Conference on complex analysis and approximation theory, Brazil

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (110)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

125

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY Proceedings of the Conference on Complex Analysis and Approximation Theory Universidade Estadual de Campinas, Brazil, 23-27 July, 1984

Edited by Jorge MUJICA Universidade Estadual de Campinas Campinas, Brazil

1986

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

@

Elsevier SciencePublishers B.V., 1986

All rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, ortransmitted, in any form orbyanymeans, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.

ISBN: 0 444 87997 8

Publishedby: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 Vanderbilt Avenue NewYork, N.Y. 10017 U.S.A.

Library of Congress Catalogingin-PublicationData

Conference on Complex Analysis and Approximation Theory (1984 : Universidada Estadual de Campinas, Brazil) Complex analysis, functional analysis, and approximation theory. (North-Holland mathematics studies ; 125) (Notas de matemirtica ; 110) Includes index. 1. Mathematical analysis--Congresses. 2. Approximation theory--Congresses. 3. Functional analysis-Congresses. I. Mujica. Jorge, 194611. Title. 111. Series. IV. Series: Notas de m a t d t i c a (Rio de Janeiro, Brazil) ; no. ll0. QAl.N% no. 110 510 8 C5151 86-4504 cw99.61 ISBN 0-444-8'7997-8

.

PRINTED IN THE NETHERLANDS

V

FOREWORD

These are the Proceedings of the Conference on Complex Analysis and ApproximationTheory held at the UniversidadeEstadual de Campinas, Brazil, from July 2 3 through July 27, 1984. It contains papers of research orexpository nature, and is addressed to research workers and advanced graduate students in mathematics. Some of the papers are the written and expanded texts of lectures delivered at the conference, whereas others have been included by invitation. The organizing committee of the conference was formed by Msrio C. Matos, Joao B. Prolla and Jorqe Mujica (chairman). We gratefully acknowledge financial support fromthe Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico (CNPq), the Fundayso de Amparo 5 Pesquisa do Estado de Sao Paulo (FAPESP)and the UniversidadeEstadual de Campinas. We a l s o thank Miss Elda Mortari for her excellent typing of the manuscript.

Jorge Mujica Campinas, November 1985

This Page Intentionally Left Blank

vi i

CONTENTS

......

1

. . . . . . . .

25

V. AURICH, Local analytic geometry in Banach spaces. F. BEATROUS and J. BURBEA, Reproducing kernels and interpolation of holomorphic functions.

J. BLATTER, Metric projections of c onto closed vector

...................... new theory of generalized functions . . . . .

sublattices J. F. COLOMBEAU, A S.

DINEEN, The second dual of a JB* triple system

.......

47

57 67

R. DWILEWICZ, Holomorphic approximation in the theory of Cauchy-Riemann functions.

...............

C. FRANCHETTI, Approximation with subspaces of finite codimension H.

G.

...................... GARNIR, Microhyperbolic analytic functions . . . . . . . .

71

83

95

A. F. IZE, On a topological method for the analysis of the asymptotic behavior of dynamical systems and processes

109

M. C. MATOS, On convolution operators in spaces of entire functions of a given type and order

129

R. MENNICKEN and M. MOLLER, Normal solvability in duals of LF-spaces

173

L. A . MORAES, A Hahn-Banach extension theorem for some holomorphic functions

205

....................... . . . . . . . . . .

.......................

.................

L. NACHBIN, A glance at holomorphic factorization and

uniform holomorphy.

. . . . . . . . . . . . . . . . . . 221

viii

CONTENTS

P. P. NARAYANASWAMI, Classification of (LF)-spaces by some Baire-like covering properties.

. . . . . . . . . . . . 247

S. NAVARRO and J. SEGUEL, Nonarchimedean gDF-spaces and continuous functions.

. . . . . . . . . . . . . . . . . 261

0. W. PAQUES and M. C. ZAINE, Pseudo-convexity, u-convexity

and domains of u-holomorphy D.

..............

PISANELLI, The proof of the inversion mapping theorem in a Banach scale.

.................... M. VALDIVIA, On certain metrizable locally convex spaces . . . . INDEX.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273

281 287 295

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) @ Elsevier Science Publishers B.V. (North-Holland), 1986

1

LOCAL ANALYTIC GEOMETRY IN BANACH SPACES Volker Aurich Mathematisches Institut der Ludwig - Maximilians -Universitat Theresienstrasse 3 9 D - 8 0 0 0 Miinchen, West Germany

ABSTRACT After motivating analytic geometry in infinite dimensional spaces we give a survey on the local theory of SF-analytic sets and holomorphic semi-Fredholm maps. Moreover the notion of minimal embedding codimension is introduced. It allows to derive quantitative results although the dimension may be infinite.

1. MOTIVATION Analytic geometry in t n deals with the geometrical properties of the sets of solutions of analytic equations defined in an open subset of tn. Since many interesting equationsinanalysis like differential and integral equations are defined in infinite dimensional spaces and are analytic or even polynomial it seems reasonable to develop a concept of analytic geometry in infinite dimensional topological vector spaces. Although in applications mostly real solutions are considered one hopes that as in finite dimensionsthe complex analytic case yields a simpler and more complete theory. But without further restrictions this is not true: every compact metric space occurs as the set of solutions of a complex quadratic polynanialecpation in a suitable Banach space [ 2 1 . Hence in order to obtain sets of solutionswithnice geometric properties additional conditions have to be imposed on the regularity of the equation. We compile some sufficient conditions in the case spaces.

of

Banach

Let E and F be complex Banach spaces and R a domain in E . We F h o Z o r n o r p h i c if it is complex Frechet differencall a map f : R tiable or equivalently if it is complex analytic. D f ( x l denotes the x. We say that a linear operator T : E F is differential at +

+

s ; > Z i t t i r i g if its kernel and its image are complemented subspaces

of

AUR I CH

2

and F respectively: T will be called a s e m i - F r e d h o l m o p e r a t o r if it is splitting and continuous and if its kernel or its cokernel (or

E

both) are finite dimensional. Obviously every Fredholm operator semi-Fredholm. The i n d e x of T is i n d T := d i m Ker T - c o d i m I m T . Let X be a subset of

R. Then we define:

1.1. X is a n a l y t i c in R iff X ing property : (A)

is

is closed and satisfies the follow-

For each x E X there exists a neighbourhood plex Banach space H , and a holomorphic map such that x n u = h - l i o ) .

U , a comh : U + H

1.2. X is a c o m p l e x s u b m a n i f o l d of R iff it is analytic and moreover the maps h in (A)canbechosen to have surjective and splitting differentials. Because of the implicit function theorem we obtain the usual notion of a complex submanifold as a closed subset which is locally in appropriate biholomorphic coordinates an open piece of a complemented linear subspace.

1.3. X is f i n i t e Z y d e f i n e d iff it is analytic and moreover the Banach spaces H in (A) can be chosen finite dimensional. The finitely defined sets have been investigated intensively by Ramis [ 6 ] and later on by Mazet in locally convex spaces [ 5 1 . Because these sets have finite codimension induction on the codimension can be used to show that they have nice local geometric properties (see the next section). Anexample for afinitely defined set is the s e t of non-surjective linear Fredholm operators in E ( E , F ) .

1.4. X is f i n i t e d i m e n s i o n a l a n a l y t i c iff it is analytic and moreover the maps h in (A)canbechosen to have splitting differentials with finite dimensional kernels. Againby the implicit function theorem it iseasy to see that an analytic set X is finite dimensional iff it is locally contained in a finite dimensional complex submanifold of some open set in R (Notice that in general a finite dimensional analytic set X is not even

3

LOCAL ANALYTIC GEOMETRY

locally contained in a linear subspace). Therefore all local results about analytic subsets of t n hold also for finite dimensional analytic sets in Banach spaces. Because an analytic subset in a finite dimensional manifold always finitely defined we can state that 1.5.

X

is

i s f i n i t e d i m e n s i o n a l a n a l y t i c iff i t i s a n a l y t i c and more-

o v e r t h e maps h i n ( A ) can be chosen such t h a t t h e i r a r e Fradholm o p e r a t o r s .

differentiaZs

If we call a holomorphic map a F r e d h o l m map or F-map (semi-FredhoZm map or SF-map)

iff all differentials

operators then the finite dimensional sets which are locally the fiber of a importance of nonlinear Fredholm maps differential operators with Dirichlet

are

Fredholm

(semi-Fredholm)

analytic sets are preciselythe holomorphic Fredholm map. The is well known, e.9. elliptic boundary conditions or maps of

the form identity-compact map are Fredholm, and in many are polynomial hence analytic.

cases

they

Very often an equation @ (xl = 0 depends on a parameter y and Y one would like to know how the set of solutions changes when the parameter varies. For example the parameter can be the right side of a fix) = y ; hence it is natural that y varies in an infinite dimensional space. Consider g as an additional variable and put @ ( x , y l := 0 ( z l . Then the shape of @ - ' t o ) determines

differential equation

Y

how the sets Z := {x : @ (xl = 0 1 depend on y. If OY is a hoY Y lomorphic Fredholm map and depends holomorphically on y then @ is a holomorphic semi-Fredholm map. Thus the local bifurcation theoryof holomorphic F-maps is related to the local theory of the so-called SF-analytic sets.

1.6.

X is S F - a n a l y t i c iff it is analytic and moreover the maps h in SF-operators.

( A ) can be chosen such that their differentials are

With the aid of the implicit function theorem it can be shown that an analytic set is SF-analytic iff it is locally contained in a complex submanifold where it is finitely defined [ l ] Therefore the SF-analytic sets have essentially the same nice local properties as the finitely defined sets. Some of them will be presented in thenext section.

.

The above definitions are also meaningful when R is a complex manifold because all occurring notions are local and invariant under

4

AUR I CH

hiholomorphic maps.

2. LOCAL PROPERTIES OF SF-ANALYTIC SETS At first we recall some fundamental properties of arbitraryanalytic sets [ 6 ]

.

Let X be an analytic set of a domain R in the Banach space E . The c o d i r n e n s i o n of X in x E X with respect to R is defined as 0 - codimxX

:= sup { n

E

B

U {O,w}:

there exists an affine

complex subspace H of E with dimension n such that x is isolated in X n HI. Ramis showed that this definition is invariant under biholomorphic maps 16, p. 70, 7 4 1 , hence it generalizes via chartsto complex Banach manifolds s2. If no confusion can arise we write simply

c o d i m X X . Suppose x E

H is an affine subspace of E , d i m H 5 c o d i m 5 X , and x is isolated in X n H . Then for every n E iN with dim H 5 - n 5 nodimxX there exists an affine subspace G such that H C G, d i m G = n , and x is isolated in X n G [ 6 , 11.3.1.11. The set of these G's is open X,

in the Grassmannian 1 6 , p. 8 9 1

,

codim X n Y = min X

for X and Y analytic and

x

E

therefore codimxX,

codimxY)

X n Y. The function x

codimxX

is

upper semicontinuous [ 6 , 11. 3.3.11. A point

x

E

X

is r e g u l a r iff X is near x

a complex submani-

fold, otherwise x is called s i n g u l a r . The set X* of regular points is not always dense in X, but every point x where codimxX m is a cluster point of X* and for every cluster point x the following equation holds codim X = l i m inf codim X X

Y + X

YE

Y

x*

Because a finitely defined analytic set X has everywhere finite codimension x* is dense in X. The closures of the components of x* are again analytic and form a locally finite decomposition of X into

5

GEOMETRY

LOCAL ANALYTIC

i r r e d u c i b l e components. A l s o t h e germ o f a f i n i t e l y d e f i n e d a n a l y t i c

s e t c a n b e decomposed i n t o f i n i t e l y many i r r e d u c i b l e germs of f i n i t e l y d e f i n e d sets w i t h t h e u s u a l u n i q u e n e s s sult in [ 6 I 2.1.

16, p . 60 ]

.

A f u n d a m e n t a l re-

is

The l o c a l p a r a m e t r i z a t i o n of f i n i t e l y d e f i n e d a n a l y t i c s e t s . Let

b e a f i n i t e l y d e f i n e d a n a l y t i c s u b s e t of a d o m a i n R i n p o s e 0 E X and X i s i r r e d u c i b l e i n 0 . L e t E = El x E 2 X

pological decomposition such t h a t 0 i s i s o l a t e d i n and d i m E z = codinioX < m. T h e t i e a c h n e i g h b o u r h o o d o f p r o d u c t of t w o b a l l s ramified

E2)

x

contains

the

0 s u c h that i s an a n a l y t i c a l l y c o v e r i n g map w i t h f i n i t e l y many s h e e t s i n t h c foZZowiny sense:

Bl

C

t h e canonical projection

and

be a t o -

X n ({O) 0

Sup-

E.

El

TI :

B2

X n (Bl

n i s finite i.e. compact nonempty K C Bl arid (a)

-1

Ez

C

X

B21

centered a t

+

Bl

f K l i s c o m p a c t a n d n o n e m p t y f o r eve-y

m := s u p { c a r d

TI

-1

(xl

:

x

E

is

Bl]

finite. (b)

The b i f u r c a t i o n s a t

S

:= cx

E

BI

: card a - l i x l

f i n i t e l y d e f i n e d n o w h e r e d e n s e a n a l y t i c s u b s e t of i s c o m p l e x s u b m a n i f o l d of .irlX

n ((Bl

x Be)

S/

+

c o v e r i n g map w i t h m

+

Bl

(Bl

- S

-

Bl

< m) i s a

. X n ( ( B l- S l

x

Bzl

S l x B 2 and i s d e n s e i n X n ( B 1 X B 2 ) . i s a l o c a l l y b i h o l o m o r p h i c unramified

s h e e t s . 1 6 , 11.2.3.7,

11.2.2.4,

11.2.2.121.

Since SF-analytic sets a r e l o c a l l y f i n i t e l y defined s u b s e t s

of

s u b m a n i f o l d s t h e y e n j o y t h e above m e n t i o n e d p r o p e r t i e s . I n p a r t i c u l a r

we o b t a i n t h e f o l l o w i n g consequences. 2.2.

fold

Let

COROLLARY.

X

be an S F - a n a l y t i c

subset of

a Banach

mani-

R. (a)

i s l o c a l l y c o n n e c t e d b y complex a r c s i . e .

X

f o r each

3:

E

t h e r e are a r b i t r a r i l y small neighbourhoods U o f x such t h a t f o r e a c h y E U t h e r e e x i s t s a h o l o r n o r p h i c map y f r o m t h e open u n i t d i s k D i n t o R w i t h x , y E y ( D I C X. X

(b)

I f

f z i n c t i o n on (c)

X If

X

i s i r r e d u c i b l e t h e n e v e r y non c o n s t a n t

holomorphic

i s open. X

i s i r r e d u c i b l e t h e n t h e maximum p r i n c i p l e h o l d s i . e .

a h o l o m o r p h i c f u n c t i o n on X i s c o n s t a n t i f i t s m o d u l u s a t t a i n s a loc a 1 maximum. (d)

If X

i s c o m p a c t and

R i s holomorphically separable then

AURl CH

6

i s finite.

X

(a) Choose for each irreducible component of X at x a local parametrization as in 2.1. Moreover choose a complex line L through r(x1 and n ( y l . Then IT -1 (L1 is a onedimensional analytic set. The uniformisation of its normalization is isomorpic to D .

PROOF.

By (a) there is through every x E X a complex curve f o y is not constant and hence open.

fb)

such that (c)

y

follows from (b) and (d) from (c).

The next proposition will serve to define the minimal embedding codimension of an SF-analytic set in a point. This notion will allow to prove and to use in the following sections codimension formulas which correspond to the dimension formulas in finite dimensionalcmplex analysis. Let X be analytic in a domain 0 of E . For x E X let Xx be the germ and I, the ideal of germs of holomrphic funcTxX := T X x := { u E E : u E Ker Dk(x1 f o r euery tions vanishing on X,. h E Ixl is called the t a n g e n t s p a c e of X in I. 2.3. LEMMA. I f X i s f i n i t e l y d e f i n e d t h e n E - codimT X < 0 x = f o r e v e r y x E X. I n p a r t i c u l a r T X X i s c o m p l e m e n t e d .

-

codim X X

PROOF. Put p := codimxX and x = 0. Then dim H n X > 0 for every ( p + 11-dimensional linear subspace H of E and therefore 0 } # T (H n X) X

C

H n TxX.

Hence codim TxX 5 p . ~

2.4. PROPOSITION. L e t X x be t h e germ o f a n S F - a n a l y t i c s e t a t x E 0 . Denote S ( X x ) t h e s e t o f a l l germs o f c o m p l e x s u b m a n i f o l d s a t x i n w h i c h Xx i s c o n t a i n e d and f i n i t e l y d e f i n e d . S ( X x l i s nonempty and p a r t i a l l y o r d e r e d by t h e i n c l u s i o n . Moreover

SIX,)

(a)

Each germ i n

(b)

Sx

(C)

G i v e n t w o m i n i m a l germs

E

S(Xx)

i s minimal i f f

a b i k o l o m o r p h i c mapping germ t i t y on

PROOF.

Xx

c o n t a i n s a m i n i m a l germ.

Mx

Ox : M x

TSx = TXx

and

* Nx

.

Nx i n

S(X,)

there exists

which i n d u c e s t h e

iden-

.

Obviously

S ( X x ) is nonempty. To prove (a) and (b) let M

be

LOCAL

7

ANALYTIC GEOMETRY

a complex submanifold of a neighbourhood of x in

E

a finitely defined representative X of TzM - c o d i m T x X 2 M - c o d i m x X .

TxX

Suppose

#

TxX

Xx.

Then there exists

TxM.

Then

u

E TSM,

which contains C

TxM

and

f,

u f 0 , and

IX with D f x ( x ) u # 0. We may assume that f, has a representative f with nonvanishing derivative on M . Then S := f - ' ( O ) is a sub= I T M codim T X ) - 1. manifold of M which contains X , and TXS-codim TX X 2 X

After finitely many steps be minimal since T x X C TxM

we arrive at T x X = TxS. This S must for every submanifold M with S x C M x .

To prove (c) choose a topological decomposition E = T X x $ H and representatives M and N of M, and N x . Locally they are the graphs of mappings TXx H. Let ~i be the canonical projection E TXx. +

@

:= ( I T I N ) - '

Let

M(Xx)

Then

M(XX) = {Mx C Mx

+

o (TIM)

is biholomorphic at x and

be the set of minimal germs in

: MX

QX

I

Xx

= id.

S I X x ) . Because of 2.4.

is the germ of a complex submanifold at x with Xx

T X x = TMz} and

and

erncodirnxX

:

= emcodim

X

X

:= Mx - codirn X x

is independent of Mx E M ( X X ) and will be called the minimal c o d i r n e n s i o n of X in x.

embedding

This notion should not be confused with the embedding codimension t c c o d i r n x X in [ill. In general they do not coincide. The above considerations show that an SF-analytic set X is near a point x E X always the zero set of a holomorphic SF-map f with K e r D f ( x ) = and c o d i r n I m D f ( x ) m.

TxX

(Local parametrization of SF-analytic sets). L e t X b e SP-anai n E . Suppose 0 E X and X i s i r r e d u c i b l e i n 0 . Choose a t o p o l o g i c a z d e c o m p o s i t i o n E = T X x H and l e t p : E ToX be t h e c a n o n i c a l p r o j e c t i o n . T h e n p ( X o l i s a f i n i t e Z y d e f i n e d anaZ y t i c germ i n T o X and c o d i m p ( X o I = erncodimx . 2.5.

Z y t i c i n a domain R

+

Let

ToX = El x G

= emcodim X baZls

and

0

b e a t o p o l o g i c a 2 decomposition such t h a t d i m G i s i s o l a t e d i n p l X o ) n ({O} x G ) . Put E 2 : = H x G.

T h e n e v e r y n e i g h b o u r h o o d o f x c o n t a i n s t h e p r o d u c t of t w o o p e n B 1 C El and B 2 C E 2 c e n t e r e d a t 0 such t h a t t h e canonical

8

AUR I CH

projection

7~

: X

n IB1

x

B21 * Bl

i s an a n a l y t i c a l l y ramifiedcouer-

ing map w i t h f i n i t e l y many s h e e t s i n t h e s e n s e of 2 . 2 .

PROOF. Choose germ p , : Mo

+

T0X

of

Then p induces a biholomorphic mapping T X = TMo. Therefore p(Xol is finitely defined in 0 Mo E M ( X o l .

and c o d i m p ( X o l = e m c o d i m X o . Now apply 2.1 to a representative p ( X o l and the decompostion ToX = E l x G. From 2.5 the local bifurcation theorem in [ l ] can be derived.

We close this section with some remarks on the intersection of SF-analytic sets. A closed subset X of a Banach manifold is SF-analytic iff it is locally the intersection of a complex submanifold M and an analytic set Y where M is finite dimensional or Y is finitely defined.

In general the intersection of two SF-analytic sets X and Y is not SF-analytic, simply because the intersection of complemented linear subspace is not always complemented 2.6.

fold

LEMMA.

Let

X

and

Y b e S F - a n a l y t i c s u b s e t s of a Banach

mani-

0.

(a)

If Y is f i n i t e d i m e n s i o n a l o r f i n i t e l y d e f i n e d t h e n XnY

i s SF-analytic. I f Y i s f i n i t e l y d e f i n e d and is f i n i t e d i m e n s i o n a 2 .

(b) then

X

X n Y

is f i n i t e dimensional

PROOF. (a) is obvious. To prove (b) let x E X n Y and M z E M ( X t I . Then ( X n Y I L c is finitely defined in M x , hence its codimension is finite. By 2 . 3

dim TMz = d i m T(X n Y j x + c o d i m T ( X n Y I z

Therefore X must be finite dimensional in

x.

3 . HOLOMORPHIC SF-MAPS

Let il be a domain in the complex Banach space E. Suppose 0 E 0 and assume that f : R F is a holomorphic map into another Banach +

9

LOCAL ANALYTIC GEOMETRY

space F with splitting differential D f ( 0 ) i.e. there are topoloqical decompositions E = K e r D f ( 0 l 63 M and F = I m D f t O ) 63 J . T h e n one can find a zero neighbourhood U in E such that

maps U biholomorphically onto a zero neiqhbourhood V in I m D f ( O 1 X K e r D f ( 0 l . Putting h : V J , h ( y , z l := f o @ - ' ( y , z ) - y one obtains +

Setting J, := ( i d v , h ) o 0 we can reformulate the local representation of f in the following way: There are neighbourhoods U' in E and W' in F x Ker D f ( 0 l such that $ : U' W' is a holomorphic --f

embedding, $ l o ) is an isolated point of for every x E U'. (cf. [ 9 1 1 .

J,W) n J ,

and f(x) = nF o '(XI

This local representation holds in particular for holomorphic SF-maps. If the differential in a point x is (semi-)Fredholm then automatically all differentials in a neighbourhocd are (semi-) Fredholm as well. This follows from 3.1. 3.1.

LEMMA.

The s e t

S F ( E , F I of

o p e r a t o r s is

semi-Fredholm

open

in

d: ( E , F I .

PROOF.

Let

ip E

SF(E,F)

and

decompositions. I := I m T . neighbourhood U of T in

S

E

U. Then

IT=

: S(L)

F = I m T 6 3 J betoplogical

v I o TIL is isomorphic there is a such that v I o S I L is isomorphic

Since efE,Fl

S E U. We want to show

for every Let closed.

E = K e r T 63 L ,

--f

U C SFIE,FI. I

is isomorphic, hence

S(Ll

is

c o d i m I < a. Then I m S has finite codimension and is therefore complemented. Choose a linear subspace M 3 L such that E = M 63 K e r S is an algebraic decomposition. Then SIM + I m S is bijective and induces an isomorphism between M / L and the space I m S / S l L l which is finite dimensional because c o d i m S ( L l = c o d i m I C m . Consequently M is closed and K e r S is complemented.

F I R S T CASE:

SECOND C A S E :

dimKey2T
-

then

f(Ul i s

f i n i t e l y d e f i n e d and

c o d i m f ( x l f I U I = emcodim X I

(c)

flu

+

f I U l i s open in x

o n t o neighbourhoods o f

i.e.

f

ind D f ixl.

maps n e i g h b o u r h o o d s o f x

f Ixl.

