corduneanu, Inequalities for Almost Periodic Measures, Mathematical Inequalities & Applications, Volume 5, No. 1 (2002), 105–111. [5] S.O. Corduneanu, Inequalities for a Class of Means with Parameter, Buletinul Institutului Politehnic din Ia¸si, Tomul LIII (LVII), Fasc. 5 (2007), 77–84. [6] N. Dinculeanu, Integrarea pe Spat¸ii Local Compacte, Editura Academiei R.P.R., Bucure¸sti, 1965 (Romanian). [7] W.F. Eberlein, Abstract Ergodic Theorems and Weak Almost Periodic Functions, Trans Amer. Math. Soc. 67 (1949), 217–240. [8] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. I, Springer-Verlag, Berlin, G¨ ottingen, Heidelberg, 1963 [9] J. Gil de Lamadrid, Sur les Mesures Presque Périodiques, Astérisque 4 (1973), 61–89. [10] W. Rudin, Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, Number 12, Interscience Publishers – John Wiley and Sons, New York, London, 1962. Silvia-Otilia Corduneanu Department of Mathematics Gh. Asachi Technical University of Ia¸si 11 Carol I Blvd. Ia¸si, Romania e-mail: [email protected]


How Summable are Rademacher Series? Guillermo P. Curbera Abstract. Khintchin inequalities show that a.e. convergent Rademacher series belong to all spaces Lp ([0, 1]), for finite p. In 1975 Rodin and Semenov considered the extension of this result to the setting of rearrangement invariant spaces. The space LN of functions having square exponential integrability plays a prominent role in this problem. Another way of gauging the summability of Rademacher series is considering the multiplicator space of the Rademacher series in a rearrangement invariant space X, that is, an rn ∈ X . Λ(R, X) := f : [0, 1] → R : f · an rn ∈ X, for all The properties of the space Λ(R, X) are determined by its relation with some classical function spaces (as LN and L∞ ([0, 1])) and by the behavior of the logarithm in the function space X. In this paper we present an overview of the topic and the results recently obtained (together with Sergey V. Astashkin, from the University of Samara, Russia, and Vladimir A. Rodin, from the State University of Voronezh, Russia.) Mathematics Subject Classification (2000). Primary 46E35, 46E30; Secondary 47G10. Keywords. Rademacher functions, rearrangement invariant spaces.

1. Introduction: a problem on vector measures The problem which originated the research that we are going to present arises from the study of vector measures and the space of scalar functions which are integrable with respect to them.

Partially supported D.G.I. #MTM2006-13000-C03-01 (Spain).

136

G.P. Curbera

In the note Sequences in the range of a vector measure, by R. Anantharaman and J. Diestel, [1], [2], the following vector measure was considered: A ∈ M([0, 1]) −→ ν(A) := rn (t) dt ∈ 2 , A

where M([0, 1]) is the σ-algebra of Lebesgue measurable sets of the interval [0,1] and (rn ) are the Rademacher functions (see Section 2). According to the integration theory of Bartle, Dunford, and Schwartz, [8] (see also [24]), the space L1 (ν) of functions which are integrable with respect to ν is the set of all f : [0, 1] → R such that, for every A ∈ M([0, 1]), the sequence of Rademacher–Fourier coefficients of f χA belongs to 2 , that is, f (t)rn (t) dt ∈ 2 . (1.1) A

The problem we were interested in was identifying the functions in L1 (ν). A similar problem, related to the Hausdorff-Young inequality, is to describe the space Fp (T), for 1 < p < 2, of all functions f ∈ L1 (T) such that for every A ∈ B(T) 1 f (t)e−int dt ∈ 1/p . 2π A Recently, Mockenhaupt and Ricker have shown that Lp (T) Fp (T), [22]. Thus answering a problem posed by R.E. Edwards in the 1960s. The underlying measure in this case is 1 A ∈ B(T) −→ ν(A) := e−int dt ∈ 1/p . 2π A

2. The Rademacher system We briefly recall the main properties of the system. It was defined by Hans Rademacher in 1922 in Section VI, Ein spezielles Orthogonalsystem, of [26]. The functions of the system are rn (t) := sign sin(2n πt),

t ∈ [0, 1], n ∈ N.

The system is uniformly bounded and has a strong orthogonality property: 1 rn1 (t)p1 rn2 (t)p2 . . . rnk (t)pk dt = 0, ni = nj , 1 ≤ i < j ≤ k, 0

unless all pj are even, in which case the integral is equal to 1. It follows that the closed linear subspace generated by (rn ) in L2 ([0, 1]) is isometric to the 2 , ∞ ∞ 1/2 an rn = a2n , n=1

2

n=1

How Summable are Rademacher Series?