PROOF. We may assume that X lies in fi with minimal codimension and that g is a holomorphic SF-extension of f to R. Because of 2.6(b) g - 1 ( f i x ) ) is finite dimensional and 3.2 implies d i m K e r D ( x l < m. 9 Assume x = 0 and g I x ) = 0. Choose a local representation of g in terms of $, U', and W' as in the beginning of section 3. Since 0 is isolated in f-'(OI n X there is a ball B in Ker Cg(0l with center 0 such that $ ( X I and { O } x a B C F x K e r D g ( 0 I do not meet. $(X) is closed in W' and { O l x 2B is compact because Ker Dg(0) is finitedimensional. Therefore there exists a zero neighbourhood V in F and positive reals r c s such that $ ( X I and V x ( B I 0 , s l - B(O,rl/ are disjoint. Hence the projection T ~ I + I X / n ( V x B ( O , s I ) V is finite. Putting U = U' n g - ' ( V l n X and observing r F o J, I U = f / U we obtain that f I U V is finite. This prove (a). -+

-+

( b ) and (c) follow from a theorem on the projection of analytic

14

AURl CH

nF[V

sets applied to in [ 6 ] ) .

x

B(O,sl

-+

V (see prop. 11. 3.7 and 111. 2 . 2 . 1

This result corresponds to a well-known theorem of finite dimensional complex analysis (see e.g. [3, Th. 3.2(b), p . 1331). The usual dimension formula

can he transformed into the codimension formula in 4.1. If E and F are finite dimensional and X is embedded into E with minimal codimension in x then i n d D f ( x l = dim E

-

dim F

and

= d i m E - indDflxl - dim

f(U!

f (2)

= codimXX - i n d Df ( x ) . 4.2. THEOREM. SF-anaZytic ind f

> -

Let

f

: X

+

Y

b e a f i n i t e h o l o m o r p h i c SF-map betueen

s e t s i n Banach m a n i f o l d s . then

m

Then

flXl

is

nnazytic.

If

f ( X l i s f i n i t e l y d e f i n e d and

Y - codim f (XI = min Y

emcodimxX

-

emcodim Y

Y

-

ind Of ( x ) : x E f-' (y) }

.

f(X) is closed since f is proper. Let y E ftX) and f-'(y) = ( xl , . . . , x n } . Because f is proper there are neighbourhoods V of y and U j of x j such that the U j are pairwise disjoint and f-'(V) PROOF.

= U { U j : j = 1 , ..., n } . Moreover we may assume that V lies in a domain 5 of a Banach space with minimal codimension in y . According to 4.1 we can make and Uj smaller such that each f l l J j ) is analytic in x , and if i n d D f ( x j l > - m then 2

Hence

-

f(Xl n V

c o d i m f(U .) = emcodimx X - i n d D f ( x .) Y 3 d i

is analytic and the above formula holds.

LOCAL A N A L Y T I C GEOMETRY

In finite dimensions emcodim X X - emcodini Y Y - i n d D f ( x ) = d i mY y

-

d i mX X

and therefore the above formula is transformed to

dim f l X l = max { d i m X X Y

5.

:

3: E

f-llyl}

LOCAL FACTORIZATION

In this section it is shown that a holomorphic SF-map can be locally factored into a finite map and a projection as it is thecase in finite dimensions (see e.g. [ 3 ] ) . As a consequence the fiber dimension is semi-continuous.

5.1. PROPOSITION. L e t SF-anaZytic

s e t s and

U of

neighbourhoods

x

f

:

E

X.

x, V

X

+

be a holomorphic

Y

SF-map

Then t h e r e a r e a r b i t r a r i l y

of

f i x ) , a domain

a n d a f i n i t e h o l o m o r p h i c SF-map

X : U

W

open

in a Banach space G, s u c h t h a t t h e follow-

V x W

+

between

small

i n g diagram commutes:

u

x(UI is a n a l y t i c in
- m and hence V ' is so in and from V

-

codim

fizl

then X ( U I is finitely defined in V x W V . The codimension formula follows from 5.1

= (V

V'

x

Wl

-

codim

x i z ) V'

x

W = codimX(ZIX(U).

If X and Y are manifolds then the embedding codimersions vanish and the stated formula follows from i n d Df ( z l = d i m Ker Of ( z l

-

codim I m D f

(2).

6. A CRITERION FOR OPENESS Immediately from 5.1 there follows a criterion for openess holomorphic F-maps. 6.1. PROPOSITION. L e t SF-analytic s e t s . I f emcodim then

f( X I

Y

-

f

: X

+

Y

be a h o Z o m o r p h i c

emcodim X = dimzf X

-1

F-map

( f l x l l - ind Df(xl

f i s open i n x . For manifolds the converse implication holds a l s o .

of

between

19

LOCAL ANALYTIC GEOMETRY

6 . 2 . THEOREM. folds.

Let

f : X

b e a h o l o m o r p h i c F-map

Y

+

between

mani-

T h e n f is o p e n if a n d o n l y if

(Notice that i n d D f l x ) is constant if X is connected). Suppose that f is open and that x E X. We may assume that X and Y are domains in Banach spaces E and F . Define K:=KerDf(x) and I := I m D f ( x 1 . Choose local coordinates such that E = I X K, F = I x J , and f ( y , z ) = i y , h ( y , z 1 1 for 1 y , z l E U C I x X where h : U I and x = 1 0 , O ) . We show that h i y , l is open for each y. Let V be open in K and W be an open neighbourhood of y . Then f ( W x V l is open, hence

PROOF.

+

is open in

{yj x J.

In particular

g := h ( 0 ,

*

I

:

X n Ker D f ( x )

+

is open.

J

The

criterion for openess in finite dimensions (see e.g. 13, p. 145 1 ) im-1 plies dim Ker D f ( x l = d i m J + d i m g ( g ( 0 l l . Since g-l I g ( 0 l l = f-l(f(xll

n U

dimXf-I 1 f ( x l I

7.

and

ind Df(x)

= ' d i m Ker D f i z l

- dim J

we

obtain

= i n d Of 1x1.

THE SINGULAR S E T OF A HOLOMORPHIC FREDHOLM MAP

The singular set S i f ) of a holomorpnic map f : X Eanach maniforlds X and Y is the set of all points x f the differentials D f ( x ) are not surjective. For Fredholm set is finitely defined analytic and its codimension can mated as in finite dimensions (see e.g. [ 3 , p. 97 1 ) +

.

7.1.

Let

LEMMA.

R. S u p p o s e

B

A

and

i s near

B

x

between in which maps f this be esti-

b e a n a l y t i c s u b s e t s of a Banach A n B

Q - codimxA

PROOF.

Y

X

2

manifoZd

a s u b m a n i f o z d . Then

B - codimzA n B .

We may assume that R is a domain in a Banach space

E and

B

20

AUR I CH

i s near x

a complemented l i n e a r subspace of

l i n e a r subspace

LEMMA. L e t

f o l d s and

f

+

f(xl

2

> -

:= X

PROPOSITION.

Let

Define

x

B

Then

Y.

f o r every

x

f-liZ).

E

Y , B := g r a p h f, A := X x Z

and

B

b e Banack s p a c e s .

P

and a p p l y

codimTFo

s u b s e t of

U

F(E,FI.

T E Fo

of

Fo

$-'(E01

indT 2 0.

with

~

i s a f i n i t e l y d e f i n e d analytic

I

t h e r e e x i s t s a holomorphic map

of t h e s e t

E(K,J)

Fo

More p r e c i s e l y i t i s shown t h a t i n a n e i g h b o u r h o o d

K = Ker T, J

that

T E

I n [ l ] i t i s proved t h a t

PROOF.

that

f o r every

1

of

defined a m Z y t i c

of a l Z Z i n e a r P r e d h o i m o p e r a t o r s . M o r e o v e r

F(E,FI

5 indT +

7.1.

Fo

The s e t

n o n s u r j e c t i v e l i n e a r Fredholrn o p e r a t o r s i s a f i n i t e Z y s u b s e t of t h e s e t

a

x

b e a h o Z o m o r p h i c m a p b e t w e e n Banachrnani-

Y

codimxf-l(z)

Q

PROOF.

is

such t h a t

Then t h e r e

E.

c o d i mXA

R - codirnxA 2 dirnC.

n C. Therefore

: X

-

dimC = B

with

B

b e a n a n a l y t i c s u b s e t of

Z

codim

7.3.

of

A n B

i s isolated i n 7.2.

C

:

U

such

E(K,Jj

+

i s a complement o f f m T and F , i s t h e preimage L o of n o n s u r j e c t i v e o p e r a t o r s i n E ( K , J ) . N o t i c e

i s a f i n i t e d i m e n s i o n a l v e c t o r s p a c e . Now suppose i n d T

2 0 . Then d i m R 2 d i m J and t h e n o n s u r j e c t i v e l i n e a r o p e r a t o r s from K t o J a r e e x a c t l y t h o s e l i n e a r o p e r a t o r s t h e r a n k of which i s s t r i c t l y smaller than irreducible

dimJ

i

dimJ.

According t o

a n a l y t i c s u b s e t of

E(K,J)

THEOREM.

Let

f : X

Banach m a n i f o l d s w i t h

-t

Y

indDf(x)

5 codim E

dimK

an

-

W e may assume t h a t

and

Then

+

Now a p p l y 7 . 2 and 7 . 3 .

i

I.

2

0

f o r every

X and

3:

E X. l ' h e n

codirn,SIf)

S(fl i s

(indDf(x1 t l

X and Y are domains i n Banach s p a c e s E F ( E : , F ) i s holomorphic and S(f! = ( D f ) - 2 ( F o 1 .

PROOF.

Df : X

= ind T

b e a h o l o m o r p h i c FredhoZm map b e t w e e n

a f i n i t e l y defined a n a l y t i c subset o f f o r every x E Sif).

F.

t h e y form

1 = i n d T + 1 . With 7 . 2 o n e o b t a i n s c o d i m T F o = codirnT+-l (Lo)

7.4.

13, p.98 1

w i t h t h e codimensicn

LOCAL A N A L Y T I C GEOMETRY

7.5. X

--f

Let

COROLLARY.

and

X

Y

21

b e c o n n e c t e d Banach m a n i f o l d s and f :

a f i n i t e h o l o m o r p h i c F r e d h o l m map w i t h n o n n e g a t i v e i n d e x . Then

Y

S(f),

:= f ( S ( f ) ) ,

C

flx

o n e and

f-l(C)

and

- f-7(Cl

Y

+

-

codimension

are a n a l y t i c subsets with i s a c o v e r i n g map.

C

i s c o n s t a n t and b e c a u s e o f 4 . 1 i t v a n i s h e s , and f i x ) i s open. T h e r e f o r e f i s s u r j e c t i v e . According t o t h e theorem ind Dfixl

PROOF.

1121 t h e s e t of c r i t i c a l v a l u e s

of Sard-Smale hence

C

by 4 . 2

is

C

meager i n

Y,

# Y, f - ' ( C ) # X, and S l f ) X. By 7 . 4 c o d i m Slf! = 7 and C = f l s i f ) ) i s a n a l y t i c and o n e c o d i m e n s i o n a l . f IX - f - ' ( C )

i s l o c a l l y biholomorphic because t h e d i f f e r e n t i a l s

are

s i n c e t h e i r i n d e x i s z e r o . The map i s a l s o f i n i t e ,

hence

isomorphic

it

is

a

c o v e r i n g map.

8 . GRAPH THEOREMS Graph t h e o r e m s c h a r a c t e r i z e t h e r e g u l a r i t y of a map b y g e o m e t r i -

c a l p r o p e r t i e s o f i t s g r a p h , f o r example c o n t i n u i t y

by

closedness.

Recall t h e following c h a r a c t e r i z a t i o n s of d i f f e r e n t i a b i l i t y .

1. A map

f : X

+

between complex ( o r r e a l

Y

Cr-)

Banach m a n i f o l d s

i s holomorphic ( o r r e a l C r - d i f f e r e n t i a b l e ) i f and o n l y i f its g r a p h I- i s a complex ( o r r e a l c r - ) s u b m a n i f o l d uf X y Y arid i f f o r e v e r y (.c,yl E r t h e t a n g e n t s p a c e o f I' i n Ix,yl i s a topol o g i c a l complement o f

2. and

If

X

X

(0)

T Y Y

x

(2,

x

T Y. Y

Y are ( l o c a l l y f i n i t e d i m e n s i o n a l )

and

i s normal t h e n R e m m e r t p r o v e d t h a t a map

morphic i f and o n l y i f i t s g r a p h

dim

TxX

in

Y )

r =

dim X X

r

ix,YI

f o r every

reduced ccmples spaces

f : X

is analytic i n E

r

[ 8

1

+

Y

i s holo-

x

Y

and

X

if

.

i t i s shown t h a t i l l i n f i n i t e d i m e n s i o n s t h e a n a l y t i c i t y of t h e g r a p h i s t o o weak t o g u a r a n t e e t h a t t h e map i s h o l o m o r p h i c . In [ 1 ]

There e x i s t s

d

map from the open u n i t d i s k i n t o a Banach s p a c e which

i s a homeomorphism o n t o i t s image dnd h a s dn a n a l y t i c g r a p h

but

n o t h o l o m o r p h i c . Holomvrphy c a n , however, b e c h a r a c t e r i z e d

by

is the

SF-analyticity o f t h e g r a p h . 8 . 3 . THEOREM.

space

E

Let

f

:

R

--f

F

i n t o t h e Banach s p a c e

b e a map f r o m t h e a o m a i n F

and l e t

following properties are equivaient:

r

R i n a Banach

be i t s graph. Then t h e

22

AUR I CH

(i)

f

(ii) r = E ; and

i s holomorphic

is a n S F - a n a l y t i c s u b s e t of

emcodim

P

r = dim(T r P

(iii) For e v e r y h o l o m o r p h i c map Q-'(Ol

and

p

0 : U

DzOipI

R

E -+

H

x

x F

and moreover

n ( { O } x F I I f o r every F

p

E

t h e r e are a neighbourhood

i n a Banach s p a c e

H

such t h a t

i s a Fredkolm o p e r a t o r w i t h i n d e x

r.

7~ ( T

E

U and

P

rl

a

n U =

0.

The equivalence of (i) and (iii) is proved in [ 11 . (i) implies (ii) because of 8.1. Thus it remains to show that (ii) implies

PROOF.

.

(i)

Let p = ( x , y I E r . Then there are a neighbourhood U of p , a complex submanifold W of U and a topological decomposition F = such that r n U C M , d i m F l = enicodim r = M - c o d i m r, T M = F 1 %3 F 2 P P P E X F I , and p is isolated in r n ( { X I X F I I . Making U smaller we can achieve that the projection E * F E x P I induces abiholomorphic map h : M * V onto a domain V in E x F l . The set A := h f r l is a finitely defined analytic subset of V . Making again V smaller -+

we may assume that A is the finite union of finitely defined analytic sets which are irreducible in p . Because of 2.1 we can find one of them, say A i t such that the projection E x F 1 * E induces an analytically ramified covering map from A j onto a neighhourhood of x. Since r is the graph of a map this covering has only one sheet and furthermore A j must be the only irreducible component of A. Because A is a submanifold outside of the bifurcation set it is the graph of a locally bounded map g : Ei * F 1 from a neighbourhood W of x in E into F I which is holomorphic outside of proper analytic subset of W. The Riemann removable singularity theorem [ 6 , p. 241 implies that g is holomorphic everywhere. Hence f I W = h-' o g is holomorphic.

REFERENCES [ 11

V. AURICH, B i f u r c a t i o n of t h e s o l u t i o n s of h o l o m o r p h i c F r e d h o l m e q u a t i o n s and c o m p l e x a n a l y t i c g r a p h t h e o r e m s , Nonlinear Anal. 6 (1982), 599 -613.

121 A. DOUADY, A r e m a r k o n Banach a n a l y t i c s p a c e s . Symposium Infinite Dimensional Topology, Ann. of Math. Stud. 69 (1972), 41 - 42.

23

LOCAL ANALYTIC GEOMETRY

[ 3 ]

G.

FISCHER, Complex A n a l y t i c G e o m e t r y , L e c t u r e Notes i n

Math.,

v o l . 538, S p r i n g e r , B e r l i n , 1 9 7 6 . [ 4

I P . MAZET, Un t h e o r e m e d ' i m a g e d i r e c t e p r o p r e , L e l o n g 1 9 7 2 / 7 3 and 1 9 7 3 / 7 4 ,

in:

Ssminaire vol.

L e c t u r e N o t e s i n Math.,

4 1 0 and 4 7 4 . S p r i n g e r , B e r l i n , 1 9 7 4 and 1 9 7 5 . 15

I P . MAZET, E n s e m b l e s a n a l y t i q u e s c o m p l e x e s dans l e s e s p a c e s

lo-

c a 2 e m e n t c o n v e x e s , T h g s e de d o c t o r a t d ' g t a t , P a r i s , 1 9 7 9 . 16

17

1

J. P.

1 J. P .

RAMIS, S o u s - E n s e m b l e s

A n a l y t i q u e s d ' u n e V a r i e t e Banachique

C o m p l e x e . E r g e b . d e r Math. 5 3 , S p r i n g e r , B e r l i n ,

1970.

R A M I S and G .

directes

RUGET, Un t h z o r e m e s u r l e s

a n a l y t i q u e s b a n a c h i q u e s , C.R. 995 [ 8

Sci. P a r i s

267 ( 1 9 6 8 ) ,

- 996.

1 R. REMMERT, Holornorphe und meromorphe A b b i Z d u n g e n komplexer Rawne, Math. A n n .

19

Acad.

images

133 (1957), 328-370.

1 G. RUGET,

A p r o p o s d e s c y c l e s a n a l y t i q u e s de d i m e n s i o n i n f i n i e , I n v e n t . Math. 8 ( 1 9 6 9 1 , 2 6 7 - 3 1 2 .

1101 B.

SAINT-LOUP,

[ 111 W .

SCHICKHOFF,

Un t h e o r e m e d ' i m a g e d i r e c t e e n g g o m g t r i e a n a l y t i q u e s complexe, C.R. Acad. Sci. P a r i s 2 7 8 ( 1 9 7 4 ) , 555 - 5 5 6 .

W h i t n e y s c h e T a n g e n t e n k e g e l , Multiplizit;itsverhalten,

Normal-Pseudo-F2achheit

und Aquisingularitatstheorie

R a m i s s c h e Rzume, S c h r i f t e n r e i h e des Math.

j%r

I n s t . der Univ.

Miinster, 2 . S e r i e , H e f t 12, 1 9 7 7 . [ I 2 1 A.

J . TROMBA, Some t h e o r e m s on F r e d h o l m m a p s , SOC. 3 4

(1972)

,

578

- 585.

P r o c . Amer. Math.

This Page Intentionally Left Blank

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) @ Elsevier Science Publishers B.V. (North-Holland), 1986

25

REPRODUCING KERNELS AND INTERPOLATION OF HOLOMORPHIC FUNCTIONS Frank Beatrous and Jacob Burbea Department of Mathematics and Statistics University of Pittsburgh Pittsburgh, Pennsylvania 15260 USA

ABSTRACT. The problem of characterizing restrictions to small sets of holomorphic €unctions satisfying certain norm estimates,is studied from the point of view of functional Hilbert spaces and reproducing kernels. Characterizations are obtained for boundary value distributions of various Hilbert spaces of holomorphic functions along certainthin subsets of the boundary of the unit ball. In addition,boundary values of bounded holomorphic functions along certain thin subsets of the boundary are characterized by an analogue of the classical PickNevanlinna condition.

INTRODUCTION In this paper we address, in several different contexts, the problem of characterizing the restrictions of various classes of holomorphic functions to certain subsets of their domain, or of the boundary of their domain. The problems to be considered fall into two broad classes: characterization of restrictions of Hilbert spaces of holomorphic functions (e.g. L 2 Bergman and Hardy spaces), and characterizations of restrictions of bounded holomorphic functions. The Hilbert space case has been considered previouslybyseveral authors, most notably Aronszajn [ l ] . A novel feature of the present paper is the characterization of restrictions to certain subsets of the boundary, which may in fact be quite thin. In this case the boundary values may not be defined in the classical sense, so we are forced to consider distributional boundary values. For the second problem, we attempt to characterize restrictions of bounded holomorphic functions to a small set E by an analogue of the Pick-Nevanlinna condition in the unit disk A , which may be formulated as follows: A function f on a subset E of A satisfies the Pick-Nevanlinna condition if the kernel K ( s , 5 ) ( 1 - f i z l is

fx))

26

BEATROUS AND BURBEA

- -1 positive definite on E x E, where K ( z , < 1 = ( 1 - 2 s ) is the Szeg6 kernel of the unit disk. The classical Pick-Nevanlinna Interpolation Theorem states that f is the restriction of a function in the unit ball of H m ( A ) if and only if f satisfies the Pick-Nevanlinna condition on E . The higher dimensional analogue of this problemhas been considered previously by Korsnyi and Puksnski [14], FitzGerald and Horn [ l l ] , Hengartner and Schober [13], Hamilton [l2], and by the authors [2,3]. In Section 2 we will show that, for appropriatelychosen sets E, and for a wide range of kernels, the restrictions to E of bounded holomorphic functions are characterized by a direct ana-

logue of the Pick-Nevanlinna condition. It was observed by Korhyi and Puksnski[l4] that in the case of functions of several complex variables it is essential to place some restriction on the size of the set E. Roughly speaking, E must be large enough to be a set of uniqueness for an appropriate class of holomorphic functions. In Section 2 we investigate the extent to which this condition might weakened, obtaining a slight improvement on a earlier result Hamilton 1 1 2 1 in this direction.

be of

The paper is organized as follows. In Section lweintroduce the necessary background on positive definite kernels and functional Hilbert spaces. In Section 2 we specialize to spaces of holomorphic functions, and address the interior interpolation problems alluded to above. Finally, in Section 3 we introduce distributional boundary values along certain submanifolds of the boundary of the unit ball, and address the boundary analogues of the results in Section 2.

1. POSITIVE DEFINITE KERNELS Let H be a Hilbert space of complex valued functions on a set X. We will say that H is a f u n c t i o n a l H i l b e r t s p a c e on X if it has the property that for every x E X the linear functional f -+ f ( x ) is continuous on ff. It follows that for each x E X there is a unique function K x E H with the property that f f x : ) = ( f,Kx) for every f E H. The kernel Klx,y) = K ( X I defined on X X X is called Y the r e p r o d u c i n g k e r n e l f o r the functional Hilbert space H . It is p o s i t i v e d e f i n i t e in the sense that for any pair of finite sequences XI’. ,xn E X and a l J . . .,a E C we have Z a . a .K(xl,xjl 2 0 . Conn 2 3 versely, it can be shown that every positive definite kernel is the

..

reproducing kernel for some functional Hilbert space. Moreprecisely, we have the following result which is due essentially to Aronszajn

27

REPRODUCING KERNELS

Let

THEOREM 1.1.

be a p o s i t i v e d e f i n i t e k e r n e l o n

K

X

on

There i s a unique f u n c t i o n a l H i l b e r t space

(i)

Then

X.

X

with

X

as i t s reproducing kernel;

I(

Ho

(ii) The S p a c e 8 cli K f *

,xl) i s d e n s e i n

C

ff;

on X

(iii) A f u n c t i o n f negative constant

is in

(iV)

H i f and o n l y i f 2 C K(x,y) -

t h w e i s anon__

such t h a t the kernel

p o s i t i v e d e f i n i t e , and i n t h i s c a s e ,

of a l l s u c h

f=

c o n s i s t i n g o f f i n i t e sums o f t h e f o r m

t h e norm o f

f

fix) ffy)

i s

the

is

infimum

C; I f

i s a n y s u b s e t of

E

with reproducing kernel

KIE x F

X

is

t h e n t k e f u n c t i o n a l Hilbert space

fflE.

T h e following interpolation result is an immediate corollary of

Theorem 1.1.

COROLLARY 1.2. let

K b e a p o s i t i v e d e f i n i t e k e r n e l on

Let

b e a n a r b i t r a r y s u b s e t of X .

E

t i o n on E

Let

f

and

X x X

b e a c o m p l e x v a l u e d func2

s u c h t h a t f o r some n o n - n e g a t i v e constant C the kernel C Klx,y)

-

f ( x I f ( y ) i s p o s i t i v e d e f i n i t e on

F

E

~

H w i t h IIPII 5

C

and

F

1

E x E.

Then t h e r e i s a

function

= $.

For our interpolation results for bounded functionswewill need

a generalization of a classical result of Bergman and Schiffer [ 6 1 on analytic continuation of functions of two complex variables. The classical result may be formulated as follows:

K b e t h e Bergman k e r n e l of a d o m a i n D i n S and E be an open s u b s e of D . L e t f : E x E S be any function Let

THEOREM 1 . 3 .

let

+

satisfying

(1.1)

/hi BjflZi’Sji

,...,

,...

zl,. ..,zfl, cM E E and a l , .. . , a N J 8, T h e n t h e r e i s a u n i q u e h o l o m o r p h i c f u n c t i o n F on D x D

whenever with

f

on

the points

E

8.

which agrees

and such t h a t t h e i n e q u a Z i t y ( 1 . 1 1 p e r s i s t s and C j are allowed t o vary over t h e s e t D.

E x E,

zi

, ,6

when

28

BEATROUS AN0 BURBEA

We will present an abstract formulation of this result which admits a particularly simple proof based on Hilbert space considerations, and which yields a significant improvement of the classical result when specialized to the classical setting. For our purposes it will be more convenient to work with a slightly modified version of the Bergman-Schiffer inequality. Let K, and K 2 be positive definite kernels on X, and X 2 respectively. We will say that a kernel L : X I x X 2 '6 is s u b o r d i n a t e tothe pair (Kl,KZJ if there is a non-negative constant C such that -+

whenever x 1 ,..., xu E x1, y , yM E X, and a ,,..., a N , B l , . . ., ,6 5 . We denote the infimum of such C by IIL II . In the special case X I = X 2 and K, = K 2 = K, we will say simply that L is s u b o r d i n a t e to K. Note in particular that it follows from Theorem 1.1 that if H i is the functional Hilbert space with reproducing kernel K. then, when3 ever (1.2) holds, the functions L ( , y ) and L ( x , . ) are in H, and H2 respectively for any fixed x E X I and y E X2. Thus for any f 6 ff, we may define a function Lf on X, by

,...,

,

denotes the inner product of tl . ) . j 3 The next result establishes a (conjugate linear) isometrybetween

(Here

(

)

the space of kernels satisfying (1.2) and the space tinuous linear operators from ff,to

B ( H I J ti2) of con-

H2.

i s s u b o r d i n a t e t o ( K l , K 2 1 . Then f o r any He, and t h e l i n e a r mapping 1 : ff, + H2 h a s norm a t m o s t C where C i s t h e c o n s t a n t L : ff, H2 i s a l i n e a r operator o c c u r r i n g in ( 1 . 2 ) . C o n v e r s e l y , i f i n B f H I , H 2 1 t h e n t h e k e r n e l L d e f i n e d by LEMMA 1.4.

f

E

fil'

Assume t h a t

the function

Lf

L

defined by (1.31 i s i n

+

s a t i s f i e s (I. 21 w i t h

C

= II 1 II.

Moreover, i f

XI = X2

and

t h e n t h e o p e r a t o r L i s s e l f - a d j o i n t ( p o s i t i v e ) i f and o n l y k e r n e l L i s hermitian ( p o s i t i v e d e f i n i t e ) .

PROOF.

We first assume that L

K,

i f

= K2 the

satisfies (1.2) - We .will show that for

29

REPRODUCING KERNELS

f E H I we have L f H2 and ItLfII 5 C It f II, where fined by (1.3). By Theorem 1.1 it suffices to show that

any

yl

whenever form

f

,...,

yM

X2

6

and

,..., RM

B,

-

= B a i K I ( * , x . ) are dense in

E

Lf

is de-

6. But functions

of

the

H I , so it suffices to verify

the above inequality for functions of this form. But in this case, the inequality reduces to (1.2). For the converse, let linear map, and let L g

and

L : H i,

H,6

+

be an arbitrary

be defined by (1.4). Writing f

continuous

=L:zjK I (

= B J j K 2 ( * , y j ) , we have, by the Cauchy-Schwartz

-

,x.) 3

inequality

and the reproducing properties of the kernels,

which is (1.2) with

C = II L I I .

The proofs of the remaining assertions are straightforward will be omitted. We are now in a position to formulate the main result section, which is essentially contained in [ 3 ]

of

and

this

.

EI

X

j x

K be a p o s i t i v e d e f i n i t e k e r n e l o n X . and l e t E: j 3 Assume t h a t L o : E l x E 2 + 6 s a t i s f i e s (1.2) on

Let

THEOREM 1.5. C

( j = 1,Z).

Then t h e r e i s a unique

E2.

(i)

:

L

XI

x

X2

+

6

such t h a t

L I E l X E 2 = Lo ;

(ii) L

s a t i s f i e s ( 1 . 2 ) on

XI

x X2

and

IIL II = tILott;

f . 6 H. with f v a n i s h i n g i d e n t i c a l l y on 3 3 j f , ) I = ( L ( x , i, f,) = 0 f o r e v e r y x E X1

(iii) W h e n e v e r we h a v e y E X2.

kerneZ

(LI

(Here K.). 3

,y),

-

E . 3

and

H i s t h e functionai! H i l b e r t space w i t h reproducing j M o r e o v e r , if X I = X2, El = E 2 and K I = K 2 , t h e n L i s

h e r m i t i a n ( p o s i t i v e d e f i n i t e ) on

XI

m i t i a n ( p o s i t i v e d e f i n i t e ) on

x

El

x XI El.

i f and o n l y i f

Lo

i s her-

30

BEATROUS AND BURBEA

PROOF. +

Let

HZIEZ

R

j

..

Hj

+

Hj l E

be the restriction mapping. Let Lo : Hl

be the linear operator induced by 1 = R i LORl

1.4, and set

IB1

j :

HI

H,.

+

Lo

according

to

Lemma

the

Using the fact that

R 3t R j

is an orthogonal projection in H and R j R i is the identity mapj' ping on Hi I we conclude that 1 I 1 II = II Lo I1 5 C . By Lemma 1.4, the

lEj

= LKl( * , x ) ) ( y )

function L ( x , y ) over, for ( x , y )

E

El

X

E2

and (i) is verified. (Here

satisties ( 1 . 2 ) on

Xl

X

X2.

More-

we have

(

-,

*

denotes the inner product

)

in

Ei

HIE.). Moreover, since the kernel

L(x,y)

represents the operator

1

3

in the sense of (1.3), item (ii) follows from Lemma 1.4. To verify (iii), let E XI

f j

E

H

with

fjlEj

= 0. Then for any

x

we have

Similarly,

For uniqueness, we use once again the observation that R*R j

j

is

and R . R ? is the identity mapping an orthogonal projection in H j' 3 3 Thus if L ' : X I x X 2 + S satisfies (i)-(iii), and if L f,

HI

since vanishes on y E E,

=

(

L(x,

.

El.

Thus we obtain

l,f,),

f,

-

L ' ( x , y ) = L ( x , y ) whenever x E XI and

S i m i l a r l y , f o r any x E XI *

we have

so it follows that

and f, E H , L' = L.

we have ( L ' ( x , *),f2)

31

REPRODUC I N G KERNELS

F o r t h e r e m a i n i n g a s s e r t i o n s , it i s o n l y n e c e s s a r y , Lemma 1 . 4 ,

again

by

1 is self-adjointorpositive

t o verify t h a t the operator

Lo.

whenever t h e same i s t r u e o f t h e o p e r a t o r

But t h i s i s clear from

1 so t h e p r o o f i s c o m p l e t e .

t h e d e f i n i t i o n of

To r e c o n c i l e t h e preceding r e s u l t with t h e c l a s s i c a l r e s u l t

of

Bergman and S c h i f f e r , i t i s o n l y n e c e s s a r y t o r e p l a c e t h e k e r n e l K 2 by i t s complex c o n j u g a t e . Thus w e h a v e t h e f o l l o w i n g c o r o l l a r y , v e r s i o n s o f which h a v e a p p e a r e d i n 1 9 1 ,

COROLLARY 1.6.

Let

(j = 1 , 2 ) .

E . C X . 3 3

whenever

X

i

x I , . ..,xN

E

[131,

[ 1 2 1 and

El, y l

Fo

,... , y N

a unique

F

:

El

:

X

XI x X2 * 6

i

and l e t

6 satisfies

+

al,... , a N ,

and

E2

E

E2

[41.

X

be a p o s i t i v e d e f i n i t e k e r n e l on

Assume t h a t

E 6 . Then t h e r e i s

[81,

. , BM

Bl,-.

w i t h t h e following proper-

ties:

y

and

E

-

X2;

,y),fl)

(F(

H

fj E

( i i i )W h e n e v e r

we h a v e

-

...,

s a t i s f i e s ( 1 . 5 ) whenever x I , xN E XI and Y I J * .,YM E x2. F( , y ) E HI and F ( x , ) E H2 f o r any f i x e d x E X I

(ii) F

I n particular,

with

j

= (F(x,

f . v a n i s h i n g i d e n t i c a l l y on

E

3

= 0

),f,)

i’

x E XI, y E X2.

f o r every

A s a n a p p l i c a t i o n o f t h e a b o v e r e s u l t s , w e give an abstract version

o f t h e P i c k - N e v a n l i n n a I n t e r p o l a t i o n Theorem. W e w i l l s a y t h a t a s e t

E

C

X

i s a s e t of u n i q u e n e s s f o r a f a m i l y of f u n c t i o n s

t h e r e s t r i c t i o n mapping

the associated E

C

X

f : E

K be a p o s i t i v e

Let

THEOREM 1 . 7 .

functional Hilbert

+

6

definite

space

particular,

Then t h e r e i s a f u n c t i o n

K(x,y)(l

IF]

5

1

on

-

on

X

such t h a t

-

k e r n e l on

if

H,

:

g

-

f ( x l f ( y ) ) i s positive

E

F E H

x

E

-

XI, and l e t

such t h a t F I E = f

F ( x ) F ( y ) ) is p o s i t i v e d e f i n i t e o n X

X.

F o r any n o n - n e g a t i v e

E

H contains the constants. Let

{gK( ,x) be such t h a t t h e k e r n e l K(x,y)(l

d e f i n i t e on E x E .

F

is injective.

b e a s e t of u n i q u e n e s s f o r

and s u c h t h a t

PROOF.

* FIE

R : F

constant

C we can w r i t e

x

X. In

BEATROUS AND BURBEA

32

Since

1

E

H, it follows from Theorem 1.1 that the kernel

2

1

C K(x,y)-

is positive definite for C 2 I1 1 II , and clearly the kernel f( x ) f ( y l is positive definite on E x E . It follows from Schur’s Lemma that the last term on the right of (1.6) is positive definite for C , I I l I l . But the first term on the right is positive definite by hypothesis. Thus it follows from (1.6) and Corollary 1.2 that there is a function F E H with F I E = f. Moreover, by Theorem 1.5 there is posi-

K tive definite kernel L ( x , y ) on X x X with L subordinate to and L ( x , y 1 = K ( x , y l l l - F ( x ) F ( y l ) for x , y E E . Since E is a set of uniqueness for CgKl * , x ) : g E H, x E XI, we have LI ,y) = = K( , y ) ( l -F( 1 F ( y ) ) on X for any fixed y E. Similarly, for any fixed x E X, we obtain L ( x , 1 = K ( x , l ( 1 - F(xlF( 1 on X . Since L is positive definite, the theorem is proved.

-

-

~

~

A remark is in order concerningthe unusual hypothesis on

Theorem 1.7. An example due to Korsnyi and Puksnski [14] shows

E

in that

some such condition is essential. However, we do not know whether it is enough to take E to be a set of uniqueness for H. We will address these questions in certain special cases in the next section. We conclude this section with an alternate formulation of the positivity condition of Theorem 1.1. Let K be a positive definite kernel on X x X and let H be the associated Hilbert space. Wewill call a function

H

on X a m u l t i p l i e r on

‘p

pf E tl

if

whenever

f E f f .

PROPOSITION 1.8.

Let

H b e a f u n c t i o n a l H i Z b e r t s p a c e on X w i t h re‘p b e a f u n c t i o n on X . Then t h e following

p r o d u c i n g k e r n e l K and l e t conditions are equivalent:

(i)

~p

is a m u l t i p l i e r on

H;

(ii) The r n u Z t i p Z i c a t i o n o p e r a t o r a c o n t i n u o u s l i n e a r o p e r a t o r on

(iii) For some t i v e d e f i n i t e on X

C 2 0 x X;

M

d e f i n e d by ( M q 1 f = pf is

‘p

H;

the kernel

K(x,y)(C

2

- ‘ p ( x l q ( y 1 ) i s posi-

~

(iv) The k e r n e l

K (x,y) = ‘p(y)K(x,yl Ip

Noreover, i n t h i s case, t h e k e r n e l

Kg

skbordinate

is

represents the operator

o f p a r t l i i 1 ( i n t h e s e n s e o f Lemma 1 . 4 1 ,

and

IIM II ‘p

t o K. MIp

is t h e infimwn o f

33

R E P R O D U C I N G KERNELS

the constants

C in ( i i i l .

The equivalence of (i) of (ii) is an immediate consequence of the Closed Graph Theorem.

PROOF.

C

2

For the equivalence of (ii) and (iii) note that the operator I - MpM; (where I is the identity operator on H ) is represented

-_

- p ( x ) p ( y ) ) . Thus by Lemma

1.4 the kernel K(x,y) ( C 2 - p ( x I l p ( y l l is positive definite if and only if the operator C 2 I - M 14" is positive, i.e. if and only if lIMpll 5 C.

by the kernel

K(x,y)fC'

rptp

The equivalence of (ii) and (iv) is again a consequence of Lemma 1.4 since the kernel Kp represents the operator MP

.

Finally, we have COROLLARY 1.9. = Ix

E

x

:

If p is a m u l t i p l i e r on ff

K(x,x) # 0 )

with

then

sup { l l P ( x ) / : x

E

i s b o u n d e d on Y

p

Yl

5

IIMpII.

2. SPACES OF HOLOMORPHIC FUNCTIONS

In this section we will specialize to subspaces of the space o ( D ) of holomorphic functions on a domain D in r n or, more generally, in a complex manifold. In this case we will say that a kernel Kiz,SI is s e s q u i - h o l o m o r p h i c if it is holomorphic in the first variable and conjuqate holomorphic in the second. One easily checks that the reproducing kernel of any functional Hilbert space of holomorphic functions is sesqui-holomorphic, and that conversely the functional Hilbert space associated with a sesqui-holomorphic kernel consists of holomorphic functions. We begin with some examples. If p is a positive measure on D, we will denote by tfp the vector space L 2 ( d u 1 n U f D I of all holomorphic L 2 functions on D . In addition, if D is a domain with C 2 boundary (ora product of such domains), and if 1~ is a positive measure on the (distinguished) boundary of D, we will let ff, denote the space of all functions in the Nevanlinna class NfDI having boundary values in L 2 ( d p l . Under appropriate conditions on p , the space ff, is a functional Hilbert space on D, and we will denote its reproducing kernel by K p . Thus if 5 denotes the euclidean surface measure on the distinguished boundary a o D , then H o is the usual Hardy space H ' l D l . In the case when p is a measure on D which is absolutely continuous with respect to Lebesgue measure, the space ff )J

34

BEATROUS AND BURBEA

is called a w e i g h t e d Bergman s p a c e s . B = B

in g n we introduce, for~n -1 dV ( a ) = x * r ( n + q ) r ( q l - ' ( l 1 ~ 1 ~dvlzl, ) ~

In the case of the unit ball q > 0

the probability measures

4

where dV denotes the Lebesgue measure in b n . Then as q +. '0 the measures dV converge w * , as measures on B, to the normalized sur4 2 face measure d o on a B . Thus we define H o to be the Hardy class H ( B I . The espace H = f f y is then a functional Hilbert space of holomor4

4

= (1 - ( a,< ))-in+q). phic functions on B , with reproducing kernel K (.z,C) 4 This can be easily verified as follows. For z E E n and a 6 Zy a a "1 " n multi-index, we let z = z l . . . z . By integration in polar coordinates, one checks that

(za,zB)

4

=

a! I'(n + q l r ( n + q + ] a l ) - ' ,

where we have used the notation a ! = a , ! ...an.' and la1 = a1 + ... + a Thus ( 2 " ) forms a complete orthogonal system and it follows that

.

The definition of the spaces can be extended to the case of H4 negative values of q . For any q E B we define a function h on 9 the unit disk by

h

(2.1)

where F

4

(XI =

II

(1

-

4'0

F ( 1 , 1 ; 2 - 4 : A),

q

5

0,

is the usual hypergeometric function defined by

Here we have used the notation For any

Al-9,

q

E

E

(am) = a ( a + l l

we define a kernel K

4

on B

x

... ( a + m - l / = f ( a + m l / r " a ) .

B

by Kq(z, 51 = h

((

n+q

z, 5 ) ) .

Then K is sesqui-holomorphic and positive definite on B, so it is 4 the reproducing kernel for a Hilbert space H q of holomorphic functions on B . In fact, it can be shown that for q < I , H q is the space of all holomorphic functions on B with square integrable partial derivatives up to order ( 1 - q ) / 2 . (Note that in the unit ball, derivatives of fractional order can be defined in terms of power

35

REPRODUCING KERNELS

series. We refer to [ 5 ] for details). we now turn to the problem of interpolating bounded holomorphic funcfunctions. Let D be a domain and let f be a complex valued tion on a subset E of D. We wish to determine whether f can be extended as a bounded holomorphic function on D. Of course,for functions of one complex variable a celebrated theorem of Carleson assures us that every bounded function on E has a bounded holomorphic extension provided that E satisfies a uniform Blaschke condition.In higher dimension, however, the situation is considerably more complicated. In fact, it follows from a result of Varopoulos [18] that, for n L 2, there is for every p > 0 an analytic variety E satisfying a uniform Blaschke condition in B n such that the restriction of H P ( B n ) to E does not contain H C O ( E ) . Thus it is natural to consider the problem of characterizing the functions on restrictions of bounded holomokphic functions on B . Let K be a positive definite kernel on a set

E

which

are

X, and let E be

an arbitrary subset of X. We will say that a function f on E satisfies the P i c k - N e v a n l i n n a c o n d i t i o n on E (with respect to the kernel K ) if the kernel K l x , y ) l l - f l x ) f ( y l l is positive definite on E x E . By specializing Theorem (1.7) to the present setting we obtain: THEOREM 2 . 1 .

Let

K be a p o s i t i v e d e f i n i t e s e s q u i - h o l o m o r p h i c kernel

H on a domain D such t h a t t h e a s s o c i a t e d f u n c t i o n a l H i l b e r t space contains the constants. L e t E C D b e a s e t of u n i q u e n e s s for {gK( * , 5 1 : g E H, 5 E D?, and l e t f : E 6 be a f u n c t i o n s a t i s +

f y i n g t h e Pick-NevanZinna function

F

on D w i t h

c o n d i t i o n on E .

FIE = f

and

T h e n t h e r e is a holomorphic

lF(z)l

5

1

for all

When D is the unit disk and K is the Szegs kernel K ( z , 5 1 = ( 1 - z?)-', this result is a special case of the Pick-Nevanlinna Interpolation Theorem. In this case, the

z E D.

given by classical condition

that E be a set of uniqueness is not required. In general some such assumption is essential. Korhyi and Pukgnski [141 gave an exampleof a function on a 2 point set in the bi-disk that is not the restricK tion of any function in the unit ball of Hm. For the kernels 1-I introduced above, it is clear that it suffices to assume that E is a set of uniqueness for the holomorphic subspace of L 1 fdul. Moreover, is a multiplier on if for each fixed 5 E D the function K( * , 5 1

HK, then it suffices to assume that E is a set of uniqueness for HK since in this case HK contains the class of functions used in the

36

BEATROUS AND BURBEA

theorem. (This is the case for the Bergman and Szega kernels of any strictly pseudoconvex domain (see [ 7 ] ) , as well as for any explicitly computable example in the unit ball). Our next result shows that the conditions on E cannot be relaxed much. PROPOSITION 2.2. tion

Let

B = B

and a s s u m e

f E H 2 ( B ) and a s u b s e t n E

(i) ii)

B

i s a s e t of u n i q u e n e s s f o r

f

s a t i s f i e s t h e Pick-Nevanlinna

fl

E

2

n

2 . There are

a func-

such t h a t :

E

spect t o t h e Szeg; kernel f o r

iii)

of

p > 2;

for every

Hp(3)

c o n d i t i o n on

E

w i t h re-

B;

h a s no bounded h o l o r n o r p h i c e x t e n s i o n t o

Note that if E were a set of uniqueness for tradict Theorem 2.1.

H

2

B.

( B l this would

con-

It will be more convenient to prove the preceding result in more general setting.

a

LEMMA 2 . 3 . L e t K be t h e reproducing k e r n e l o f a f u n c t i o n a l H i l b e r t s p a c e U on a s e t X and a s s u m e t h a t K ( x , x l i s n o n - v a n i s h i n g on X . L e t F b e a s u b s e t o f H w h i c h c o n t a i n s a l l bounded f u n c t i o n s i n H, and assume t h a t t h e s e t s o f u n i q u e n e s s f o r H and F do n o t coincide. T h e n t h e r e i s a s e t of u n i q u e n e s s E f o r F and a f u n c t i o n f E U s a t i s f y i n g t h e P i c k - N e v a n Z i n n a c o n d i t i o n on E w h i c h d o e s n o t a g r e e on E w i t h a n y b o u n d e d f u n c t i o n i n U.

PROOF.

We use a variation on an argument of Hamilton 1 1 2 1

. Let

Eo

be any set of uniqueness for F which is not a set of uniqueness for U , and let g be a non-zero function in H which vanishes identically on E o . Let E = E U I x o } where x o is any point of X satisfying g l x o ) # 0. Letting C = IIg II, it follows from Theorem 1.1 that the kernel C 2 K ( x , y ) - g ( x ) g f x I is positive definite, so it follows that ~

whenever

xl,

. . . ,x N

E

x

and

a*,

. . . ,aIv E

~

~ f(e x )=t ~ - ~ ~ ~ x ~ , ~ ~ ~ - ~ /

Then it follows from (2.1) that K ( x , y ) ( l - f ( x l f ( x l ) is positivedefinite on E X E , i.e. f satisfies the Pick-Nevanlinna condition on E . Assume, by way of contradiction, that there is a bounded function ~

37

REPRODUCING KERNELS

F

in ff which agrees with f on

= 0

E. Then it follows that F I E 0

and f(xo) # 0, contradicting the fact that E o is a set of uniqueness for the family of bounded functions in H, and the proof is complete.

PROOF OF PROPOSITION 2.2. According to a theorem of Rudin [17] there is a function in H ’ ( B ) whose zero set is a set of uniqueness for H p ( B ) for every p > 2. The result follows immediately from Lemma 2.3 by taking F = U C H p ( B ) : p > 2 1 .

3. CAPACITIES AND BOUNDARY BEHAVIOR

In this section we will introduce an abstract notion of capacity. It is not our purpose here to develop a comprehensive theory,but rather to set up a convenient context for an abstract discussion of boundary behavior. Let )J be a complex Bore1 measure on define the q - e n e r g y llu II of p by

E

C

x. For any

q E Z? we

9

The kernels

K

4

KO

are defined in the last section. In particular,

is the Szeqo kernel and

KI

is the Bergman kernel of

B.

For the remainder of this section we shall denote the norm and by

II

spectively. The ambiguity of the symbol II

I1

inner product in the Hilbert space

H

4

IIq 9

and

(

,

)

4

re-

should cause no dif-

1 we ficulty. In addition, for any function f on B and any 0 r will denote by fr the d i Z a t i o n of f defined by f r i z ) = f ( r z I .

IIpII

LEMMA 3.1.

is in

ffq

.

4


-n

then

q > -n.

C ~ p - ~ l E >l 0 .

Writing K q ( z , < ) = Z e a l q l z a ? " sure 1-1 supported on a B ,

PROOF.

aB.

Moreover,

Then

if

Cap ( E l i s a d e c r e a s 9 Cap ( E l > 0 f o r some 4

we have, for any Bore1

mea-

39

R E P R O D U C I N G KERNELS

For q > - n it follows from (2.1) that c c l ( q ) = In + q l I a l / c 1 ! which for each fixed multi-index a is an increasing function of q. Thus by ( 3 . 2 ) we have that II 1~ II is an increasing function of q for q > - n 4 and the first assertion follows immediately. For the second assertion, note that c a ( - n l = ~ a ~ ! / +{ Z~) a ~ ! } and a ~ so for any q > - n there is a positive constant M = M(n f q ) such that c a ( q ) 2 Me,(- n i for every multi-index a. Thus it follows from (3.2) that 1 1 1 ~ 1 1>~ M l l u ! l f n , 4 and hence Cap-n(El MCap ( E ) > 0 . This concludes the proof. 9

Let E C aB be a Borel set with positive q-capacity and let A be a linear functional on the vector space E ( E i . We will say that 4 A is the boundary v a l u e of a holomorphic function f on B i f for every u E E ( E l we have 4

Our next result asserts that, in the weak sense described aboverevery function in pacity. THEOREM 3 . 3 .

H

9

Let

has boundary values along any set of positive q-ca-

q

E

iR,

t i v e q-capacity and l e t

f

let E

Hq

E

be a Bore2 s e t

.