137

which write as Rad (L2 ) = 2 . It also follows that the system is not complete. In fact, the set of all finite products of different Rademacher functions constitute the Walsh system, which is complete. An important property of the Rademacher system is related to the almost everywhere convergence of Rademacher series. Namely, ∞

an rn (t) converges a.e. ⇐⇒

n=1

∞

a2n < ∞.

n=1

The reverse implication was proved by Rademacher in 1922, [26], and the direct implication by Khintchin and Kolmogorov in 1925, [15]. Regarding the closed linear subspace generated by (rn ) in other Lp spaces, it is easy to see that Rad (L∞ ) = 1 , since n n n ai ri = sup ai ri (t) = |ai |, i=1

∞

t∈I(a1 ,...,an ) i=1

i=1

where I(a1 , . . . , an ) is the dyadic interval where ri = sign ai for 1 ≤ i ≤ n. For other values of p, Khintchin proved in 1923, [14], that there exists constants Ap , Bp such that ∞ ∞ ∞ 1/2 1/2 Ap · a2n ≤ an rn ≤ Bp · a2n . n=1

n=1

p

n=1

(This formulation in terms of Lp -convergence and square summability was given by Paley and Zygmund in 1930, [25].) It follows that the closed linear subspace generated by (rn ) in Lp ([0, 1]), p = ∞, is isomorphic to 2 ; we write this as √ Rad (Lp ) ≈ 2 . Regarding the constants, Bp ≤ p. The best constants for these inequalities where found by Szarek in 1976, for p = 1, and for general p by Haagerup √ in 1982. Asymptotically, we have Bp ∼ p. Concerning best constants, it is worth mentioning [18], where they are discussed for Kahane’s inequalities, i.e., the vector version of Khintchin inequalities. The power series expansion of the exponential function together with Khintchin inequalities allow to prove that 1 a r 2 n n a2n < ∞ =⇒ exp < ∞, for some λ > 0. λ 0 It follows that Rad (LN ) ≈ 2 , where LN (= Lψ2 ) is the Orlicz space associated to 2 the function ψ2 (t) = et − 1 and consisting of all functions f such that 2 1 |f | exp < ∞, for some λ > 0. λ 0 The space LN is ‘close’ to L∞ in the sense that L∞ LN ⊂ Lp , for all 1 ≤ p < ∞. Are there any other function spaces on [0, 1] where the Rademacher functions generate a subspace isomorphic to 2 ? A precise answer was given by Rodin and Semenov in 1975, [27], in the context of rearrangement invariant (r.i.) spaces. These

138

G.P. Curbera

are Banach function spaces where the norm of a function f depends only on its distribution function, λ → m({t : |f (t)| > λ}). For the definition and properties of these spaces, see [10], [17], [19]. Rodin and Semenov proved that, if X is an r.i. space over [0,1], then Rad (X) ≈ 2 ⇐⇒ (LN )0 ⊂ X, where (LN )0 is the closure of L∞ in LN . The proof is based on the Central Limit Theorem and the role of the function log1/2 (2/t) in the space LN : f ∈ LN ⇐⇒ f ∗ (t) ≤ M · log1/2 (2/t), where f ∗ is the decreasing rearrangement of f (i.e., the right continuous inverse of its distribution function). The above equivalence is due to the fact that, as the 2 function ψ2 (t) = et − 1 increases very rapidly, the Orlicz space LN coincides with the Marcinkiewicz space (see Section 4.3) associated to log−1/2 (2/t), [21]. The situation when Rad (X) is complemented in X was characterized, in terms of (LN )0 , by Rodin and Semenov, [28], and independently by Lindenstrauss and Tzafriri, [19, Theorem 2.b.4].

3. A problem on function spaces The problem of identifying the space L1 (ν) of functions satisfying (1.1) can reformulated as follows. Describe the space of all functions f : [0, 1] → R such that 1# # # # an rn (t)# dt < ∞, (3.1) #f (t) · 0

for every (an ) ∈ 2 . At this stage, it is reasonable to abandon the original vector measure. Thus, we change notation and label this space by Λ(R). The space Λ(R) is Banach function space for the norm f Λ(R) := sup f · an rn . (3.2) (an )∈B2

1

Note that, due to Khintchin inequalities, the space Λ(R) satisfies ) Lp ⊂ Λ(R) L1 p>1