Then f o r e v e r y

e x i s t s , and m o r e o v e r ,

Note in particular that the theorem asserts that when q < - n , has boundary values along any Borel set in a B , and thus, any f E H q in particular, f has pointwise boundary values. In fact it can be shown that Hq is contained in the Zygmund class when( n+q i /2 ever q < - n , and H - n contains unbounded functions. For these

'-

matters we refer to [ 5 ]

PROOF OF THEOREM 3 . 3 .

and the references given there.

Let

f E Hq

and

E

E ( E i . By Lemma 3.1, 9

40

BEATROUS AND BURBEA

and the proof is complete. Our next result gives a class of subsets of

aB

with

capacity. Recall that a smooth submanifold M of

aB

is

positive inter-

an

p o l a t i o n m a n i f o l d if its tangent space at every point is contained in

the complex tangent space to a B . We will say that M is non-tangentiaZ at a point p E M if its tangent space at p is not contained inthe complex tangent space to a B at p . Let

THEOREM 3 . 4 . of

q

every

E

IR.

Moreover,

u h i c h is e v e r y w h e r e f

E

and a s s u m e t h a t

E C aB

E

contains a submanifold

Cap ( E l > 0 f o r 4 a smooth s u b m a n i f o l d o f aB

t h a t i s n o t an i n t e r p o l a t i o n m a n i f o l d . Then

aB

i f

E

i s itse2.f

non-tangential,

f f q , t h e boundary v a l u e o f

f

then f o r every

along

E

q

E

B

and a n y

i s a d i s t r i b u t i o n on

Theorem 3.4 is an immediate consequence of the following which for q = 0 is essentially due to Nagel [15] For k E Z, k we will denote by A ( B ) the space C k ( E ) n U l B ) .

.

LEMMA 3 . 5 .

Let

M b e a s m o o t h s u b m a n i f o l d of

aB,

E.

lemma U

{-I

of r e a l d i m e n s i o n

m , w h i c h i s e v e r y w h e r e n o n - t a n g e n t i a l , and l e t 2, be the space of smooth on M w i t h c o m p a c t s u p p o r t . T h e n f o r e v e r y q E 1R we h a v e

m-forms K~

:

D

+

A ~ ( B I .

By a partition of unity argument we may assume that M iscontained in a small open subset U of as. We assume that U is sufficiently small that there is a cube Q about the origin in Bm and a diffeomorphism @ from Q onto M. After shrinking Q if necessary, we may assume that @,(a 1 I has a non-zero complex normal component at each point of Q. PROOF.

Let h be an arbitrary holomorphic function on the unit disk, p and let H be a primitive for h . Then for any z E B and any cz 2, we have

REPRODUCING KERNELS

41

where w and w ' are smooth. (Here d t denotes the euclidean volume n element in B ) . For any P E D, integrating by parts in the tl variable yields,

for some

E

D.

is defined by K ( z , < l = h ( ( z,

or

01

i s t h e D i r a c measure a t x) i s a n e q u i v a l e n c e r e l a t i o n f o r p : R B by s e t t i n g , f o r (x,x') E R,

and w e d e f i n e a f u n c t i o n

p(x,x')

-+

0

if

0 = 6 ( G = 6x,IG

a

if

o # 6 X l ~=

X

=

~ A ~ , I f Go r

some

c1

> 0.

50

BLATTER

We have then glx) = p ( x , x ' ) g ( x ' ) for all g E G and all (x,x') E R and X o = {I E X : g(x) = 0 for all g E GI is either empty or an exceptional R-equivalence class. Kakutani's Stone-Weierstrass theorem states that

is the smallest closed vector sublattice of CIXI which contains G , which is to say that if G is a sublattice of ClX), then L i G I = G and Stone's Weierstrass theorem is the fact that L ( G ) = G if G is a subalgebra of ClXl (note that p 2 = p if G is a subalgebra of C(X)!). Obviously L(G) = C(Xl iff Xo = 0 and R(x) = for every x E x. The result we alluded to- above is the Blatter [ 2 1 .

following theorem

of

Suppose G is a sublattice of ClX) with the property that R is upper semi-continuous (i.e. the quotient map of X onto X/R is closed) and that all R-equivalence classes are compact (this iswhat we meant by a "nice" sublattice). Then G is proximinal iff

furthermore, if G is proximinal, then for all

f l ,f 2

E

C(X),

where H is the Hausdorff metric for the set of all non-empty closed and bounded subsets of C(X), and finally, G is Chebyshev (if and) only if G = C(X) or G = { O } or G = I R g for some non-negative zero-free f E C ( X I . In order to see how this theorem settles our problem on continuous selections we note that Hausdorff continuity of P implies its lower semi-continuity and thus, by the consequence of Michael's theorem mentioned above, P has continuous selections whenever G is proximinal and a unique one iff G is Chebyshev. We also note the simple fact that if G is a subalgebra of CfX), then R is upper semi-continuous, all R-equivalence classes are compact and, since p 2 = p , y = 1. Finally we note the not so simple fact (see Blatter [ 2 1 ) that unless X is finite, C ( X ) contains a "nice" sublattice G

51

METRIC PROJECTIONS OF c

which i s n o t p r o x i m i n a l ( s o a s t o l e t i t b e k n o w n t h a t t h e above t h e o -

r e m is not a hoax).

3 . CLOSED VECTOR SUBLATTICES OF c

I n t h i s f i n a l Section we set

X = JV

c o m p a c t i f i c a t i o n of t h e p o s i t i v e i n t e g e r s

(= A l e x a n d r o f f o r o n e - p o i n t

UV)

, so

c . To s t a r t w i t h , t h r e e examples o f s u b l a t t i c e s

C l X l becomes

that

G

c.

of

G = { g E c : g ( % n ) = Z g ( 2 n - 1 ) f o r a l l n E W 3 . For t h i s G, t h e R - e q u i v a l e n c e c l a s s e s a r e t h e s e t s i 2 n - 1 , Z n ) for n E ill and Xo = { m } ; a l s o p ( n , n ) = I, p ( Z n - I , 2 n ) = 1 / 2 and p ( Z n , 2 n - I ) = 2 for n A7 a n d p(m,m) = 0 . Thus y = 1 / 2 and t h e r e f o r e (in c,

EXAMPLE 1.

compact c l a s s e s a l o n e g u a r a n t e e t h a t the r e s u l t i n Section 2, G

i s u p p e r s e m i - c o n t i n u o u s ) by

R

is a "nice" s u b l a t t i c e

of

c

n

E W ] .

which i s

proximinal.

G = { g E c : g ( 2 n ) = ng(2n- 1 ) for a l l

EXAMPLE 2 . G,

= i, p ( 2 n - 1 , n ) = l / n 0.

For t h i s pin,??.) W and p i m , m / =

t h e R-equivalence c l a s s e s a r e t h e same a s i n Example 1 b u t y = 0

Thus

and

pin,2n - 1 )

and t h e r e f o r e

G

= n

n E

for

i s a " n i c e " s u b l a t t i c e of

c

which

is not proximinal. G = {g E c : g ( 2 n ) = g(2)/2n

EXAMPLE 3 .

n E a7).

for a l l

For t h i s

o f t h e e v e n and t h e odd

and

G , t h e R-equivalence c l a s s e s are t h e s e t s

Xo =

p o s i t i v e i n t e g e r s and

" n o t s o n i c e " s u b l a t t i c e of

a two-dimensional

g(2n - 1 ) = g ( l ) / ( 2 n - l ) {m}.

Thus

G

is

c.

H e r e now i s o u r c u r i o s i t y . If

THEOREM.

G

i s a p r o x i m i n a l s u b l a t t i c e of c ,

then

P

admits

c o n t i n u o u s s e l e c t i o n a n d a u n i q u e one ( i f a n d ) o n l y if e i t h e r or

G = c

zero-free

PROOF.

G = B g f o r some n o n - n e g a t i v e o r e l s e h a s a single z e r o a t m . or

Suppose

G

g

c

E

is a proximinal s u b l a t t i c e o f

c.

(1) A r a t h e r l e n g t h y argument of B l a t t e r [ 2 ] lowing s e n s e . T h e r e e x i s t c l o s e d - J e c t o r s u b l a t t i c e s with t h e p r o p e r t i e s t h a t

a

{a}

w h i c h is e i t h e r

tells us that

falls only s l i g h t l y s h o r t of being a "nice" s u b l a t t i c e i n

c

G=

G'

the and

G

fol-

G"

of

BLATTER

52

and that G' is a proximinal "nice" sublattice of c , that G " are lattice-disjoint (i.e. Ig'l A Ig"l = 0 whenever g ' g " E G") and that G" is finite-dimensional.

G' E

and G'and

( 2 ) It is a mildly intricate exercise in vector lattices (the proofs that I know of all require Xakutani's M-space theorem) to prove that for any n E IN, any n-dimensional A r c h i m e d e a n vzctorlattice is isomorphic to iRn with its natural order and therefore there exist n E W and n at most one-dimensional pairwise lattice-disjoint vector sublattices G; . . . G; such that G" = G; + . . . ,+''G

whence G = G' + G"1 + (3)

+ GI:.

Set suppiG'I =closure {x

and, €or

...

g'(xl # 0

E

X

g'

llsupp(Gfl

:

for some

g'

E G')

f E c'IX1,

d'ffl

= i n f i IIf -

: g'

E G'}

and

Define d;(f) ... dA(fl every f E C ( X / ,

and

P;(fi

...

P'(fl analogously. Then, €or n

and therefore

we not that this inclusion, in general, becomes false whithout previous passage to the supports. (4)

Using the fact that

G'

the

is a proximinal "nice" sublattice

METRIC

53

P R O J E C T I O N S OF c

of c and the result in Section 2, it is easy to see that G'IsuppiG') is a proximinal "nice" sublattice of Cfsupp(G'ii. Since all the elements of G ' vanish off s u p p l G ' I , we clonclude, invoking again the P' has a result in Section 2 and the discussion thereafter, that continuous selection S ' . The G'!lsupprGl) are at most one-dimensional vector sublattices of C ( s u p p ( G ' ! ) ) and we conclude, using the result of Lazar, Morris and Wulbert in Section 1 and the fact that vanish off s u pp(Gl/, that the P: have conall elements of :G tinuous selections 5;. Appealing now to ( 3 ) we have that

s =

b '

+ s"I +

...

+ s;

is a continuous selection for P. (5) With the existence part now out of the way, we turn to the uniqueness part of our theorem. Suppose G is different from I 0 1 and

c . If d z m G = I, then by the first Blatter and Schumaker result in G = B g Section 1, P possesses a unique continuous selection iff with g as claimed. Suppose then, in addition, that dim G 2 2.We need to show that P admits more than one continuous selection. To do this,

f,

we first define a function If

s u p p t ~ )F

f

f

0

E

c

is such that

I s o } for some xo % c : llgll 5 11). Also, if

= 1

P(fol = b a Z I ( G I ( = {g

Obviously and if

iii, we set

as follows.

E c

llf - foil

5 r , then

-

s u p p (G) .

0 < r < I/2

( 1 - 213) ba'YZ(G) C P ( f I :

Let g E ( 1 - 2 r l DaZl(G). If x E 6 - s i i p p ( G ) , then = Ifo(xl I 5 drfi and if x E supp(Gl, then

Ifo(xJ

- gix)/

If s u p p ( G / = W , then either 2 5 d i m G < m or dim G = m. In the latter case, dirn(G') = m and, by the uniqueness part of the re(otherwise sult in Section 2, we may assume that GI = C(supp(G'll P' of ( 4 ) would have more than one continuous selection and therefore also

P ) ; the rest of the assumptions made so far, imply

2 < d i m G" < c a r d

(%

then that Now, iooking at bases of non-negarespectively, it is easy to see that in

- supp(G')l.

tive atoms for G and

G"

either case there exist a non-negative g o E c which is positive at and x 1 of IN and a closed vector subtwo distinct points x - l lattice

Go of G which is lattice-disjoint from

Ego

such that

54

BLATTER

G = B g

f

fo = 1 i x l l - 1 h - l >and show as b e f o r e t h a t

W e set

Go.

P ( f o ) = b a l l ( G o ) and t h a t ( 1 - 2 r ) b a l L ( G o )

and e v e r y With tions for

f E c

- foII

llf

such t h a t

5

P ( f ) f o r every

C

O 0 such that +

61

GENERALIZED FUNCTIONS

as soon as

EXAMPLES.

(2)

(Case of

=

R(p,xl

0
0

E

and (ii) holds. In general there is no linear map T satisfying (i) and (ii) (since in the construction of T it is essential to use a selection of the best approximation map M V ) . However T can be chosen linear when MY does possess a linear selection P v (in this case IIr - P ~ I I= 2 ) . (b)

(c) As a matter of fact the principle of local reflexivity guarantees the existence of a linear T E satisfying (i') and (ii) (see Section 5 below).

85

A P P R O X I M A T I O N WITH SUBSPACES OF F I N I T E C O D I M E N S I O N

I t i s u s e f u l t o reproduce h e r e t h e proof of t h e

following

in-

t e r e s t i n g c o r o l l a r y o f Theorem A d u e t o P h e l p s :

A s s u m e that

THEOREM B ( P h e l p s ) . is r e f l e x i v e . T h e n

V

PROOF.

1

i n (id I *

C

W

=)

d

d

C

* X /W

V

is p r o x i m i n u 2 in X

is r e f l e x i v e . L e t

b e a Hahn-Banach e x t e n s i o n t o

and

'3

V1

and t h a t X / V

be a functional of

'3;

since

V Y

-

i s p r o x i m i n a l b y Theorem A t h e r e i s a y E X s u c h t h a t fly 11 = 11 @ 11 1 = 11 Q I1 and Q(fI = f i y l , f E V T h i s i m p l i e s , a g a i n by Theorem A , t h a t W is proximinal. PROBLEM.

.

Under what c i r c u m s t a n c e s t h e p r e v i o u s c o n d i t i o n i s

suffi-

cient f o r proximinality ? L e t u s s t a t e t h i s problem f o r m a l l y :

X b e l o n g s t o t h e c l a s s A i f whenV i s a s u b s p a c e of f i n i t e c o d i m e n s i o n i n X , t h e n t h e s t a t e m e n t

W e s a y t h a t a Banach s p a c e

ever

is t r u e . Similarly w e say t h a t t h e subspaces

(ak)

V with

(codirnW

5

X b e l o n g s t o t h e class

codimV > k

k,

id 3

V * W

Ak,

0

5

k , if for

the statement

proximinull * V

proximinal

is true. A 3 Ak

Note t h a t

3

Ak-z.

So t h e a b o v e p r o b l e m h a s t h e f o l l o w i n g f o r m u l a t i o n : which Banach

s p a c e s a r e i n t h e class

A , o r i n t h e classes A k ?

A.

i s e x a c t l y t h e c l a s s of r e f l e x i v e s p a c e s : i n f a c t (ao) i s j u s t a w e l l known c h a r a c t e r i z a t i o n o f r e f l e x i v i t y ( X i s r e f l e x i v e i f and o n l y if e v e r y s u b s p a c e of f i n i t e c o d i m e n s i o n i s p r o x i m i n a l ) . Note t h a t

t h e only

W

3

V

i s X i t s e l f which i s t r i v i a l l y p r o x i m i n a l ,

The f o l l o w i n g examples show a g r e a t v a r i e t y of s i t u a t i o n s c e r n i n g t h i s problem.

con-

86

FRANCHETTI

Let X be the space co of null sequences and V a subspace of finite codimension. Then if every hyperplane containing V is proximinal so is V , i.e. c is in the class A l .

EXAMPLE 1.

This result is proved in [ 2 ] . In fact W = f-l(O) is proximinal if and only if f (as an element of E l ) has only a finite number of 1 iscontained nonzero coordinates: since V < m, this implies that V' 1 in R'(v,I for some v. Hence ( V I * is contained in i l m ( v ) C c o and Theorem A gives the desired result. Let X be the space C(Q) , where Q is a compact Hausdorff space, and V a subspace of finite codimension. Then if everysubi.e. space of codimension 2 Containing V is proximinal so is V ,

EXAMPLE 2 .

C I Q I is in the class

A,

.

This result is a consequence of the famous Garkavi's characterization of the proximinal subspaces of C ( Q I of finite codimension in 1 term of the measures of the annihilator V : V

is proximinal if and only if:

(1)

For every

decomposition into

p E

V1

two

\ I01

the carrier

closed sets

(ii) For every pair of measures 3 1 ~ \~ S1t p , )

pl

S(pii p1,

p2

S ( p ) admits a Hahn-

and E

S(u)-=S(p)

V1 \

{O)

\ S(V)'.

the

set

is closed.

(iii) For every pair of measures p I J u2 E V 1 \ I01 the measure is absolutely continuous with respect to u 2 on the set S ( p 2 1 . See for example [ l o ] paq. 302.

Note also the following example due to Phelps (see again paq. 309). Let

Q = [O,l]

and

V

= f-I1 (0)

17 f,l(O),

[lo]

where

n=I

n=l

(here 6 ( x l = x i a ) ) . Then the hyperplanes containing V are proximinal but V is not (Garkavi's condition (iii) is not fulfilled). Note that this shows that

C(Q)

B

A1.

The following interesting result (see [ 8 ] ) spaces even (a) does not hold.

shows that in

many

87

APPROXIMATION WITH SUBSPACES OF F I N I T E CODIMENSION

EXAMPLE 3 (Indumathi). Let X be any space L 1 ( T , V ) where ( T , V ) is a positive measure space such that ( L I ( T , v ) ) * = L m ( T , v ) and d i m X = m . For any n 1. 2 there exists a V with c o d i m V = n such that V is not proximinal but every

W

2

V,

W # V , is proximinal; i-e., X 9 A .

It would be interesting to characterize the classes A and

Ak.

3 . THE (F) PROPERTY when c o d i m V < = minimal projections are exaustively discussed (existence, non existence and formulas for the relative projectioncon1 stant) only when V is a hyperplane is the sequence spaces c o and (1 , see [ 3 1 ) . It is shown in particular that in co property (E)implies (F). In 1 2 I this is generalized to any subspace of finite codirnen1 sion. The same result on hyperplanes is true in R , see [ 3 I .

No theory whatsoever on the (F) property literature.

is available

in

the

4. Dn-SPACES

We have seen that if c o d i m V = n any projection P : X V is of the form P = I - Q,, where Q p is a projection onto a (variable) +

n-dimensional subspace of

X.

If we want to see how good is

Px

as a linear approximation of

x we compute as follows: IIz

- Pz II

= II

iz

- vl +

PV

- PzlII

= II (I - P ) ( x - V l l l < II 1 -

-

PI1 II x

-

VII.

i.e.

II x Recalling that a projection P such that III - PI1 is a minimum is termed c o m i n i m a l , we see that the smallest error is given when P is cominimal. In the case that III - Q I1 = 1 f l l & p l l J P is minimal if and only P if it is cominimal. This is an interesting situation which arises in

88

FRANCHETTI

some "very flat" spaces. Let us give the following

..

DEFINITION 1. X is called a Dn -space ( n = I, 2,. if every operator A : X +. X of rank less than or equal to n is such that

X

is called a D-space if (1) holds for every finite rank operator A .

We shall see in the next section the relevance of theDn-property with respect to the (E) and (F) properties. In [ 9 1 it is proved that C(Q), perfect, and L l ( S , 2, p l , for a wide both D-spaces, generalizing previous Pichugov for the special cases C [ 0 , l

with Q Hausdorff, compact and class of measure spaces S are results of DaugavetandBabenko] and Ll [ 0 , l ] respectively.

It can be shown (see [ 7 ] , [ 9 ] ) that no finite dimensional space and (see [ 7 1 ) no uniformly non square space can be D1. Some new positive results for the D2-property in Ll-spaces are proved in [ 5 1 .

5. RELATIONSHIP BETWEEN THE (E) AND (F) PROPERTY recalled in

The following conjecture has been proposed in [ 4 ] , [ 2

1 and in Ill] (pag. 84, Problem 5 . 6 ) .

CONJECTURE.

Assume that d i m X / V = n . Then, for

V , (E)

.

* (F)

This conjecture has been recently disproved by D. Amir 1 1 1 . We shall reestablish the conjecture for the restricted class of Dn-Banach spaces. First of all note that the reverse implication is not general, as it was well known. The example below is taken from [ 3 ] -1

IIfII,

Let X be the space co, V = f = I ; then: if

Ifi(

sup

if sup

If&

2

1/2

< 1/2

Taking €or example

then then

(0)

AfV,Xl

f = (f,,

= 1

and every

> 1

f = (1/2,1/4,1/8,

( F ) (a minimal projection P is given by

in

.

with

)\(V,X)

true

and for

... )

... ,fn,... I, V V,

has

(F);

(E) * (F).

then V = f-'(O) P = I - f 8 z,

has with

89

APPROXIMATION WITH SUBSPACES OF F I N I T E CODIMENSION

..

.

= (2,0,0,. I ) b u t n o t ( E ) Note t h a t i n any c a s e (E) =+ (F) i s i n f a c t t r u e f o r any f i n i t e c o d i m e n s i o n a l s u b s p a c e of co). z

(this

W e g i v e n o w a n o u t l i n e of Amir's counterexample. Y b e a Banach s p a c e w i t h u n i t b a l l

Let

a c l o s e d bounded convex s e t A c e n t e r s of [7

1).

i n which t h e r e exists E ( A ) of the Chebysbev

i s empty ( a n example o f such a s p a c e can b e found

A

A f t e r n o r m a l i z a t i o n it can b e assumed t h a t d i m A = 2 .

in

Let X =

be normed by t h e u n i t b a l l

Y x B?

Let

B

f o r which t h e set

M = Y

{O};

x

i s a p r o x i m i n a l h y p e r p l a n e o f X I t h e metric pro-

M

P (x,al

= (x,O). If

P : X + M i s a projection then i s any. I t c a n b e shown t h a t f o r any such P o n e always h a s IIP 11 > r ( A ) ( t h e Chebyshev r a d i u s of A ) Hewever f o r any E > 0 s e l e c t i n g y E E E ( A ) ( t h e nonempty s e t of E j e c t i o n being

M

P ( x , a ) = (x

-

c e n t e r s of

A ) one o b t a i n s a p r o j e c t i o n

ay,O/

conclude t h a t

where

y

Y

E

.

PE

with

llPEll 5 r I A ) + € . W e

= r l A / and t h a t t h e r e i s no minimal p r o j e c t i o n

A(M,X)

M.

onto

L e t u s prove t h e following easy

PROPOSITION 1.

have t h a t PROOF. Since X

If X

A(V,X)

=

2

-

i s a D -space f o r every hyperplane 1 and (El IF).

P :X V i s a projection then i s a Dl-space w e have:

If

+

The norm of

11 f 1 I II z I1 =

i s 2 i f and o n l y

P 1I

i . e . i f and o n l y i f

if V

there

P = I - f

exists z

E

V

8 z,

in X

we

f(zl = 1.

X such

that

is proximinal.

A s l i g h t g e n e r a l i z a t i o n of P r o p o s i t i o n 1 g o e s a s f o l l o w s :

i s a D - s p a c e , codim V = k Ik 5 n ) a n d V has n a l i n e a r m e t r i c p r o j e c t i o n P t h e n 2 = IIP II = A ( V , X ) ; s o t h a t the impZication ( E l * I F ) i s true. PROPOSITION 1'.

PROOF.

I f

X

I t i s e a s y t o see ( [ 4

I

)

IIPII = I I I - (I - P)II

t h a t i n t h i s case

= I

f

(II-

= 2;

ID - PI1 = I : hence

90

FRANCHETTI

o f c o u r s e f o r any o t h e r p r o j e c t i o n

Q

2

w e have I I Q I I

2.

W e s h a l l g i v e a r e s u l t which c o u l d b e a s u p p o r t t o t h e c o n j e c t u r e

* ( F ) when c o d i r n V 5 n

(E)

and

V

i s i n a Dn-space.

L e t u s f i r s t p r o v e a t h e o r e m on a p p r o x i m a t i o n o f g e n e r a l p r o j e c t i o n s with s p a c i a l ones. L e t Y be a d u a l s p a c e , Y = X*; l e t

Y , and

s p a c e of P : Y

has a representation

F

+

F be a n n - d i m e n s i o n a l

a basis f o r P =

.

= 6ij

$ . ( f .I z 3

,..., f , l

[ flJf2

n Z i=I

Qi 8 fi, w i t h

THEOREM 1.

Given any p r o j e c t i o n P, f o r e v e r y

projection

Po : Y

+

F

with a representation

sub-

Any p r o j e c t i o n

F.

Qi

E

X** and

u > 0. t h e r e is another n Pu = I: z i 8 f;, with i=l

zi

E

X

X**

C

and

z . ( f .1 2 3

3

= 6ij,

and s u c h t h a t

W e s h a l l u s e t h e p r i n c i p l e of l o c a l r e f l e x i v i t y

PROOF.

f o r example [ 6 I Let

.. . , $ x

every

6 > 0

X , see

(pag. 3 3 ) :

t h e r e i s a l i n e a r map

Given t h e s y s t e m there is a solution any n - t u p l e

la,,

$i(vP) =

poE w i t h

. . .,an)

L e t the projection lip

of

Y b e a n n - d i m e n s i o n a l s u b s p a c e of X** s p a n n e d by a b a s i s 1 ; G a f i n i t e d i m e n s i o n a l s u b s p a c e o f X*, t h e n for

[$,,$,,

that

= f .(zil

ci

Tg

, i =

lIp,II

: Y

1,.

< 1 + k

+

.. +

X

such t h a t

f o r every

,n E

E

>

0

i f and o n l y i f f o r

w e have

P = C z. $ i

8

fi

b e given. L e t 9 E X * be such

and IIP I1 5 IlPpIl + u. L e t G b e any f i n i t e d i m e n s i o n a l X c o n t a i n i n g { f l J f 2 , . . ., fn,P}. F o r e v e r y 6 > 0 l e t 96

11 = 1

s u b s p a c e of

b e a l i n e a r map s a t i s f y i n g ( i ) and ( i i )and s e t

z:

= T6Qi .

Define

91

APPROXIMATION WITH SUBSPACES OF FINITE CODIMENSION

P6

Pg =

by

zi

6 zi 8 fi

.

is a p r o j e c t i o n b e c a u s e of

P6

(ii).Ke have

6

llP611 y IIP*Vll = IIZ z i ( V I fi II i

so f o r e v e r y

IIy,ll

=

and

1

6 w e have

IIP61yE)II > IIP611

Qi(gI

(2)

W e h a v e , f o r any n - t u p l e

Helly’s

Therefore +

= ci

la,,

-

E.

6

Set ci=yEIziI

i = I,

E.

y, E X*

with

and consider the system

..., n .

..., a n ) ,

theorem w i l l now g i v e a s o l u t i o n

< 1 + 6 +

IIP I1

IIPgII > II P II - cr. S e l e c t now

X*

gE E

( 2 ) w i t h Ilg,ll

of

W e have now

IIPII > IIP611 / ( 1 + 6) and t h e r e i s a

6

such t h a t

IIP611


0). For a vector distribution

-EX

if

. . ,TN),

= (T

T

(Nl N

u

WFaT =

WFaTi

A - X

with

C - 0

we set

.

i=l HYPERBOLICITY AND MICROHYPERBOLICITY 3 . Let us first define the h y p e r b o l i c i t y of

polynomial of

2 E 8': of

P(AEa

+ ix) #

degree 0,

m , in the d i r e c t i o n

V A > 0,

homogeneous

P(Z)

V X E iRn+'.

Eo

by

98

GARN I R

Concerning that polynomial, let us recall the Garding T h e o m (1972) by which it exists an open convex cone r defined as the connected component of Eo in the open cone

and for which

If

Eo =

(

I,

I, we have h y p e r b o l i c i t y w i t h r e s p e c t t o the time.

0

( 1 )( n l

r

The duaL c o n e of

denote by

plays an important part in the following. is convex and closed.

Trivially Y 1

of

Let us point out that it is ( s t r i c t l y ) o r i e n t e d in t h e d i r e c t i o n E o : there is a (non flat) circular cone around Eo which con-

tains

rl: x

E~

> 0,

W X E

r1

4 . This notion was recently (1975) precised by introducing the microhyperbolicity of f ( Z ) holomorphic function of 2 E S n near Xo E

lRn.

We say that E o J if

rection

f(Zl is m i c r o h y p e r b o l i c a t a p o i n t 3

&

f ( X + iXEol # 0

Xo i n t h e

di-

> 0:

if

x

E

to,

E l .

We see immediately that a homogeneous polynomials Pm ( 2 ) hyperbolic in the direction E o is microhyperbolic at every point X in

99

MICROHYPERBOLIC ANALYTIC FUNCTIONS

the same direction

P(X

+

Eo:

=

iXEo)

imP(XEo

-

# 0,

i X )

V X

Bn.

E

For microhyperbolic functions there is an important notion localization. Let us call m u l t i p l i c i t y of

the integer p ( X o l

Xo

of

such

0

that Daf (Xol

2a

= p(Xo)

: ia

= 0.

: Daf(XoI

Such a multip icity is nothing else that the geometric plicity of the point X o on the real surface

We may compute more simply the multiplicity here extraneous) E o by 2

- E:

I a I =P ' 0 such that

property:

[ E C K .

More specially we have a s t a b i l i t y property: if fixed, we have

i s

E E

fX

and that expresses that the function studied is automatically microhyperbolic at the same point Xo for every E E r

.

fxO

From this follows that, in spite of possible changes of multiplicity of X, the Garding cone r is i n n e r c o n t i n u o u s with respect to

x c x

fX

:

MICROHYPERBOLIC ANALYTIC FUNCTIONS

r

3

K

compact

=. r

f X

X

rl

I

%

r

%

Xo

.

is o u t e r c o n t i n u o u s with f X

Xo

\ 0)

V X

fX

This implies that the dual cone respect to

K,

3

101

c w

r1

o p e n cone =.

\

wx

o c w,

%

xo,

fX

f x O

which means that he multifunction in CO:

of

I?

X

%

Xo

has a closed graph

f X

APPLICATION TO THE CAUCHY PROBLEM FOR THE DERIVATION MATRIX OPERATOR WITH ANALYTIC COEFFICIENTS Let us come back to a matrix derivation polynomial with analytic coefficient 5.

Let us express the following o n l y assumption:

i s hyperbolic with respect t o

e = (1,O)

This assumption implies that microhyperbolic a t every p o i n t (Xo,X

m(X,Z) )

f o r every frozen

X.

a s a f u n c t i o n of ( X , Z )

i s

i n the direction (0,e).

As hyperbolicity and microhyperbolicity just concerns the zeros 0 of the function considered,we may replace dtm L ( X , Z ) by any other homogeneous polynomials in Z , analytic function of X , with the same zeros E for every X.

GARN I R

102

For instance, we may take 0

dtm L ( X , E ) m(X,E) = of

g.c.d.

E(x,El

t h e c o f a c t o r s of

We may apply to rn(X,Z) the previous properties of hyperbolic functions.

the

micro-

p(Xo,so)

Of

(Xo,Z)

Specially, we may define the multiplicity which, here, may be computed simply by I

D,k r n ( X o , E "t

1

I

" I

p,

k

if

k = p.

=O'

# 0,

We may also define the localization IXo,L

2

if

rn

(Xo,=o)

IX,Zl

of r n ( X , Z I

at

). To this localization we may associate the Garding cone

and, moreover, its dual cone

r i t x o , ZI

rrnlXo,Z)

which is a nonvoid

cone,

0

convex, with closed graph (then closed), strictly increasing Et .

oriented

to

6. Let us now state an essential result for the propagation singularities of the operator L (X,DxI studied.

TO the cone

the

of

1 r,,,(x,Z) considered as a multifunction of (X,Z) let

us associate the multidifferential hamiltonian system

EIO) = E

.

This is a generalization of a differential system in which to every point iX,Z) we do not give the gradient of the unknown functions, but a conical multifunction containing that gradient. When the multifunction of the second member is convex, nonvoid, with closed graph, strictly oriented to the positive time, we may prove

MICROHYPERBOLIC ANALYTIC FUNCTIONS

103

that it exists a "solution" of this system which is a multifunction of the data defined as the set

i by a lipschitzian curve

of the (X,zI which may be connected to ( X o , E

whose tangent a.e. stays in

(-

I

It may be proved that when such a curve from ( X o , Z

0

K+ (XO,Z

1

rm(x,B) #

0

there is at least

one

I with t monotonically increasing when its

parameter varies from 0 to +

K;,o,B

ii r,' X, B

-. I

I

is a closed graph multifunction of

I

is directed to the increasing time but perhaps

(Xo,Z

not

strictly (in which case, we have to suppose that the closed timesections are compact).

r;xo,B0 )

We shift from

to

-

r-( X o , C

by changing the sign of the

second member of the multidifferential equation. Let us notice that (a)

(X,Zl

E

K

+ =)

(X0,T (b)

Ix,Z)

E

K

:

K+i x , x ,

C K:XO,x

)

~

~) 0 , lX0,E ~

(Huygens Principle) 0

)

E K:x,3.,

(Coming-backPrinciple)

0

In this conditions, we have

of

We may interpret geometrically this result from every singularity [ M I , escape singularities of I described by the multifunction

solution of the system mentioned.

7. Let us mention some particular cases where it is easy to integrate the multidifferential system.

104

GARN I R

A. If all the point ( X , = ) we have

are of multiplicity 0 or 1 of rntX,=),

and so the system becomes

We find back the differential equation of Hamilton by replacing the variable s by As for every A 0. This condition is realized when

m(X,Z)

# 0,

which means when the given operator is of p r i n c i p a Z t y p e . In this case, as we know, the singularities are proparated along the bicharacteristics solution of the Hamilton system issued frmthe singularities of M such that rn(X,LI = 0. B. In the case of an operator with constant coefficients is independent of X and we have

and so

Then the multidifferential system is

MICROHYPERBOLIC ANALYTIC FUNCTIONS

and as

Z

= X 0'

105

reduces to D ~ EX

rrn.1;.

I

X(0l =

'

xo .

As the second member is a constant multifunction, the solution is trivially

We see that from every singularity of M comes a beam of singularities of 31 which is or the point (X,zl itself, or a ray, or a beam which becomes more and more complicated when the multiplicityof A increases. I

0

8. A direct majorization of is not trivial. We may get an estimation of

[n l a IX

as the projection

I a as

of

WFaI

follows.

It is easy to see from its simplified expression that P(xo,x

)

= Pxo(=o)J

0

multiplicity of

go in

rn(Xo,X)

when

Xo

is frozen.

But as

we have

From there

So X is arbitrary and we have for every

=

the simplified

GARN I R

106

multidifferential system

of the previous type. It has as a solution a multifunction with closed graph directed to the increasing time denoted by

Then finally, we have

BIBLIOGRAPHICAL NOTES

9. The recent book of L. Hormander [ 2 ] contains (among others) the classical theory of hyperbolic polynomials (11, p. 112) and a concise study of microhyperbolic functions (I, p. 317). The directproof of the uniformity property is due to Hormander and seem to have been For a found simultaneously by P. Laubin in his Pn.D. thesis [ 3 1 . detailed study of the analytic wave front set of a distributionbased on the only Fourier-Bros-Iagolnitzer criterium, see also H. G. Garnir and P. Laubin [ i ] . The application to partial differential equations through multifunction theory is taken with some improvements from S. Wakabayashi I 4 I . The detailed proofs are in course of publication.

REFERENCES [ 11

[2

H. G. GARNIR and P. LAUBIN, F r o n t d ' o n d e a n a L y t i q u e d e s d i s t r i b u t i o n s . Sgminaires de LiGge, 1981-1982.

1 L. HORMANDER, A n a l y s i s of L i n e a r P a r t i a Z D i f f e r e n t i a l Operators. Springer, Berlin, 1983.

107

MICROHYPERBOLIC A N A L Y T I C FUNCTIONS

[ 3 ]

P. LAUBIN, A n a L i s e rnicroLocaEe d e s s i n g u 1 a r i t g . s analytiques. Bull. SOC.

[ 4

1

S.

Roy. Sci. Lisge 5 2 ( 1 9 8 3 1 ,

103 - 2 1 2 .

WAKABOYASHI, A n a L y t i c s i n g u z a r i t i e s of s o J u t i o n s o f t h e p e r b o Z i c Cauchy p r o b l e m . Proc. Japan 449 - 452.

Acad.

59

hy-

(1983).

This Page Intentionally Left Blank

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) 0 Elsevier Science PublishersB.V. (North-Holland), 1986

ON A TOPOLOGICAL METHOD FOR THE ANALYSIS OF THE ASYMPTOTIC BEHAVIOR OF DYNAMICAL SYSTEMS AND PROCESSES A. F. 126 Instituto de Cigncias Matemsticas Universidade de S a o Paulo em S Z o Carlos 13.560 Sao Carlos, S!? Brazil

1. INTRODUCTION Waiewski principle [23] plays an important role in the study of ordinary differential equations. Its applicability is largely due to the fact that in a finite dimensional euclidean space,theunit sphere is not a retract of the closed unit ball. Since this isno longer true in infinite-dimensional Banach spaces the direct extension of Waiwski's principle to processes or semidynamical systems on infinite dimensional Banach spaces has a very limited applicability. Since in finite-dimensional spaces the fact that the unit sphere is not a retract of the closed unit ball is equivalent to the fact that every continuous mapping of the unity closed convex ball has a fixed point, the main idea of this work is to develop a method based on fixed point index properties instead of retraction properties. Our fixed point formulation, Corollary 1, is essentially equivalent, in finite dimension, to Wakewski theorem. Although in infinite dimension, Wakewski theorem is no longer applicable, Theorem 1 and Corollary 1 are applicable and give deeper results since fixed point index methods have proved to be very useful in the solution of differential equations either in finite or infinite dimensional spaces. After that we go further and give a formulation of Theorem 1 using Leray-Schauder degree theory or the fixed point index for compact or condensing maps. These generalizations, Theorems 2 , 3, 4 , 5 and 6 are stronger even in finite dimension than Waiewski theorem. After Wakewski paper several papers arised applying Waiewski principle to the asymptotic behavior of ordinary differential equations, C. Olech [171 , A. Pliss I 2 0 1 , Mikolajska [141 , N. Onuchic 1181, A. F. 126 [111 and others. Kaplan, Lasota and Yorke 1121 applied also Waiewski method to boundary value problems and C. Conley [ 3 I applied Waiewski method to a boundary value problem for a difusion equations in biology. Since our approach uses Waiewski basic ideasin

I ZE

110

connection with fixed point index theory it should give, even finite dimensions, much better results and can be applied also boundary problems in Hilbert spaces.

in to

2. DYNAMICAL SYSTEMS AND PROCESSES of

Let X be a topoloqical space, X + = X x lR' such that

and let

IT

wx =

if

m

be a mapping from A

A C X x JR+

[ O,wl,

a subset

X. We put

into

s u p Ix does not exist. n ) is a l o c a l semidynamical sys-

DEFINITION 1. We say that ( X , l R + , A , t e m if and only if (a) The map x + f i x , sense that for every x E X

x

E

X, is lower semi-continuous in

If wz < m, then for every 11 > 0 bourhood V of x such that

y € V + w

>fA

Y

there exists a

the

neigh-

- q

X

if w = m then for every C E R* there exists a neighX bourhood V of x such that

y€v'w TI

> c

is continuous

= x

TI(x,O)

If

Y

t

E

Ix

n(n(t,x),sj

for every and

x

E

X

s E 1 7 1 ( X , tthen ,

= v(x,s

+ tl

for every

s

+ t

E

t

Ix, s

E

IEc E ITI(X,tJ.

Autonomous differential equations on Banach spaces, autonomous functional differential equations are examples of semi-dynamical

111

DYNAMICAL SYSTEMS AND PROCESSES

systems. Dafermos [ 4 I introduced a generalization of dynamicalsystems as to include also non autonomous differential equations in Banach spaces or nonautonomous functional differential equations. D E F I N I T I O N 2. 1 4

1 . Suppose X is a Banach space

U ( o ,t )

is a given mapping and define by U(cr,t)x = u ( o , x , t ) .

A p r o c e s s on x is a mapping the following property

i)

u

ii)

U(0,O)

iii)

Ula + s , t l U ( o , s /

:

X

u : B

+

X

x

x

a

E

+

x

for

x

B+

B,

t

E

I?+

satisfying

is continuous

= I

(identity) = Ulo,s + t l .

A process is said to be an a u t o n o m o u s p r o c e s s or a s e m i d y n a m i c a l s y s t e m if U(o,t) is independent of 0 , that is, T l t i = u ( G , t ) , t,G. Then Let

0

(x,a )

-- m

T(t)x A

c B

if

is continuous for (t,xl x

x

sup I

x

B+ and

+

B '

X

X.

X. We define

does not exist.

(X,Ol

Then if the map (x,a)

u : A

E

of (x,0) is continuous in the sense Definition 1, u defines a local process. A local semi-dynamicalsystem is an autonomous local process. -+

w

If X is a bounded metric space we define the m e a s u r e o f ndnc o m p a c t n e s s of A to be i n f { d > OIA can be covered by a finite number of sets of diameter less than or equal to d ) . If X is a Banach space and A a bounded subset of X , A inherites a metric from X and we can give the same definition of the measure of non-conpactness of A.

Let X I and X 2 be metric spaces and suppose f : X I + X 2 is a continuous map. We say that f is a k - s e t - c o n t r a c t i o n if given any < kyIiA). Of bounded set A in X I , f ( A I is bounded and y 2 ( f l A l ) -

I ZE

112

course, yi denotes the measure of non compactness in X i , i = 1,2. We assume that 0 5 k < 1. If f is a k-set contraction we define y ( f ) = i n f {k 2 O ( f is a k-set contraction}. We say that f : X X is a l o c a l s t r i c t s e t - c o n t r a c t i o n if for every x E X there is a neighbourhood N(x) such that f(N(x) is a kx-set-contraction. -+

M. Furi and A . Vignoli [ 8 ] and B. N. Sadovskii gave a slight generalization of k-set-contraction. Given a continuous mapping

f : X I --* X 2 we say that f is a c o n d e n s i n g map if for every bounded set A C X I such that y l ( A I f 0, y g ( f l A l l y l ( A ) . We say that f is a l o c a l c o n d e n s i n g map if every x E X has a neighbourhood N(xl such that f l N ( x l is a condensing map. If f is a k-set-contraction f is condensing but the converse is not true in genera1:see Nussbaum [16]. If f is linear the two concepts are equivalent. There are several examples of processes described by functional differential equations and partial differential equations of the e m lution type that are compact or a-set contractions. EXAMPLE 1. Let r > 0, C = C ( [ - r, 0 1 , Zin the space of continuous functions defined in [ - r , O l . If x E C ( [ U - r , u + A l , I R n l . A > 0, u E IR define x t ( 9 ) = x ( t + 8). Let Q C 23 x C, s2 open, and let D , f : Q --* ?i be continuous functions, D is linear and 0

D t 4 ) = 4co) -

j

apit,ei$tei

-r

where is a matrix function of bounded variation for 8 E [ - r , O ] . A f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n o f t h e n e u t r a l t y p e is a relation of the form (1.1) Let

x ( t , $ ) be the solution of

d dt

D(t,xtl

= 0, 4 = 0.

We say that D is an u n i f o r m l y s t a b l e o p e r a t o r if there areconstants K > 1, c1 0 such that Iz(t,$)

I 5

-clft-tof

Ke

>

t,to*

The solutions of this equations describes a process U(a,tl@= x t l o , 0). If D is an uniformly stable operator and t > P , U is a weak a-set contraction, that is, for every bounded set A C iR x C for which

113

DYNAMICAL SYSTEMS AND PROCESSES

U(Al is bounded is the equation

Y(U(AI1 5 k Y f A I . When B ( t , @ 1 = @ ( a 1 equation (1.1) x = f ( t , x t ) and if t > r l the process U is compact

191.

Another general form of a neutral equation €or which there is a reasonable existence and continuation of solutions theory, 1 5 1 ,is the equation

= $ ( 8 ) E L p ( [ - ~ , O ] ,Bn) where ~ ~ ( =8 $ )( e l E C([-r,O],lRn), and f satisfies a uniform Lipchitz condition with respect to $ in n L p ( [ -r, 0 1 , B I , 1 5 p 5 m. The process described by these equations is also a k-set contraction. The following example is given in [ 2 2 1 . Let X be a Banach space and A : D ( A l * X be a closedl densely defined linear operator in X. A is called seetoria2 if there are constants @, M, a with 0 < 4 < n / Z , M .>_ I , a E IR such that the sector S = {A E C 1 X # a , @ < u r g 1 A - u 1 < r3 is contained in @,a pfA) the resolvent set of A , and 11 (A - a)-'II 5 M / ( h - a ) for all A E S Q , a . If A is sectorial, then there is a k 0 such that

EXAMPLE 2 .

RecrfA

f

A;"

X

0

A' IIu

kI) > 0. Let

Al = A

f

k l . For

is bounded and injective. Let

0


o

pro-

cesses o r l o c a l s e m i d y n a m i c a l s y s t e m s . D E F I N I T I O N 3.

Suppose u

t h r o u g h (o,x) E lR x X

The o r b i t

+

i s a p r o c e s s on i s the set i n R

X. x

X

The t r a j e c t o r y - r + i o , x ) d e f i n e d by

y f a , x ) t h r o u g h ( u , x ) i s t h e s e t i n X d e f i n e d by

D Y N A M I C A L S Y S T E M S A N D PROCESSES

i s a p r o c e s s on X t h e n a n i n t e g r a l of t h e p r o i s a continuous function y : B X s u c h t h a t f o r any

DEFINITION 4 . cess on 5 E

115

W

u

If

+

R,

An i n t e g r a l

I s an i n t e g r a l t h r o u g h ( o , x ) E R x X

y

if

y ( o l = X.

W e assume i n t h e f o l l o w i n g t h a t t h e i n t e g r a l t h r o u g h e a c h ( o , x l E

R

i s unique.

X

x

W e define -1

T

(21 = {(a,y)

Po =

If

( 5 , ~ )E

E

R

x

X13t > 0

R x X

and

such t h a t z E y+(o,xl,

U ( o , t ) y = xj. we define

3 . MAIN RESULTS Let w

# @

W e put:

R b e a n o p e n s e t of R x X, w a n o p e n s e t of R , w aw = n (R - w l t h e b o u n d a r y o f w w i t h r e s p e c t t o

and

w

C

R, a.

116

IZ E

The points of S are cal ed e g r e s s p o i n t s , the points of called s t r i c t e g r e s s p o i n t s .

S*

are

+

Given a point P o = (U,x E w, if the trajectory ‘I ( 0 , ~ )of the process is contained in w for every t > 0, we say that the trajectroy is a s y m p t o t i c with respect to w; if the trajectory is not asymptotic with respect to w then there is a t > 0 such that (U + t , u t ~ , t i x )E a w . Taking t p = l m i n t > 0 1 ( 0 + t , ~ ( a , t ) x E) a m )

we have

The point

is

C ( P o l is called the c o n s e q u e n t of Po = (u,x) E

Define G to be the set of all C(po) E S * .

Po. I*)

such that there

C ( P o ) and

Consider the mapping

K

: S*

U

G

+

S*

defined by

K I P o ) = C(Po) if

P o E w,

K ( P O l = Po

p0

if

E

s*.

The proof of the following is standard,see for example [231,[19]. LEMMA 1.

T h e mapping

K : S*

U

G

+

S*

i s continuous.

To prove the following theorem, we will need to know the basic properties of the fixed point index theory as well as the extensions made by Nussbaum I161 for k-set contraction and condensing maps. We shall say that a topological space X is an a b s o l u t e n e i g h b o u r h o o d r e t r a c t (ANR) if given any metric space M, a closed subspace A C M and a continuous map f : A X there exists an open neiqhbourhood U of A and a continuous map F : U + X such that F ( a ) = f ( a ) for a E A. X is called an a b s o l u t e r e t r a c t (AR) if F as above can be defined on all of M. A theorem of Dugundji [ 6 1 asserts that any convex subset of a locally convex topological vector space is an AR. Let A be the category of compact metric absolute neighbourhood retracts --f

117

DYNAMICAL SYSTEMS AND PROCESSES

-

(ANRs). Let A E A, G be an open subset of A and f : G A be a continuous function which has no fixed points on aG. Then there is which satisfiesthe followa unique integer valued function i,(f,G) ing four properties: [ 1 ] +

-

1. Additivity property: If f : G A has no fixed point on aG and the fixed points of f lie in G I U G2 where GI and G2 are two disjoint open sets included in G, then +

In particular if f has no fixed points in that iA(f,G) = 0.