The attempts to identify the space Λ(R) with any of the classical Banach function spaces (Lp , Orlicz, Lorentz, Marcinkiewicz, Zygmund, Lorentz–Zygmund, . . . ) fail. The reason is revealed by the following result. Theorem 3.1 ([11, Theorem]). Λ(R) is not a rearrangement invariant space. The strategy for proving the result is building sequences of sets (Bn ) and (Dn ) with m(Bn ) = m(Dn ) = 2nn and such that χBn Λ(R) −→ 0. χDn Λ(R)

(3.3)


139

This contradicts Λ(R) being r.i., since χBn and χDn have the same distribution function, so their norms in any r.i. space should be the same (or equivalent). Let us sketch how these sets are built. Note that, according to (3.2), n n ∞ sup χE · ai ri ≤ χE Λ(R) ≤ sup χE · ai ri + χE · ai ri . (ai )∈B2

i=1

1

(ai )∈B2

1

i=1

n+1

1

n

Let Δ1n , . . . , Δ2n be the dyadic intervals of order n. Let aij be the value of the Rademacher function ri on the interval Δjn . The values of r1 , r2 , . . . , rn over the n intervals Δ1n , . . . , Δ2n are shown in the following matrix: Δ1n r1 1 .. ⎜ .. . ⎜ ⎜ . ri ⎜ ⎜ ai1 .. ⎜ . . ⎝ .. 1 rn ⎛

Δ2n 1 .. .

... ... .. .

n

. . . Δ2n ⎞ . . . −1 .. ⎟ .. . . ⎟ ⎟ . . . ai2n ⎟ ⎟ .. ⎟ .. . . ⎠

Δjn a1j .. .

ai2 .. .

. . . aij .. .. . . −1 . . . anj . . . −1 ' Choose columns J1 so that, for Bn = j∈J1 Δjn , we have that sup

χBn

(ai )∈B2

n

ai ri Λ

i=1

is small; and choose columns J2 so that, for Dn = sup (ai )∈B2

χDn

n

' j∈J2

Δjn , we have that

ai ri Λ

i=1

is large. From this, together with an adequate control of the norm of the tails of the series, we deduce (3.3).

4. The Rademacher multiplicator space It was a suggestion of S. Kwapie´ n to the author to consider the above result substituting the role played by the space L1 ([0, 1]) in the definition of the space Λ(R), namely: an rn ∈ L1 , Λ(R) = f : f · an rn ∈ L1 , for all by Lp ([0, 1]). We went a step further and considered an arbitrary r.i. space. 4.1. The space Λ(R, X) Definition 4.1. Let X be an r.i. space on [0,1]. The Rademacher multiplicator space for X is Λ(R, X) := f : f · an rn ∈ X, for all an rn ∈ X .

140

G.P. Curbera

an rn ∈ X Note that when X = L1 , or X = Lp for finite p, the condition corresponds to (an ) ∈ 2 , due to Khintchin inequalities. However, this is not the case for a general r.i. space. Some examples are in order: ⎧ 2 ⎨ , for X = LN , due to Rodin and Semenov’s result. Rad (X) ≈ q ,∞ , for X = Lψq with ψq (t) = exp(tq ) − 1, q > 2. ⎩ 1 , for X = L∞ . In general, for X an arbitrary r.i. space Rad (X) is (isomorphic to) a sequence space which is an interpolation space between 2 and 1 (that is, for every bounded linear operator T satisfying T : i → i , i = 1, 2, we have T : Rad (X) → Rad (X)). The converse to this result is also true. Theorem 4.2 ([3]). Every interpolation space between 2 and 1 is a space Rad (X) for some r.i. space X. Theorem 3.1 can be extended to this more general setting. Note that the space Λ(R, X) is a Banach function space for the norm f Λ(R,X) := sup f · an rn : an rn ≤ 1 . X

X

Theorem 4.3 ([11, Theorem], [5, Theorem 2.1]). If X is an r.i. space such that the lower dilation index of its fundamental function ϕX satisfies γϕX > 0, then Λ(R, X) is not rearrangement invariant. Recall that the fundamental function of an r.i. space X is defined by ϕX (t) := χ[0,t] X for 0 ≤ t ≤ 1. In particular, for X = Lp , 1 ≤ p ≤ ∞, we have ϕX (t) = t1/p . The lower dilation index γϕ of a positive function ϕ is γϕ := lim

t→0+

log sup 0<s≤1 log t

ϕ(st) ϕ(s)

.