G, this is meant to say

2 . Homotopy property:

Let I denote the closed unit interval [ 0 , l l . If F : G X I A ( A belongs to A of course) is a continuous map, and Ft(x) = F(x,tl has no fixed points on aG for 0 5 t 5 1 then +

-

If G = A then i , ( f , G ) = A(f), the Lefchetz number of f, equals z ( - 1 ) trace (f*,), where f * K : HK(A) HK (A) is the vector space homomorphism of HK(AI to HK(A) and HK(A) is the Cech homology of A with rational coefficients. 3 . Normalization property:

+

4 . Commutativity property: Let A and B be two spaceswhichbelongs to A . Let f : A B be a continuous map. Let V be an open subset of 3 and g : V + A a continuous map. Assume fg has no fixed points on a V . L e t U = f - l f v l . Then gf has no fixed points on a l l and -+

-

Let G be an open subset of a Banach space X and g : G -+ X a continuous map such that g ( x l # x for x E aG. Assume that g is compact, that is, g(Gl has compact closure. Leray and Schauder [ l 1 defined a fixed point index for g and consequently a degree for I - g, I the identity function. We shall denote this degree by deg(I-g,G,OI. It turns out that the Leray-Schauder degree satisfies all the four properties of the fixed point index listed above. Sowe can define the fixed point index by

I ZE

118

i,(g X =

where

1

X n G,X

GI = deg(I - g,G,OI

n

g(G).

fi

In the following we indicated by

z u sI

Assume t h a t t h e r e e x i s t s s e t s

U S I J

Z # @

to

s

ii)

Z

iii)

There i s a r e t r a c t i o n

=

i s a compact

f o r every

izI@

v)

open, S 1

C S

ANR.

r : S1

*

P

U,Z

r

Z r- S 1

E

Z n SI

-P

Q1 :

Z

S1

.

+

2 n S1

such t h a t

.

n w ) # 0.

p0 = ( 0 , ~ )E z n w CIP I d o e s n o t e x i s t , t h a t i s

Then t h e r e e x i s t s a t l e a s t one p o i n t that either

and

C aw

s*

T h e r e i s a c o n t i n u o u s map

iv)

w

satisfying the conditions

i)

@(PI # P

c

K

'

THEOREM 1.

Z c w

the restriction of

CtPoI

E

S

-

SI

or

+

T

such

10,x)

w.

PROOF. C(Po) E

Assume that the theorem is not true. Then for every P S1 and then Z n w C G . Then

E Znw,

From (i) S = S* and from Lemma 1 the map K is continuous and the restriction U of K to Z U S 1 is also continuous. From condition (iii) there is a retraction r : S I + ~ n ~ l .

Po

Then the map R = r into C ( P o ) E Z n Sl

--

U : Z

U S1

--+

Z n S1

is continuous and takes

.

From condition (iv) the map @ takes C ( P o I into Q1(CfPo)) = C' ( P o ) # C ( P o l and then the composite map @ * r : Z Z is continuous and @ - r c(P) # P for every P E Z n w. From property 2 of the fixed index i z ( @ * II fi, Z n w l = 0, what is a contradiction with (v). Then there exists at least one point P o E Z n w such that the trajectory through P o is asymptotic with respect to w, that is, + T ( 0 , ~ )C w, or C I P o I E S - S l .

-

-

-

+

DYNAMICAL S Y S T E M S A N D P R O C E S S E S

REMARK.

tory

T

w

U

= Sl the only final conclusion is that the trajecis asymptotic with respect to w.

When

S

+ (a,x)

Assume t h a t t h e r e e x i s t s e t s

COROLLARY 1.

Z # @

S,

w

s = s*.

(b)

Z

(c)

Z n S

(d)

T h e r e is a c o n t i n u o u s map

C

aw

and

Z C

is c o m p a c t a n d c o n v e x , i s a r e t r a c t of

f o r every

S

Z n S

0 :

Z

-+

S

such t h a t

P E Z n S. Po E Z n w

Then t h e r e e x i s t s a t l e a s t one p o i n t trajectory

open, S

satisfying the conditions

(a)

@(PI # P

119

-r+(o,x) i s contained i n

such t h a t t h e

w.

The proof follows easily since a compact convex set is an

-c

ANR

and then (b) implies (ii). Since @ * r : Z --t Z is continuous @ * r * u" has a fixed point in Z and then iz(@ r u", Z n W ) # 0 what implies (v). In the applications, the following form of Theorem 1 is more useful since we can use the Ascoli-Arzela Theorem to prove the compactness of E . Assume t h a t t h e r e e x i s t s e t s

THEOREM 2 . Z C w U S

2

I'

# @,

s = s*.

ii)

ii

iii)

T h e r e is a r e t r a c t i o n

r

:

T h e r e i s a c o n t i n u o u s map f o r every

- -

P E Z n

z

n

W)

s1

Sl Q1

+.

:

c

C S C

aw

and

Z n Sl

2 n SI

-+

Z n Sl

such t h a t

.

# 0,

A

= 75a-r

Then t h e r e e x i s t s a t l e a s t one p o i n t that either C(PoI E S - S1 or C O O ) -r+(D,Z)

SI

i s compact.

iA(@ r i7,

V)

open,

Z closed convex s a t i s f y i n g the conditions

i)

iv) @(PI # P

w

- -

u"(z).

Po =

( 0 , ~ )E

does

not

n w such exist, that i s ,

w.

The proof follows as in Theorem 1, since 0 and r are continuous and then 0 - P is a compact map such that @ r * c(P) # P for

-

I ZE

120

every tion.

P

E

i A( @

and then

Z

. r - c,

Z n

=

W)

Assume t h a t t h e r e e x i s t s e t s

COROLLARY 2 .

open,

w

S

C

aw

and

such

that

Z # 0, such t h a t

Z c w u S,

=

(a)

S

(b)

Z

(c)

Z n S i s a r e t r a c t of S .

S*.

v"

i s c l o s e d bounded c o n v e x and

(d) T h e r e i s a c o n t i n u o u s map @(P) # P f o r e v e r y P E Z n S .

is compact.

n S

CJ : 2

+

Z n S

Po = (a,xl E Z n

Then t h e r e e x i s t s a t l e a s t one p o i n t that

what is a contradic-

0

C ( P o ) does n o t e x i s t , t h a t is, -c+(a,x)

C

such

G,

w.

Theorems 1 and 2 are general enough to cGver most of the application and if we restrict ourselves to finite dimension, Corollary 1 is essentially equivalent to Waiewski Theorem [ 2 3 1 . However we can give a more general formulation of Theorem 1 and 2 . Actually Theorem 1 and 2 are Corollaries of Theorems 3 and 4 respectively although we prefered to prove them independently. THEOREM 3 .

and

# P

Z

C

w

Assume t h a t t h e r e e x i s t s s e t s U

S l ,

i)

s = s*.

ii)

Z

U

Sl

# 0

Z

i s a compact

.

sl ( @ G , z

i

n

W)

#

@ : SI

jectory

C ( P o ) E S - Sl

T+(G,x)

52,

Sl

C

S C aw

or

--t

Sl

s u c h that

@(PI

Z n w

such

0.

Po = ( a , x )

Then t h e r e e x i s t s a t l e a s t one p o i n t that either

open i n

ANR.

iii) T h e r e e x i s t s a c o n t i n u o u s map f o r every P E Sl iv)

w

satisfying the conditions

C(Pol

E

does n o t e x i s t , t h a t is, the t r a -

i s asymptotic w i t h respect t o

w.

The proof follows as in Theorem 1. If the theorem is not for every Po E Z n w. Then,

C ( p 0 ) E Sl

z

=

(z

n

s 1 ) u (z

n

W)

c

s

u

G.

true

DYNAMICAL SYSTEMS AND PROCESSES

121

From Lemma 1 the map K is continuous and the restriction v" of K to Z U Sl is also continuous. The map 0 * K : Z U S l +. Z U S l is continuous and O * KIP) # P for every P E Z U Si . Then i z sl (@ U,Z n w) = 0, a contradiction and the theorem is proved.

-

"

THEOREM 4 . and Z C w

Assume t h a t t h e r e e x i s t s e t s U

Z # @,

S l ,

Z

U

Sl

w

open i n

R, S l

C

S

C

aw

c l o s e d convex s a t i s f y i n g t h e condi-

tions:

# P

s*.

i)

s =

ii)

T h e r e e x i s t s a c o n t i n u o u s map

f o r every

P

E SI

iii)

v"

iv)

iI0i7, z

0 : Sl

--*

Sl

such that U P )

.

i s compact. n w # 0,

A =

7 5 1 0 . u(z u

Then t h e r e e x i s t s a t Least one p o i n t

sl)). Po =

( 0 , ~ )E

Z

n w

t h a t e i t h e r , C(Po) E S - S I o r C I P I d o e s n o t e x i s t , t h a t i s , t r a j e c t o r y T + ( O , P ~ ) i s a s y m p t o t i c w i t h r e s p e c t t o w.

PROOF. Assume that the theorem is not true. Then every P o E Z n w and then 2 n w C G. Then

C(Pol E S l

such the

for

From Lemma 1 the map K is continuous and the restriction u" of K to

Z U Sl is also continuous. Since 0 is compact the map K that takes Po into C(Po) is compact. The transformation O f i is also compact and Ou"(P) # P for every P E Z V S l Hence i I @ i , Z n w l = 0, what is a contradiction. Then there exists at least one point P o E Z n w such that either C(Po) E S - S I or the trajectory through Po is asymptotic with respect to w.

.

For delay differential equaticns and some integral equations the operator 8 is compact. However most process described by neutral functional differential equations and differential equations of the evolution type as in Examples 1 or 2 of Section 2, the processis not compact but is an a-set contraction or a condensing map. We extend in the following Theorems 1 and 2 for a-set contractionor condensing maps. Following Nussbaum [161 we will give an outline of the theory of fixed index for a-set-contractions and condensing maps.

I ZE

122

X E

Let X be a closed subset of a Banach space B . We shall say that n F if we can write X = u Ci, where Ci are closed convex sets i=I

in B . The metric on X will alwaysbethat which it inherites from Let

G C B

and

g : G * B

a continuous map.

B.

Assume that the

set

is compact. Let us write

and

where

-

Co

denotes convex closure. It is easy to see that

and K , ( g , G ) is closed and convex. If G is bounded and g : G is a k-set contraction, k < I , Kuratowski's results [131 also

+

X

im-

is compact. Finally, assume that g is a local plies that K , ( f , G ) strict set contraction. By this we mean that for every point x E G there is a neighbourhood N ( x ) and a real number 0 5 k z < 1 such that f I N ( x ) is kz-set-contraction. Using these assumptions we can find a bounded open neighbourhood G I of S such that g : G I X is a k-set contraction, k < 1. Let us write K: = K , ( ~ , G ~ I n X . K: is a compact metric ANR, G I K: is an open subset of and g : G I n K: + K: ia a continuous function satisfying the necessary condin K:) is defined. ( R . Nussbaum [ 161 ) . We define tion, so i K * ( f , G l +

KL

m

iX(g,G)

= i *Ig,GI

n K:).

All the usual index property carry through

Km

to this setting. Let G be a bounded open subset of a Banach space B , g : c * B, I the identidy on X and g : G B a k-set-contraction, k < 1 . A s -+

sume that g(xf # x on a G , 4 = K , ( q , G ) define the L e r a y - S c h a u d e r d e g r e e for g

is compact convex so we can as

OYNAMICAL SYSTEMS AND PROCESSES

deg(I - g,G,OI

123

= iA(g,G n A /

A similar definition of index can be given for condensing maps.

X is a conSuppose X E F , G is an open subset of X and g : G tinuous map. We shall say that g is an a d m i s s i b l e map if only if -+

(1)

S

= {a E

G

I

g(xl = x )

is closed and bounded.

(2) There exists a bounded open neighbourhood U of S with U c G and a locally finite covering { C j I j E J } of X by closed convex sets such that

-

/z

(a)

g

is condensing,

(b)

I - g / v

(c)

gtV/ n c

is a closed map, is empty except for finitely many

j

If S is empty, U

may be empty. If

as above we shall say that

(

g, U, {C. 1 j 3

g,

E J } )

NOW let g be an admissible map and let an admissible triple. Since (I - g l ( x l # 0

(I

-

g/

/Z

and

U

ECj

j

E

Ij

J.

E Jl

are

is an admissible triple. (

be g,U, I G j I j E J ) ) for x E a U and since

is a closed map,

-

If f : U X is a continuous map we shall say that f is an admiss i b l e a p p r o x i m a t i o n with respect to (g,U, { C j , j E J } ) if: +

is a k-set-contraction, k < 1 .

(1)

f

(3)

F o r all

j

E

J

and

x

E

3, if

glxl

E

C j

then f(x1

E

C

j*

Let now G be an open subset of a space X E F and let g: G X be a continuous function which is admissible. Let ( g,U, {C . 1 j E J ) ) --f

3

be an admissible triple and let f be an admissible approxirrationwith In [161 is respect to this triple. We define i X ( g , G 1 = i x ( f , U / . proved that this definition is well defined. Let X be a closed convex subset of a Banach space B , G is an open subset of X and f : G X is a continuous condensing mapsuch +

I ZE

124

that g i x l # x for x E aG. Then the fixed point index can be described in terms of the Leray-Schauder degree. First it is not hard to show that there exists 6 > 0 such that 113: - f ( x l 11 5 6 for x E aG. Select any fixed xo E X and define gtlxl

= tglxl + (1 - tlxo

and take t so close to 1 that Define

0 < t

for

1

Ilglx.) - g t f x l l l < 6

for

x E

aG.

-

K l = Co f t l G l , Kn

= Co ftlG n K n - l l

n > 1

for

and

One can prove that K is compact (possibly empty) and convex and that g t ( G n K,l C K. If Km is empty define i X ( g , G l = 0. If Km is not empty let K be any compact convex set such that K 3 Km and g t i G n K l C K. Km is itself such a set so the collection of such K is non empty. Let p be any retraction of B onto K. ( A result of Dugundji [ 6 j guarantees the existence of such a retraction), and let H be any bounded open neighbourhood of the (compact) fixed pint set of g t in G such that H C P - ' ( G i- K). Then one can prove that i X ( g , G 1 = d e g ( I - ft * P , H , 0 1 . In particular the integer on the right hand side is independent of the particular K chosen, the retraction p,H,t and xo. We say that a set A is a d m i s s i b t e i f A C F. For example A is closed convex. THEOREM 5 . Sl

C S

Assume t h a t t h e r e e x i s t s e t s

and

Z

u S l , Z # 8, Z u S l

C

w

open

cZosed

in

R,

convex,

S C

aw,

satisfying

the conditions

# P

i)

s = s*

ii)

T h e r e e x i s t s a c o n t i n u o u s nap

for every

iii)

fi

P E S1

.

i s condensing.

P : Sl

-+

Sl

s u c h t h a t @(PI

125

D Y N A M I C A L SYSTEMS AND PROCESSES

T h e n e i t h e r Po E S - S l l o , z / i s c o n t a i n e d i n w.

o r the trajectory

T

+

ia,z/

through

The proof follows as in Theorem 2. THEOREM 6 .

and

Z

C

Assume t h a t t h e r e e x i s t s e t s

Z # @,

w U S I J

i)

s = s*.

ii)

Z n SI

traction

r

iii) that

Q(P)

iv)

: S1

i s a retract of Z n Sl

+

w

.

@

f o r every

r

*

u"

SI

C

S

that i s , there exists a

SIJ

T h e r e e x i s t s a c o n t i n u o u s map

# P

R,

open i n

C

aw

closed convex s a t i s f y i n g t h e conditions

Z

P

E

Z n Sl

s1

Q :

-+

s1

z

re-

such

.

i s condensing

Then t h e r e e x i s t s a t l e a s t one p o i n t p 6 Z n w C ( P o ) E S - Sl o r t h e t r a j e c t o r y T+(u,x) through ( i n w.

such t h a t e i t h e r 0 , ~ )i

s contained

The proof follows as in Theorem 2. COROLLARY 3 .

Z # @

Assume t h a t t h e r e e x i s t

w

C

a w and 2

C

w C S,

such t h a t

(a)

s

(b)

Z

(c)

Z n s

=

s*.

is c l o s e d c o n v e x b o u n d e d and i s a r e t r a c t of

Q : Z n S

Then t h e r e e x i s t s a t l e a s t one p o i n t C ( P o l does n o t e x i s t ,

U i s condensing.

S.

(d) T h e r e i s a c o n t i n u o u s map Q l P l f P f o r every p 6 n S.

that

open, S

that i s ,

Po

T ' ( ~ , X )

Z n

S

= (t,z)

E

C

+

such

Z n w

that

such

w.

In the applications of the theorems above we must give criteria to verify the condition i , ( f , G ) # 0. I n most applications G is a closed convex subset of a Banach space X and the index can be described in terms of the Leray-Schauder degree. If D' is a closed

126

I ZE

convex subset of a locally convex topological vector space X we say t 3 0. We call that D is a wedge if x E D implies t x E D for D a c o n e (with vertex at 0) if D is a wedge and x E D , - z # O implies that - - z D. The following result is given by Nussbaum 1151. Assume that D

is a wedge in a Banach space

X,

r

and

R

are

unequal positive numbers,

and f : G D is a .condensing map. Assume that there exists h # 0 such that z - f ( z l # t h for all z E D with IIx 11 = R and all t LO and suppose that x - t f l x l # 0 for 3: E D, [ I x I I = r and 0 5 t 5 1 . Then: +

(a)

If

r < R , then

iD(f,UI

= - I , and

(b)

If

r < R , then

iD i f , U I

= + I.

A more general formulation is given in 1161 for local condensing maps and some other condition are also given to verify i D C f , U ) # 0.

Some other criteria are known. For example, suppose that h : R2 R z is of class C 1 , h l 0 ) = 0, and x = 0 is an isolated zero of h. Then i ( h , O I = 1 + 1/2(E - H I where E and H are integers associated +

with the flow x = h C x I , that is, E is equal to the number of elliptic regions and H is equal to the number of hyperbolic regions. This formula for the index is due to Poincars.

REFERENCES

11 ]

F. E. BROWDER, O n t h e f i x e d p o i n t i n d e x f o r c o n t i n u o u s mapping of ZocaZZy connc-!rd spoces. S

[2

1 C.

C.

m Brasil. Mat.

4 (1960), 253 - 293.

CONLEY, l s o l a t e d Invariant S e t s and t h e Morse

Index.

Regional

Conference Series inMathematics, Vol. 38, Amer. Math.Soc, Providence, Rhode Island, 1978. 13 ]

W a i e u s k i m e t h o d t o non l i n e a r b o u n d a r y v a l u e p r o b l e m which aris?-.s in p o p u l a t i o n g e n e t i c s .

C. C. CONLEY, An a p p l i c a t i o n

of

J. Math. Biol. 2 (19751, 241

- 249.

127

DYNAMICAL SYSTEMS AND P R O C E S S E S

14

[

1

C.

M. DAFERMOS, An i n v a r i a n c e p r i n c i p l e f o r c o m p a c t J . Differential Equations 9 ( 1 9 7 1 ) , 2 3 9 - 2 5 2 .

processes.

5 1 R. DRIVER, E x i s t e n c e and c o n t i n u o u s d e p e n d e n c e o f a n e u t r a l funct i o n a l d i f f e r e n t i a t e q u a t i o n . Arch. Rational Mech. Anal., 1 9 (19651,

149

- 166.

1 6 1 J. DUGUNDJI, A n e x t e n s i o n of T i e t z e ’ s t h e o r e m . Pacific J. Math.

1, ( 1 9 5 1 ) , 3 5 3 - 3 6 7 . [ 7

1 A. FRIEDMAN, P a r t i a l D i f f e r e n t i a l E q u a t i o n s . Holt, Rinehart and Winston Inc., 1 9 6 9 .

[ 8

1 M. FUR1 and A. VIGNOLI,

A j ’ i x e d p o i n t t h e o r e m i n c o m p l e t e metric

s p a c e s . Boll. Un. Mat. Ital. ( 4 ) 2 ( 1 9 6 9 ) , 5 0 5 - 5 0 9 . [

9 1 J. K. HALE, T h e o r y of Functional D i f f e r e n t i a 2 E q u a t i o n s . Berlin, 1 9 7 4 .

[lo 1 D. HENRY, G e o m e t r i c t h e o r y

of

semilinear parabolic

Springer

equations.

Lecture notes, University of Kentucky. 1111 A. F. I Z E , A s y m p t o t i c b e h a v i o r

of

r a t i o between t h e components

of s o l u t i o n s o f a l i n e a r s y s t e m o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s and i t s a p p l i c a t i o n s t o determination of L y a p u n o v c h a r a c t e r i s t i c n u m b e r s . Ann. Mat. Pura Appl. (4) 85 (1970), 259

- 276.

1 1 2 1 J. L. KAPLAN, A. LASOTA and J. A .

YORKE,

An a p p z i c a t i o n of t h e

ZESZYTY W a i e w s k i r e t r a c t m e t h o d t o boundary value problems. NAUK, Univ. Jagiello. Prace Mat. Zeszyt 16 (1974), 7 - 1 4 . [ 1 3 1 C.

KURATOWSKI, S U P l e s e s p a c e s c o m p l e t e s . Fund. Math. 1 5 (1930), 3 0 1 - 309.

[141

2.

MIKOLAJSKA, S u r l e s m o u v e m e n t s a s y m p t o t i q u e s d ‘ u n p o i n t maBull. t e r i e l m o b i l e d u n s l e champ d e s f o r c e s r e p o u s a n t s . Acad. Polon. Sci. 1 ( 1 9 5 3 1 , 1 1 - 1 3 .

1151 R. NUSSBAUM, P e r i o d i c s o l u t i o n s

of

some

functional d i f f e r e n t i a l equations

nonlinear 11.

J.

autonomous

Differential

128

I ZE

Equations 1 4 ( 1 9 7 3 )

,

360

- 394.

NUSSBAUM, T h e f i x e d p o i n t i n d e x f o r l o c a l c o n d e n s i n g Ann. Mat. Pura Appl. ( 4 ) 8 9 ( 1 9 7 1 ) , 2 1 7 - 3 5 8 .

maps.

1 1 7 1 C. OLECH, On t h e a s y m p t o t i c b e h a v i o r o f t h e s o l u t i o n s of

a sys-

[ 1 6 1 R.

t e m of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . Bull. Acad.

Pol.

Sci. 4 ( 1 9 5 6 ) , 5 5 5 - 5 6 1 . [ 1 8 1 N. ONUCHIC, O n t h e A s y m p t o t i c B e h a v i o r o f t h e S o l u t i o n s o f func-

t i o n a l D i f f e r e n t i a l E q u a t i o n s . In Differential Equations and Dynamical Systems (J.K. Hale and J. P. LaSalle, Eds.), Academic Press, New York, 1 9 6 7 . 1 1 9 1 A. PELCZAR, Some g e n e r a l i z a t i o n s o f t h e r e t r a c t t h e o r e m o f

W a i e w s k i , w i t h a p p l i c a t i o n t o o r d i n a r y and p a r t i a l f e r e n t i a l e q u a t i o n s of t h e f i r s t o r d e r .

T.

dif-

Annal.

Polon.

Math. 2 9 ( 1 9 7 4 ) , 1 5 - 5 9 . [ 2 0 1 A. PLIS,

On a t o p o l o g i c a l m e t h o d f o r s t u d y i n g t h e b e h a v i o r

of

t h e i n t e g r a l s o f o r d i n a r y d i f f e r e n t i a l ec7untions. Bull. Acad.

Polon. Sci. 2 ( 1 9 5 4 ) , 4 1 5 - 4 1 8 . [ 2 1 ] K.

RYBAKOWSKI, W a s e w s k i p r i n c i p Z a Sor r p t a r d e d f u n c t i o n a l f e r e n t i a l e q u a t i o n s . J. Differential Equations 117

dif-

36 (1980),

- 138.

[ 2 2 1 K. P. RYBAKOWSKI, On t h e homotopy f o r i n f i n i t e d i m e n s i o n a 2 semi-

f l o w s . Trans. Amer. Math. Soc.,to appear. [23]

T, WAiEWSKI, S u r un p r i n c i p l e t o p o l o g i q u e de l ' e x a m e n de Z'alZure asymptotique des i n t G g r a l e s des equations d i f f e r e n t i e l l e s o r d i n a i r e s . Ann. SOC. Polon. Math. 2 0 ( 1 9 4 7 )

,

279

- 313.

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY. J. Mujica (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

ON CONVOLUTION OPERATORS IN SPACES OF ENTIRE FUNCTIONS OF A GIVEN TYPE AND ORDER

M&io C. Matos Instituto de Matemhtica Universidade Estadual de Campinas 13.100 -Campinas, SP Brazil

1. INTRODUCTION In a previous article [lo] we introduced the spaces of entire and functions of order (respectively, nuclear order) k E [ I , + .I type (respectively, nuclear type) strictly less than A E (O,+ - 1 in normed spaces. The corresponding spaces for which the type is allowed to be equal to A were introduced for k E [ I,+ m ] and A E [ O , + w ) . All these spaces carry natural locallyconvex topologies and they are the infinite dimensional analogues of the spaces considered by Martineau in [ 8 ].We studied the Fourier-Bore1 transformations in these spaces andwewere able to identify algebraicaly and topologically the strong duals of these spaces with other spaces of the same kind. In this article we are qoingto study existence and approximation theorems for convolution operators in the cited above spaces of nuclear entire functions. These results generalize to normed spaces theorems obtained by Malqrange 1 7 1 , Fhrempreis [ 4 ] and Martineau [ 8 I They contain as special cases results of Gupta [ 5 1 , Matos [ 9 ] and Colombeau-Matos [ 3 1. See Matos-Nachbin [ 111 and Colombeau-Matos [ 3 I for references on similar results for locally convex spaces.

.

As usual there are three essential points to be studied

one intends to work in the solution of convolution

equations:

when the

Fourier-Bore1 transformation, the division theorems and the existence and approximation results. Since the first subject was studied in [ l o ] , we summarize the Corresponding results in Section 2 . In the third Section we get the correspondence between the topological duals of the spaces under consideration and the set of convolution operators. The division theorems are established in Section 4 andthe last Section is dedicated to prove the existence and approximation results.

130

MATOS

2. SPACES OF ENTIRE FUNCTIONS AND FOURIER-BOREL TRANSFORMATIONS We keep the notations fixed by Nachbin I 1 2 1 and Gupta [ 51, [ 6 I. Hence, if E is a complex normed space, X(El is the vector space of all entire functions in E, P ( ' E l is the Banach space of all continuous j-homogeneous polynomials in E under the natural norm 1 1 - /I and P N ( j E l is the Banach space of all continuous j-homogeneous p l y nomials of nuclear type in E l under the nuclear norm 1 1 . I I N . Now we describe the spaces with which we are going to this article. (a)

all

f

E

If p > 0 we denote by X l E l such that

deal

in

8 ( E I the complex Banach space of P

m

normed by 11 * 11 . The complex Banach space of all P a " f ( 0 ) belongs to P N l n E I for each n E W and

f

E

X(EI such that

m

under the norm I1 * I1 N , p

,

is denoted by

B N , p ( E l . If

A E to,+ ml

we

consider the complex vector spaces

equipped with the corresponding locally convex inductive limit topologies. We consider the projective limit topologies in the complex vector spaces

It is natural to consider the complex vector spaces

131

CONVOLUTION OPERATORS

with the locally convex inductive limit topologies, and

with the projective limit topologies. If p > 0 and k > 1 ( E l ) the Banach space of all (b)

Bk

N, P

k we denote by B ( E l P f E 3 C l E l such that

(respectively,

IlA

Bk

(El

N,p

are endowed with the projective limit topologies. In order plify the notations we also write

to

sim-

MATOS

132

E x p ko ( E ) = E x p k (E), 030 W e note t h a t

f

is i n

is i n

f

E

For A

w

in

[ 0,+

Expk

N,A

A E

(O,+

m ]

X ( E ) is i n

X ( E ) belongs t o

j E

(El.

i f and o n l y i f

(El

and

E W

k E [ 1, + m l and

P ~ ( J E I for

(c)

j

w e have: f

[O,+ml

W e a l s o have:

is i n

for

0,o

k E x p A I E l i f and o n l y i f

f E X ( E ) belongs t o

PN?E)

H e r e w e consider A

N,

E JC(E) is i n

It is also true that

Zjf(0)

E x p kN J 0 ( E ) = E x p

. When

Ezpk

O,A

EX^^,^,^ * (E)

k

E [ 1,+

-I

and

( E ) i f and o n l y i f

i f and only i f

d^j~(01

and

m)

w e d e n o t e by

X,(B

(0))

t h e complex

A-l v e c t o r s p a c e of a l l

f E XlB

(0))

such t h a t

A-l

-

Zim II ( j ~ ~ - l i ? f ( o ) l l l ”

j

i

5

A,

m

endowed w i t h t h e l o c a l l y convex t o p o l o g y g e n e r a t e d by t h e f a m i l y

of

m

seminorms

( P , , ) ~ , ~ , where

W e recall t h a t P.

As u s u a l

0-l

B,(OI

= +

d e n o t e s t h e open b a l l of c e n t e r 0 and r a d i u s and B (0) = E . W e d e n o t e by 3 C N b ( B A - , ( O ) I 0-1

t h e complex v e c t o r s p a c e of a l l

f E 3ClB

(0))

A-1 P ~ ( J E )and

such t h a t

Zjf(0)

E

133

CONVOLUTION OPERATORS

of

endowed with t h e l o c a l l y convex topology generated by t h e family simenorms (pa ) N , P P'A

,

where

m

E X ~ ~ , ~ (and E )

Exp

spaces, respectively. I n order t o simplify

the

W e a l s o adopt t h e n o t a t i o n s

m

N , O,A

( E l f o r the above

notations

we

also

write:

For

to,+

A E

m

]

we denote by e i t h e r

(respectively, e i t h e r

X N b ( B A - l ( 0 )I

2.1.

or

I)

m

EX^^,^ ( E l )

m

ExpA(El

or

t h e l o c a l l y con-

m

m

Expo, (2) ( r e s p e c t i v e l y , E x p

vex i n d u c t i v e l i m i t of t h e spaces for

X b ( B -I ( 0 A

N, 0,P

(El)

i n ( 0 , A ) . W e a l s o set

p

DEFINITION.

k ExpA ( E l

The elements of

(respectively,

k EX^^,^ (El)

a r e c a l l e d e n t i r e f u n c t i o n s of o r d e r

k (respectively, nuclear order k ) and t y p e ( r e s p e c t i v e l y , n u c l e a r t y p e ) s t r i c t l y l e s s t h a n A , when

.

i s i n (0,+ m J When k = I i t i s u s u a l not to I" and when A i s m w e r e p l a c e the phrase " s t r i c t l y and A E [ O,+ml the l e s s t h a n A " by " f i n i t e type". For k E [ I , + m ] k elements of Expo, A ( E l ( r e s p e c t i v e l y , ( E l ) are c a l l e d e n t i r e Ex:PN, 0 , A k E [ 1,

and A

f m ]

w r i t e "of o r d e r

f u n c t i o n s of o r d e r ( r e s p e c t i v e l y , n u c l e a r o r d e r )

k

and

type

(re-

spectively, nucleap t y p e ) l e s s t h a n o r equal t o A . W e now l i s t t h e main r e s u l t s w e a r e going t o need i n

this

ar-

t i c l e . As w e wrote i n t h e i n t r o d u c t i o n t h e p r o o f s of t h e s e f a c t s a r e i n [lo]. 2.2.

PROPOSITION.

and

Expk

N, A

(1) I f

A E

lo,+-]

k k E [ l , f mt h)e,n ExpA(B)

( E l a r e DF-spaces.

and k ( E l a r e Frgchet spaces. E x P ~ JO , A (2)

and

I f

A E [O,+ml

E [

I,+m],

then

Z X k~ ~ , ~ ( E Iand

134

2.3.

MATOS

I n a l l s p a c e s i n t r o d u c e d above t h e T a y l o r series a t

REMARK.

of e a c h of i t s e l e m e n t s corresponding space. 2.4.

(a)

PROPOSITION.

For

(b)

k

E

belongs t o

E x p j ( E l and If

(d)

PROPOSITION.

exp(9l with

and

9

E

the

E’,

a ]

and

and

9

E

the function

E’,

II9II 5

ExpIV,O , A (El i f

(1) The v e c t o r s u b s p a c e g e n e r a t e d by a l l f u n c t i o n s E’ i s dense i n

E x p ~ , (~E l, ~f o r

(2)

The i ’ e c t o r s u b s p a c e g e n e r a t e d by a l l f u n c t i o n s II9lII < A ,

k E (l,+m] k E

( l , + m ]

i s dense i n

and

A E

A

and

( O , + m ] ;

E [O,+m);

5

119 I1

and

A,

i s dense i n

exp(pl, with

A E (O,+ 531.

E ~ p i , ~ f Ef o ) r

The v e c t o r s u b s p a c e g e n e r a t e d by a l l f u n c t i o n s

(3)

ExpN, I O,A ( A ) f o r

A

E

exppgl, with

to,+-).

2 . 6 . DEFINITION. f a ) I f T i s i n o n e of t h e t o p o l o g i c a l duals of k E x p N J A ( E l ( f o r k E ( l , + m ] and A E ( O , + m ] ) , ExP~O , ,A (El

i n (I,+

m ]

f i n e d by (b) (O,+

and

ml ) its

f o r every

(c)

A E [ 0,+

m

I ) , i t s F o u r i e r - B o r e l t r a n s f o r m FT

FTf9l = T ( e x p ( 9 ) ) f o r every If

T

1 ErcpN(E),

(for

k

i s de-

9 E E’.

i s i n t h e t o p o l o g i c a l d u a l of

E ~ p k , ~ ( E l(for

A E

Fourier-BoreI transform F T i s d e f i n e d by F T f 9 ) = T ( e x p ( q l )

9 E E’ If

exp(9l

A.

(b)

E’

function

< A.

Il9II

E x pkN J A f E If o r

and

the

9 E E’, t h e f u n c t i o n e x p ( 9 )

(a)

E B’

E

9 E

9 E E’

ExpN, O,AfE)

A E [O,+ml

E x p o J A( E l and

and

(O,+m]

k

E ~ p i , ~ f E i) f

1

beZongs t o

9

= I,

k

E

k EX^^,^ (El.

and

A E [ 0,+ *l

k = I , A E (O,+

For

(c)

Exp;(E)

1,

(I,+

A

k E (I,fm],

Expk0 , A ( E l and

e z p ( 9 l beZongs t o

2.5.

For

exp ( 9 ) b e l o n g s t o

function

0

i n t h e t o p o l o g y of t h e

f

converges t o

f

with

119 11 < A.

T i s i n t h e t o p o l o g i c a l d u a l of

ExpN,O,A ( E l (for A E [O,+m))

135

CONVOLUTION OPERATORS

i t s Fourier-Bore1

transform

i s d e f i n e d by

FT m

v

for a l l

( 7 ) converges a b s o l u t e l y .

such t h a t

E E'

Here

is

Tj

B T j E P ( j E ' 1 i s given by

BT.(p) 3 = T . ( p 3 ) f o r a l l 9 E E'. A s Gupta proved i n [ 5 1 w e have IIBT.ll=llT.II. 3 3 3 I t can be e a s i l y proved t h a t f o r each T t h e r e i s p T > A such t h a t

t h e r e s t r i c t i o n of

to

T

P,(jE)

( 7 ) Converges a b s o l u t e l y f o r

= T ( e x p ( p o ) ) because T

FTIPI

t o p o l o g i c a l d u a l of 2.7.

NOTATIONS.

and

A-'

- 0

k'

€or

k

= +

E

m

9

respectively. and

(l,+ml.

k'

2.8.

ip

11




The case k = + m is done in the same way with simpler calculations since we do not have to work with multiplicative factors of the form (-

with

ke

n = O,l,

...

(b) We work as in part (a) but we use 3.10 instead of 3.9. We get: for every p > 0 there is C l p ) 0 such that IIPnIIN

where

Pn

5 C l p ) p n n ! IIfll

is as in the proof of ( a ) . It follows that

if p is chosen so that II f I1

< +

N, k , PZ-'

also get

N , k , p 2-I

a.

Hence

T

*f

E

ExpN(E).We k

147

CONVOLUTION OPERATORS

whenever

p

p < pl

i s chosen such t h a t

i m p l i e s t h e c o n t i n u i t y of

T*

in

I n any c a s e it i s c l e a r t h a t

and

11 f \I

T*


0,

and

E

>

E

(El f o r e v e r y P i n

-i,

k E (I,+

D ( E ) are independent o f

Exp!'(E')

and

P =

q

n

n

with

$'(FTl(Ol(ql

C(p,~l

E

(E))'

PNinEI,

and w e

n E iiV

(see

k T E (ExpN(E)l',then

P E PN(E) and

0, E c p, t h e r e are

F i r s t we c o n d i e r

FT

N, 0

0

and

D(E)

2

0

iN. 9 E

E'.

= Tlq')

W e have

f o r every

9 E E' we

MATOS

150

h. a v e

But w e know t h a t (see Gupta

Thus, f o r e v e r y

for

j

E W ,

E

j

f o r every

E

W.

n

and

B ~ E )2 0

Hence t h e r e i s

for a l l

there is

> 0

9 E E'

[ 61)

k

a f ~ 2 ) 0

such t h a t

1. Now w e o b s e r v e t h a t

such t h a t

Now if w e u s e ( 2 3 )

( 2 2 ) i n (21) w e g e t :

k . Hence w e c a n w r i t e

iT. F r o m t h i s f a c t it follows t h a t t h e i n e q u a l i t y o f 3 . 1 7 i s t r u e f o r P E P f n E l a n d , by t h e d e n s i t y of P InE) in P,CnEl, f o r every

n

in

f

f

it i s a l s o t r u e for

P

E

P~PE).

CONVOLUTION OPERATORS

3.18. PROPOSITION.

I f

for every

4

D(E)

2

PROOF.

0

E

> 0,

P E PNlnE)

> 0, p >

independent o f

n

T

E

1

(ExP~(E)J' t h e n

and some c o n s t a n t s

E

C(p,~l

W.

E

P =

As in 3.17 w e have f o r n

9 *P =

and

151

2 j=0

90n,

9 E E'

( ? ) T ( Q n - ' ) 9*

.

j

J

Hence

But, as it w a s remarked in the proof of 3.17, w e know

Thus for e v e r y

f o r all

9 E E'

> 0

E

and

Hence w e may write

t h e r e is

j E lN.

a(€) 1. 0

that

s u c h that

Thus using (25) i n (24) we get

0 and

MATOS

152

IIT * P l l n r , p

5 C ( p , ~ l D l e l n! ( p -

E ) - ~ .

Now t h i s i m p l i e s o u r r e s u l t f o r P E P ( n E l a n d , by t h e f we get the r e s u l t for P E PNlnEl. P i n E ) i n P,lnEl,

density

of

f

3.19. E

PROPOSITION.

> 0,

p >

p >

0,

If

P E P N f n E l and

T

there are constants

E

IEzp,(EI)',

E

C l p , ~ )2 0

t h e n f o r every and

D(E)

2

0

such t h a t

C l p , ~ ) and

n E W.

A s i n t h e l a s t two p r o p o s i t i o n s it is enough t o p r o v e there-

PROOF. sult for

Since

D ( E ) a r e i n d e p e n d e n t of

P = p

n

with

9 E E'.

I n t h i s case w e have

F T E E z p i ( E ' ) w e have

and

For e v e r y

for a l l

E

0 , there is

CIIE) 0 such t h a t

j E W . Now from ( 2 7 ) and ( 2 6 ) w e g e t

Thus w e may w r i t e m

'N,P

(2'

.*PI 5

as w e wanted t o p r o v e .

C f p , ~ l D ( c 1l

1 I l~ n l p -

E ) - ~

153

CONVOLUTION OPERATORS

3.20.

k E [I,

For

THEOREM.

E Z ~ ~ , ~ o ( rE i) n

ExPN, 0, B

+m],

( E l , with

T E ( E x p kN ( E ) ) ' and A

E

(O,+m]

either

f

and

in

B E [O,+ml,

i f

T * f E Expk

(El

we d e f i n e m

T *f =

then ue get

T

*ff

2 T*(ln!l-lc?nf(O!l n=O

Expk ( E l i n t h e f i r s t c a s e and N, A

N,O,B

i n t h e second c a s e . Moreover T * d e f i n e s a c o n v o Z u t i o n o p e r a t o r k IEI respectively. E x P N , A ( E ) and ExPN, 0, B

(a)

PROOF.

p

Hence i f every r

k

(I,+

E

m).

From 3.17 w e h a v e

i s a c o n t i n u o u s seminorm i n

i n (0,Al

Thus f o r e v e r y

there is

p < 0

in

a(rl 2 0

we consider

E

Expk ( E l w e know t h a t N, A such t h a t

for

r

and

> 0

such t h a t

p +

E


B

N, k , p = ExpN, 0, B

p - E > B

such t h a t

> 0

p

we get

(El.

w e choose

p > B

and w e g e t

( E l . Therefore

N,O,B

If w e g i v e

T*

i s a continuous operator i n

ExPN, 0 , B ( E l . I n a n y of t h e above cases t h e mapping f b d l f ( * l x i s c o n t i n u o u s f o r any x E E. S i n c e d l ( ? * P ) ( * l x = T * ( d 1 P I * I x ) for 1 a l l P E P N ( n B l , n E IN ( s e e 3.111, w e g e t d l ( T * f ) ( * l x = T * ( d f ( . l x l f o r every

f

Ezpk

E

N, A

( E l i n t h e f i r s t case and f o r every

f E Exp (El N,O,B

i n t h e second c a s e .

< C(P,EJD(E) if f

B1

is i n

N , P-E

II f I I N , p - E


A

. there i s

CIB)

?-

0

such t h a t

for a l l

z

f o r every

in

n

E.

E

W

By ( 3 3 ) and t h e C a u c h y ' s i n e q u a l i t i e s w e g e t

and

p > 0 . T h e r e f o r e , from ( 3 4 ) and Lemma

4.2,

we get

Hence

for all

B > A.

Thus t h e above l i m i t i s less t h a n o r e q u a l t o A

and

159

CONVOLUTION OPERATORS

the proof is complete. 4.2.

LEMMA.

i s attained f o r

p

= (T)k-'A-'

N p-nexplAp)k

p

T h e minimum of t h e f u n c t i o n

and i t s v a l u e i s (n-'ekA

for

)

The proof of this lemma follows the usual tecnique f o r cases.

is in

I f k E [ I , + w!, A E ( O , + m ] k E x p A ( E l i f and o n l y i f t h e r e i s p A

B > p

it i s possible t o find

4.3.

COROLLARY.

If(r)I

for a l l

in

x

2

C(BI

0

and

f E X(E),

such t h a t

i s entire i n

5 C ( B ) e x p ( B IIxII)

F

and

G(OI

@,

f o r every 4.5.

with order

A > 0

and k

B

and For

in

z E 8

m),

if

< D exp ( B I z

[Glzl A'

> A

satisfying

k E [l,+m),

[ 0,+

( s e e Martinem

t such t h a t

and

t. T h e n f o r e v e r y

COROLLARY.

A

every

k

be en i r e functions i n

G

# 0

k IF(zlI 5 Cexp(Alz1) , z E

then f

E.

4.4. THEOREM [ l ] . L e t

f o r every

for

these

satisfying

We recall the following result of Avanissian [ 11 [ S l for a proof of this result).

F /G

p>O

k nk-'.

and

B'

1.~1 2

I Ik

K.

f E E X ~ ; , ~ ( E and )

f / g

is e n t i r e i n E ,

and t y p e l e s s t h a n o r e q u a l t o

t h e r e i s K '0

> B

g

E

Expk

0, B

(El

t h e n i t i s of

160

MATOS

PROOF.

for a l l

We have

z E E,

A'

> A

and

B'

> B.

W e consider

and

B'

> B.

It is also true that

zo

E

such t h a t

E

g ( z o l # 0. N o w w e d e f i n e

for a l l

5

E 6,

A'

> A

From Theorem 4 . 4 it f o l l o w s t h a t t h e r e i s A',

(5)

2

GZ(0)

#

0 , depending only

0.

on

such t h a t

A, B', B

I(FZ/GZ)

K

I 5

I-[(l+X)/Al

lGz(o)

2 exp([ (A'II

-

((- l

+

x ) ) ~+

+ h ) 2 - 111 k - l

x

((1

+ XIB'Ik

I5 I

g(0l

for a l l

z E E,

f o r every

z

in

j/z

-

z

/I

2

K

and

X >

E l w e g e t h of o r d e r

0 . Hence, i f

k

h(z)=~f/g~(z-zo)

and t y p e less t h a n o r e q u a l

to i n f ( ( ~ ( 1+ A ) / ~

i(

B ( I + A/) k

(1( +A A )2 - l))k-l*

X > O Since

f/g

i s of t h e same o r d e r and t y p e a s

4.6.

REMARK.

h.

The method o f t h e proof of 4 . 5 is e s s e n t i a l l y t h e same

used by Malgrange i n

7

1 f o r t h e proof of t h i s theorem f o r

E =

an

161

CONVOLUTION OPERATORS

and k = 1. When Ehrempreis [ 4 ] .

n

and k < + m, The case B = 0 and E = @

this result was proved by E = 5' is due to Polya (see

, page 191; [13]). Another division theorem we need is the lowing

[ 2 ]

4.6. THEOREM.

Let

f

and

b e holornorphic f u n c t i o n s i n

g

B ( 0 1 C @ r e s p e c t i v e l y w i t h 0 < A < B and B i s holornorphic i n B A 1 O ) and 0 < r < A then

and i n f/g

for every

E

> 0

s u c h that

r +

E

BA(U)

g(0) #

0.

fol-

c

0

If

< A.

If T ( r ; h ) denotes the Nevanlinna characteristic function of the meromorphic function h we have the following inequalities

PROOF.

(i)

5 Tlr;hll + Tfr;hz)

T(r;hlh2)

(ii) If

h ( 0 ) # 0,

(iii) T ( r ; f l

w

then

T(r;h) = T(r;h-')

5 log+M(r;f) 5 ~ R + r -T ( r R;f)

if f is holomorphic in a neighborhood of number

s u p { If(zll;

If in (iii) R

+ loglh(0ll

IzI

= r)

and

is replaced by Z + E

(iv) l o g M f r ; f l 5 7 T ( r + Now we apply (iv) to

f/g

r +

B R ( 0 ) and

M ( r ; f ) is

= maxlO,tog(xll.

Zog'x E

we get

E;f).

to get

Applying ( i ) and (ii) it follows that

If we use the first part of (iii) we get

the

162

MATOS

Now the result follows if we exponentiate both sides and we use the inequality e x p ( l o g + x ) 5 I + x for x 2 0. COROLLARY.

4.7.

g ( 0 l # 0 . If

f/g

then

sup

I(f/gl(x)I

m

m

( E l , g E E X ~ ~ , ~ (wEi t )h B > A 2 0 and 0, B i s k o l o m o r p h i c i n B -1(0) C E and + 00 r > B, B

Let

5

f

E

Ig(0)

Exp

I (2+EJ/-E(l

+

sup

If

(5)

I I (Z+EJ/E

II x II =r-'+E

II x I1 =r-1 '

(1 +

sup

lgixl

1)

(2+EJ/E

I1 3: II =r-l+c for e v e r y E > 0 i n ExpOJBtEl.

r

suck t h a t

-1

+

E

< B-'.

f/g

This implies that

is

PROOF. It is enough to consider F x ( z l = flzxl and G x ( z ) = g ( z x l for every x in E and z in 6. If we apply 4.6 we get

/ z

/ z

I=l+E

]=I+€

We recall the following result proved by Gupta in 4.8.

f

g

Let

is d i v i s i b l e b y

of d i m e n s i o n o n e and f o r any c o n n e c t e d g i s n o t i d e n t i c a l l y zero, the r e s t r i c glS' w i t h t h e q u o t i e n t h o l o m o r p h i c i n

i s d i v i s i b Z e by

g

any a f f i n e s u b s p a c e component tion S',

.

U b e a n o p e n c o n n e c t e d s u b s e t of E and l e t be hoZomorphic i n U w i t h g n o t i d e n t i c a l l y z e r o . I f , f o r

PROPOSITION.

and

[51

of

S'

flS'

then f

S

S n U

of

E

where

U.

w i t h t h e quotient kolomorphic i n

Now we will be able to prove the next three division theorems. 4.9.

THEOREM.

T2 # 0

and

If

k E

[l,+m]

TlIPexp91 = 0

and

TlJT2 E

whenever

( E q k

N, 0

(E))' are such t h a t

T2*Fexpq = 0

with

Ip

E

E'

163

CONVOLUTION OPERATORS

and

P in

P I V C n E I , n E iX,

then

q u o t i e n t b e i n g an e l e m e n t of We c o n s i d e r

PROOF. S

"PI +

are e n t i r e f u n c t i o n s . I f

k = 1

and r a d i u s

FTl

if

is a z e r o of o r d e r

to

each

k > I , w e have

FTl(S

holomorphic i n

S. For

p o n e n t of equal t o

g l ( t ) = FTl(Pl

of

d i v i s i b l e by k = 1

p . This gives

j

for

4.10.

-

FT2

are i n

Expk'(E'l

k

Tl(P exp9I

and n

P E P,(nEl,

E

we m a y w r i t e For

4 . 9 w e have

[I,+ = 0

then

W,

b e i n g a n e l e m e n t of

PROOF.