This index measures the situation of ϕ with respect to power type functions; see [17, II. §1.1]. In particular, γϕX = 1/p for X = Lp , 1 ≤ p ≤ ∞. The proof of Theorem 4.3 is modeled on that of Theorem 1.1 and follows from the estimate ∞ 1/2 χ[0,n2−n ] n+1 bi rr X χBn Λ(R,X) ϕX (1/2n ) + ≤ sup . χDn Λ(R,X) ϕX (n/2n ) n1/2 · ϕX (n/2n ) (bi )∈B2 Theorem 4.3 holds, for example, for Lp -spaces with 1 ≤ p < ∞; for Lorentz L -spaces, [10, IV.4.1]; for Orlicz spaces Lφ with φ satisfying the Δ condition globally, [16]; for r.i. spaces X whose lower Boyd index is strictly positive, αX > 0, [10, III.5.12]. p,q


141

4.2. The symmetric kernel of Λ(R, X) Theorem 4.3 implies that for ‘most’ of the classical r.i. spaces the multiplicator space is not r.i. Thus, it becomes relevant to identify the largest r.i. space contained in Λ(R, X), which we call the symmetric kernel of Λ(R, X). Let us illustrate the concept of symmetric kernel with a simple example. The function space 1 1/p−1 Z= f: |f (t)|t dt < ∞ 0

is not r.i. (to see this, consider the functions f (t) = (1 − t)−1/p and g(t) = t−1/p , they have the same distribution function but, f ∈ Z and g ∈ Z). The largest r.i. space inside Z is easy to identify: it is the Lorentz space 1 p,1 ∗ 1/p−1 L = f: f (t)t dt < ∞ . 0

Note that the inclusion L ⊂ Z follows from a result of Hardy and Littlewood on rearrangements of functions; see [10, II.2.2]. The definition of the symmetric kernel of the multiplicator space follows. p,1

Definition 4.4. The symmetric kernel of Λ(R, X) is the space Sym (R, X) := f ∈ Λ(R, X) : if g f, then g ∈ Λ(R, X) , where g f means that g and f have the same distribution function. The norm in Sym (R, X) is f Sym (R,X) := sup gΛ(R,X) : g f For the identification of Sym (R, X) we need to recall the associate space of a Banach function space X. It is the space X of all measurable functions g such that g·f is integrable, for every f ∈ X. If X is r.i., then also X is r.i. The biassociate of X is the space defined by X := (X ) . Theorem 4.5 ([5, Theorem 2.8], [7, Proposition 3.1]). Let X be an r.i. space with LN ⊂ X, then Sym (R, X) = f : f ∗ log1/2 (2/t)X < ∞ . Recall that LN is the space functions with square exponential integrability. As could be expected, the proof relies on the Central Limit Theorem. Indeed, the proof is based on the inequalities: ∗ f ∗ (t) an rn (t) ≤ K (an )2 f ∗ (t) log1/2 (2/t), n

∗ ri 1/2 ∗ ∗ √ (t). f (t) log (2/t) ≤ C f (t)· lim n n 1 Some examples following from Theorem 4.5 are in order. In many problems in classical analysis the class of Lorentz–Zygmund spaces plays a prominent role;

142

G.P. Curbera

see [9], [10, IV.6.13]. For 0 < p, q ≤ ∞, α ∈ R, the Lorentz–Zygmund space Lp,q (log L)α consists of all measurable functions f : [0, 1] → R for which 1 q dt 1/q f p,q;α = < ∞, t1/p logα (2/t)f ∗ (t) t 0 with the usual modification in the case q = ∞. Note that the Lp spaces are Lorentz–Zygmund spaces for parameters p = q and α = 0. (Below, we denote by A " B the existence of constants C, c > 0 such that c·A ≤ B ≤ C·A.) • For X = Lp with 1 ≤ p < ∞, we have 1 p 1/p f Sym (R,X) " f ∗ (t) log1/2 (2/t) dt . 0

Hence, Sym (R, L ) is the Zygmund space Lp (logL)1/2 (these are Lorentz– Zygmund spaces with p = q, see [10, IV.6.11]). • For X = Lp,q (log L)α with either 1 < p < ∞, 1 ≤ q < ∞ and α ∈ R, or p = q = 1 and α ≥ 0, we have 1 q dt 1/q f Sym (R,X) " t1/p log1/2+α (2/t)f ∗ (t) . t 0 p

Hence, Sym(R,Lp,q (log L)α ) is the Lorentz–Zygmund space Lp,q (log L)1/2+α. • For X = Lp,∞ (log L)α with 1 < p < ∞ and α ∈ R we have f Sym (R,X) " sup t1/p log1/2+α (2/t)f ∗ (t). 0

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