E

we get

FTl

= h

*

FT2

TIJTz

with

t h e r e is

and

FT2

h E X(E'I

E X ~ ~ ( E ' I .

in

T1,TZ

E

k fExp,(E))

T2 *Pe x p p = 0

FT2

is

FTl

since

'

FT 1

a r e such t h a t with

p E E',

w i t h t h e quotient

k = + m

T2

b e l o n g s a l s o to ( e x p , ,

(EI

m,

I

I .

also i n ( E x p k

h E Expk'(E').

(E))'. Hence by fl, 0 Since FTl and F T 2

h E E x p ok ' ( E ' ) , by 4 . 5 .

are of t y p e z e r o w e g e t t h a t FT1

and

because

w e have

S'

We r e m a r k t h a t i n t h e e a s e

Exp;'(E'I.

being

FT21S' i s a c o n n e c t e d com-

e X ( E ' ) such t h a t

i s d i v i s i b l e by

FTI

for

H E Expk'(G'I.

whenever

T2 * P e x p p

k > 1

a]

a

d i v i s i b l e by

FTl(S' H

Therefore,

k = 1 . By 4.5 and 4 . 7 ,

B p ( O I , when

If

THEOREM. 0

in

+ tp,).

S ' , where

there is

to is

Thus

with the quotient

FT21S

w e have

S n B p ( O I . By 4.8

H

and F T 2

h E

(p +tp2)=T2(exp(pl+t921) 1

j < p.

f o r each

w i t h t h e q u o t i e n t holomorphic i n

have

2

f o r every

+ top2)) = 0

Tl(pi exp(pl

z e r o of o r d e r a t least p

#

0

j < p:

Hence

T2

and F T 2

FTI

a r e h o l o m o r p h i c . I n any case

FT2

g 2 ( t ) = PT

of

p

k > I , then

If

9'. Hence

t h e r e i s a n open b a l l o f c e n t e r and

T , ( i p i e x p (pl + t o p 2 ) ) = 0

we g e t

a f f i n e s u b s p a c e S of

t p 2 ; t E U!}.

where

with the

FT2

E ~ ~ ~ ' ( E ' I .

a one-dimensional

i s of t h e f o r m

p > 0

is divisible by

FTl

For

k

= 1, w e

m

E x p O ( E ' I a n d , e x a c t l y a s i n t h e p r o o f of 4.9

such t h a t

FTl

= h

*

FT2.

By 4 . 7 it f o l l o w s

that

164

MATOS

4.11.

k

(1) For

THEOREM.

E [ l,+m]

and

f E x p k ( E l ) ' a r e s u c h t h a t T 2 # 0 i s of N, A = 0 whenever T 2 * P e x p p = 0 w i t h p

FTl

FT2

i s d i v i s i b l e by

For

(2)

k

[

E

l,+ml

k

e l e m e n t s of

with

the

and

'

being

T2 # 0

E

n E JV, t h e n

an

TI

i f

T1,T2

T 1(P e x p p )

P E PN(nEI,

E l ,

lo,+ ml

B E

if

m)

t y p e z e r o and

quotient

such t h a t

(EX~,,~,~(E)I

A E (O,+

e l e m e n t of

and

i s of t y p e

T 2 are

zero

Tl(P e x p p ) = 0 w h e n e v e r T 2 * P e x p 9 = 0 w i t h Ip E E l , P n E W , t h e n FT1 is d i v i s i b l e b y F T 2 w i t h t h e q u o t i e n t

and

E P,(nE),

being

an

W e remark t h a t w e may w r i t e T 2 * P e x p p b e c a u s e i n a n y case i s a n e l e m e n t of I E X ~ ~ , ~ ( E1 . ) I

T2

w e also have t h a t

and

PROOF.

(1) F o r

k > 1

by 4 . 9

there is

h E E x : p k ' ( E ' ) such t h a t

i s of t y p e we get

0

and

FTl = h

*

(Expk

(E))'

N, 0 FT2. Since

FT2

(E'I

by 4.5.

(A ( k ) A ) - '

k = I , w e have

If

E

i s of t y p e less t h a n o r e q u a l t o ( X l k I A l - ' ,

FT1

h E ~ X p k ' 0,

TlJT2

FT1,FT2

E

Exp

co

= Xb(BA(OII. B y r e a -

(El)

0,A-1

s o n i n g as w e h a v e d o n e i n t h e p r o o f of 4.9 w e u s e 4.8 t o p r o v e there is

h E X ( B A ( 0 ) l such t h a t

we use 4 . 1 t o prove t h a t

h

E

FT1 = FT2

Xb(BA(OII

*

= Exp

h. Since

that

FT2 E M b ( E r )

m ( E l ) .

O,A

( E x p k ( E l ) '. By N, 0 4 . 9 t h e r e i s h E E x p k r ( B ' I s u c h t h a t FTl = h FT2 S i n c e FT2 i s of t y p e 0 and FT1 i s of t y p e s t r i c t l y less t h a n (A(k1Bl-I w e h a v e h o f t y p e s t r i c t l y less t h a t I A ( k l B l - ' by 4.5.

For

(2)

If

is

C >

Xb(EII

k > 1

k = 1 w e have

B

such t h a t

since

T2

f i r s t part we get

w e a l s o have t h a t

T1,T2

E

-

.

( E ' I = X b I T ) . Hence t h e r e B-1 F T l J F T 2E X b ( B c ( 0 ) ) ( i n f a c t w e have FT2 E

F T I JFT2 E Exp

i s of t y p e z e r o ) . Hence, a s i n h E Xb(BC(OII

such t h a t

FTl

t h e proof 1

h

FT2.

of t h e

165

CONVOLUTION OPERATORS

5. EXISTENCE AND APPROXIMATION THEOREMS FOR CONVOLUTION EQUATIONS Now w e a r e r e a d y t o u s e t h e p r e v i o u s r e s u l t s i n o r d e r t o

prove

theorems a b o u t t h e approximation and e x i s t e n c e of s o l u t i o n s of

con-

volution equations. k 5.1. THEOREM. (1) If k E [ i, + m l and 0 E A. t h e n t h e vector subs p a c e o f Expk ( E l g e n e r a t e d by t h e e x p o n e n t i a l p o l y n o m i a l s o l u t i o n s N,O of t h e homogeneous e q u a t i o n

E = Cpexplp;

0

( i . e . , generated by

0

n

P E P ~ ( ~ E lp ) ,E E ' ,

E jiv,

( ~ e x p p l= 01

is d e n s e i n t h e c l o s e d s u b s p a c e o f a l l s o l u t i o n s o f t h e

homogeneous

equation ( f . e . , dense i n

and

(2) If k E [ I,+ = ] E x pkN ( E l g e n e r a t e d by

of

0

E

Ak

then the vector

subspace

i s dense i n

0

t h e r e s u l t f o l l o w s from

PROOF.

(1) If

0 #

By 3.13 t h e r e is

0.

E

0

(Erpk (El I N, 0 i s such t h a t X l d : = 0,

T

E

'

2.3.

such t h a t

Now w e assume 0 = T*. Now, i f

(E))' t h e n by 4.9 t h e r e i s h E N, 0 k E x p k ' ( E ' I such t h a t FX = h FT. But h = FS f o r some S E (ExpNJO(El)' by 2 . 8 . S i n c e FX = FS - F T = F(S * T ) by 3.14, w e have X = S * T . I t follows t h a t X * f = S * ( T * f ) = 0 f o r f E K. Hence X t f l = (X*f)(O) = 0 f o r f E K. Thus by t h e Hahn-Banach Theorem E is d e n s e i n K .

X

E

(Ex:pk

If

(2)

If

k #

m,

0

0

t h e r e s u l t f o l l o w s from 2.3. N m we assume 0 # 0. T E ( E x pkN ( E ) ) ' such t h a t 0 = T*. If k = + m ,

by 3.13 w e g e t

s i n c e e v e r y element of

m

(Exp ( E l l '

N

3.22 t h a t t h e r e i s a n element

if

X E

k f E x p m f E ) l ' i s such t h a t

E x p i ? E ' l such t h a t

T

E

i s o f t y p e z e r o , it follows from ( E ~ p 1 f E ) l ' such t h a t 0 = T*.Now

X l E = 0, t h e n by 4 . 1 0 t h e r e i s

FX = h * F T . By 2 . 8

there is

S

h

E

i n ( E x pkN ( E ) l ' s u c h

166

MATOS

k # +

h = FS. F o r

that

a:

k = +

K . For

dense i n

above i n p a r t (1) w e g e t dense i n 5.2.

K

X = S

from 3.14 w e g e t

03

XIK = 0

i n p a r t (1) w e g e t

* T.

Thus as above

a n d , by t h e Hahn-Banach Theorem, wehave

a, from 3 . 2 3 w e h a v e X = S * T . Thus as X I K = 0 . T h u s , a s u s u a l , w e g e t t h a t L. i s

by t h e Hahn-Banach Theorem.

THEOREM.

(1) If

k

(O,+ m i and

A E

E [ l , + w I ,

t y p e z e r o , t h e n t h e v e c t o r s u b s p a c e of

0

E

Ak

is of

O,A

ExP~O , ,A (El g e n e r a t e d b y

is d e n s e i n

(2)

k

If

t h e n t h e v e c t o r subspace o f E = {Pexpq; P

E

0

B E (O,+ m l and

E [I , +m ],

Ezpk

N, B

P,( n E ) , p

E

k A,

is o f t y p e zero,

( E ) g e n e r a t e d by

E E',

n E LV,

O ( P e x p p ) = 0)

is d e n s e i n = {f

K

0 = 0

(1) I f

PROOF.

By 3.22 t h e r e i s T * . Now i f

X

T

k

= 01.

E E x ~ ~ , ~ ( E )Of ;

t h e r e s u l t f o l l o w s f r o m 2.3. E

i n (Exp

(Exp

N , 0,A

N , O,A from 4 . 1 1 t h a t t h e r e i s h

(E))

in

(El)

'

I,

T

i s such t h a t

Exp

0 # 0.

W e assume

of t y p e z e r o , s u c h t h a t 0 =

k'

Xld: = 0

it

follows

( E ' ) s u c h t h a t F X = h * FT.

(A ( k )A)-' Thus, by 2 . 8 ,

h

= FS

f o r some

S

in

(Expk

N , O,A

(E))'.

Now from

3.23

X = S * T and, f o r f i n K , w e g e t X i f ) = S(T * f) = S ( 0 f ) = S ( 0 ) = 0. Hence X l K = 0 and d: i s d e n s e i n K by t h e Hahn-Banach

we get

Theorem.

0 = 0 t h e r e s u l t f o l l o w s f r o m 2.3. W e assume 0 # 0 . By 3.22 t h e r e i s T i n ( E q k ( E l ) ' of t y p e z e r o s u c h t h a t 0 =T*.If (2)

X is

E

If

N, B (Ercpk (E))' i s s u c h t h a t X l d : = 0, w e g e t from 4 . 1 1 t h a t t h e r e N, B h i n Expk ' ( E l ) such t h a t F X = h * FT. NOW, from 2.8, 0, ( A ( k ) E)-'

167

CONVOLUTION OPERATORS

there is

(Expk

S in

N, B f E K

we prove that if

FS = h .

such that

then

X l f l = 0 . Hence

[

if 0

k

(1) F o r

5 . 3 . THEOREM.

(Ell

E

As i n the f i r s t part

i s dense i n

d:

A.k

then i t s

K.

transposed

mapping

i s such t h a t tO(Expk

(a)

N, 0

i s t h e o r t h o g o n a l of

(E))'

k

For

E

i n ( E q fl, k 0( E ) ) ' .

i s c l o s e d OF t h e weaq topology i n ( E q Nk , 0 ( E l ) '

tO(Expk (Ell' N, 0 k d e f i n e d by ExpN, ( E ) . (b)

(2)

kero

[l,+ml,

if

0

E

A

k

,

0 # 0 , then i t s

transposed

mapping :

k k (ExpNfE/I'W (ExpN(EI)'

i s such t h a t (a)

k k t O ( E x p f l ( E I l ' i s t h e o r t h o g o n a l of k e r 0 i n ( E ~ p ~ ( EI .l l

(b)

t O ( E x pkN i E ) l ' i s c l o s e d f o r t h e weak t o p o Z o g y

k ExpN(El i n

For

(3)

z e r o and

by

k

E [

I,+

m)

and

A

E

to,+

m),

0

i f

0 # 0, t h e n i t s t r a n s p o s e d mapping

(a)

( ~ x p

(b)

1

N , O,A

ExpN,O,A

(4) z e r o and

defined

by

k (EzpN(EII'.

Exp

N , O,A

(E) i n

E

A kO , A

is of type

i s such t h a t

( E l ) ' i s t h e o r t h o g o n a l of kerC i n ( E q k

N , 0,A

(Ell

( ~ x p

'

i s closed f o r the

N , O,A

(Ell

weak

topology

(El)'.

define

I.

F o r k E [ I , + - ] and B E ( O , + m ) , i f 0 # 0 , t h e n i t s t r a n s p o s e d mapping

0

E A:

i s of i s such t h a t

type

(a)

t O ( E x p kN , B ( E I I

'

i s the orthogonal o f

(b)

tO(Expk

'

is c l o s e d f o r t h e weak t o p o l o g y d e f i n e d b y

N, B

(Ell

kerO i n

( E ~ ~ , ~ ( 1E. ) I

168

MATOS

I n any c a s e t h e r e i s

PROOF.

i n t h e domain of

T

0 =

such t h a t

Now f o r each X i n t h e image of t t 0 w e have X = 0s f o r some S i n t h e domain of Hence X(fl = ( t U S l ( f l = S i O f l = 0 f o r every f i n k e r 0 . Thus t h e image of

T*.

( I t f o l l o w s from 3.13 and 3.22).

(ker 0 ) ' .

i s contained i n

X =

lows t h a t

S

*T

0 ) '

X E (ker

Conversely f o r

fol-

it

( t h i s is proved

f o r some S i n t h e domain of

as i n t h e p r o o f s of 5 . 1 and 5 . 2 ) . Now, f o r f i n t h e domain of 0 w e g e t X t f l = S ( T * f l = S ( 0 f l = C t O S l ( f l , so t h a t X = t 0s. Thus we

(ker0)'

have

C

image of

= IT

( k e r 0)'

O(Expk

N, 0

(1) For

( E l ) = Ezpk

N, 0

(2)

surjection (3)

if

k E [2,+m1,

0)

V f E ker

o

0 #

0

and

E

.Ao, k

then

(El.

For

k E [ 2,+ml,

For

k E

i f

0 #

0

0

and

E Ak ,

0

then

i s

a

k AO ,A

is

. [Z,+m]

For

and

A E lo,+

m l , if

0 # 0, 0

E

0 # 0, 0

E A,

k 0(Exp N, O,A ( E l l = Exp N , 0, A

of t y p e z e r o , t h e n

(4)

T(fl = 0

domain o f

E

( k e r o l i s c l o s e d i n t h e weak t o p o l o g y .

w e a l s o have t h a t 5 . 4 . THEOREM.

Further, since

k E [1,+

of t y p e z e r o t h e n

m ]

O(Expk

N, B

and

B E (0,+ml,

if

W e j u s t r e c a l l t h e Dieudonn&Schwartz Theorem: I f are e i t h e r F r g c h e t s p a c e s o r DF-spaces and u : E F is c o n t i n u o u s mapping, t h e n t h e f o l l o w i n g a r e e q u i v a l e n t : +

u(El

(b)

tu

E

and

a

linear

F

= F

: F'

weak t o p o l o g y of

is

( E I ) = Expk (El. N, B

PROOF.

(a)

k

+

E'

E'

i s i n j e c t i v e and

d e f i n e d by

E.

tulF'l

is closed f o r

the

169

CONVOLUTION OPERATORS

Since our spaces are either and since 5.3 holds true we just in each case. Since 0 = T* for and 3.22) , we have t O S = S * T t O ( ( tOS) (f) = Stof! = S ( T * f) =

Frgchet spaces or DF-spaces (by2.2) need to show that is injective some T in the domain of (by3.13 for every S in the domain of

*T)

0 = F(S

in the domain of we have 3.22). Since T # 0, FT # 0 and injection.

S

tOS = 0 for some

NOW, if

(S * T l f l .

= FS

FS = 0. Hence

(by 3.14 and t and 0 is an

FT = 0

*

S

5.5. EXAMPLES. In order to complete this exposition wegive examples of convolution operators of type zero. (1)

We consider

(a)

If

H~

E

( P ~ ( ~ E ) for ) ~ n in

and

k E [I,+

A E [O,+

m)

W.

we define

i* ) Then (b) If k E [ 1,+ m ) and Expk ( E l and x in E. We get N, A

(c) For k = + EXPN O,A (El and x in

m

m

(d) For k = + w know that there is 0 < use

*)

to define

A

E

Om E

Om

E

A,,,

is of type zero.

we use ( * ) of type zero.

O,+m]

AA~

and A E [ O + m) we use ( * ) m E , IIxll < A - ' . We get Om E AO,A and p

f

A E

< A

O m ( f l for x

lo,+ - 1 ,

such that in

if f is in f E

m

E, IIxII < ( p f ) - ' .

for

f in

for f in of type zero. m

E X ~ ; ~ ~ , ~we ~(EI (E). Then we

We get

Urn E A;

of type zero. (2) pose that

We consider

Hn

in (PN(nEIIr for n in

lN

and we sup-

MATOS

170

where f

is in

ExpN, 0 , A ( E l and z is in

E. We have

0 E At,A

of

type zero. (b)

where f

For

k = +

m

and

A

E [

0,+ m i

is in

of type zero.

Other examples may be given by stating convenient conditions wer IIHnII

as n varies in

D.

REFERENCES [

11

V. AVANASSIAN, Fonctions p Z u r i s o u s h a r m o n i q u e s , d i f f z r e n c e s d e u s f o n c t i o n s p l u r i s o u s h a r m o n i q u e s de

type

de

exponentie 2.

C.R. Acad. Sci. Paris, 2 5 2 ( 1 9 6 1 ) , 4 9 9 - 5 0 0 . I21

R. P. BOAS, E n t i r e f u n c t i o n s . Academic Press, New York, 1 9 5 4 .

[ 3 ] J. F. COLOMBEAU and M. C. MATOS, C o n v o l u t i o n e q u a t i o n s i n i n f i n i t e d i m e n s i o n s : B r i e f s u r v e y , new r e s u l t s and proofs. Functional Analysis, Holomorphy and Approximation Theory (J. A. Barroso, ed.) pp. 131-178. Notas de Matemstica,NorthHolland, Amsterdam, 1982. [4 1

L.

EHREMPREIS, A f u n d a m e n t a l p r i n c i p l e for s y s t e m s o f linear d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t s c o e f f i c i e n t s and of i t s applications.

some

Proceedings of the International S y m

posium on Linear Spaces pp. 1 6 1 - 1 7 4 . rusalem, 1 9 6 1 .

Academic Press

Je-

151

c.

r 6 1

C . P . GUPTA, C o n v o l u t i o n s O p e r a t o r s and Holomorphic Mappings on a Banach S p a c e . Siminaire d'Analyse Moderne, 2. Unversitg

P. GUPTA, M a l g r a n g e T h e o r e m for N u c Z e a r l y E n t i r e Funct?'ons of Bounded T y p e o n a Banach S p a c e . Notas de Matemztica, Vol. 37. Instituto de Matemstica Pura e Aplicada, Rio de Janeiro, 1 9 6 8 .

CONVOLUTION OPERATORS

171

de Sherbrooke. Sherbrooke, 1 9 6 9 . [ 7

1 B. MALGRANGE, E x i s t e n c e e t a p p r o x i m a t i o n d e s s o l u t i o n s d e s ;quat i o n s aux d e r i v e e s p a r t i e l l e s e t des e q u a t i o n s des convol u t i o n s . Ann. Inst. Fourier (Grenoble) 6 (1955-56),

[

271- 355.

8 1 A . MARTINEAU, E q u a t i o n s d i f f e r e n t i e Z Z e s d ’ o r d r e i n f i n i . Bull. SoC. Math. France 9 5 ( 1 9 6 7 1 , 1 0 9 - 1 5 4 .

[ 9

1 M. C. MATOS, On Malgrange T h e o r e m for n u c l e c i r k o l o m o r p h i c f u n c t i o n s i n o p e n b a l l s of a Banack s p a c e . Math. 2. 162 (19781, 1 1 3 - 1 2 3 , and correction in Math. Z. 171 ( 1 9 8 0 ) , 2 8 9 - 2 9 0 .

[lo] M. C. MATOS, On t h e F o u r i e r - B o r e 1 t r a n s f o r m a t i o n and s p a c e s o f e n t i r e f u n c t i o n s i n a normed s p a c e . Functional Analysis, Holomorphy and Approximation Theory I1 ( G . I. Zapata, ed.) pp. 1 3 9 - 170. Notas de Matemhtica, North-Holland, Amsterdam, 1984.

ill1 M. C. MATOS and L. NACHBIN, E n t i r e f u n c t i o n s on l o c a l l y c o n v e x s p a c e s and c o n v o l u t i o n o p e r a t o r s . Compositio Y a t h . 44 (1981) , 145 I12

- 181.

1 L. RACHBIN, Topology o n S p a c e s der Mathematik, Vol. 4 7 .

[13

1

G.

o f HoZonior>$iii>:4appings. Ergebnisse

Springer, Berlin, 1 9 6 9 .

POLYA, U n t e r s u c h u n g e n u b e r L u c k e n und SCrgklarita1,cn von Potenzr e i h e n . Math. 2. 29 ( 1 9 2 9 ) , 5 4 9 - 6 4 0 .

This Page Intentionally Left Blank

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

173

NORMAL SOLVABILITY IN DUALS OF LF-SPACES

Reinhard Mennicken and Manfred Moller Universitat Regensburg NWF I - Mathematik Universitatsstr. 3 1 D-8400 Regensburg, West Germany

0. INTRODUCTION Hormander [ 3 1, [ 4 ] proved that linear partial differential operators (LPDOs) P = P ( D ) with constant c o e f f i c i e n t s a r e s u r j e c t i v e in D ' ( R ) if and only if the open set R C lRn is strongly P -convex. This property means that for each compact set K C R there is acCmpact set K' C R such that the formal adjoint differential operator P' = P f - D l fulfills the following Froperties:

(0.2)

v u E

E'(RJ

( s i n g supp P ' u

C

K

* sing supp

u C

K').

The condition (0.1) is a preserving property for the support and states the P-convexity on R. The s i n g u Z a r i t y c o n d i t i o n ( 0 . 2 ) is a preserving property for the singular support. This property naturally cannot be formulated within the smooth space V I R ) . In the proof of his surjectivity theorem, Hormander makes essential use of the fact that VlR) is a strict inductive limit of Frgchet-Schwartz spaces. S4owikowski [161 established an abstract analog of Hormander's surjectivity theorem. He considered continuous linear operators 1 : X + Y in LF-spaces and stated a characterization of the surjectivity of the adjoint T'. A disadvantage of his theorem is that his s i n g u Z a r i t y c o n d i t i o n , i.e. the abstract analog of the preserving property ( 0 . 2 1 , is rather complicated since it has to be fulfilled for (uncountable) bases of continuous seminorms onthe LF-spaces under study. In a more general context, Palamodov [ 9 ] initiated a homological characterization of the surjectivity of linear operators. A s an application of his theory, he redemonstrated Hormander's surjectivity theorem, but only by a rather comprehensive additional argumentation.

MENN I CKEN AND MOLLER

174

Based on Palamodov's homological r e s u l t s i n

[ 9

I , Retah

1141

proved a f u n c t i o n a l a n a l y t i c c h a r a c t e r i z a t i o n of subspaces of LF-spaces t o be well-located.

I n [12] P t s k and Retah used t h i s c h a r a c t e r i z a t i o n

t o improve S3owikowski's s u r j e c t i v i t y theorem c o n c e r n i n g t h e a d j o i n t s

of c o n t i n u o u s l i n e a r o p e r a t o r s i n LF-spaces.

Indeed,

singu-

their

Z a r i t y c o n d i t i o n i s , t o some e x t e n t , e a s i e r t h a n t h a t of Ssowikowski b e c a u s e i t c o n s i s t s of a r e q u i r e m e n t f o r o n l y one continuousseminorm on

X

and

Y , r e s p e c t i v e l y . However, i t seems t o u s t o b e

less

p r o p r i a t e f o r a p p l i c a t i o n s t o LPDOs s i n c e it i s an i n c l u s i o n

aprela-

t i o n s h i p of some a r t i f i c i a l a u x i l i a r y o p e r a t o r s . The r e s u l t s of [12] are p a r t i a l l y d e m o n s t r a t e d i n [ 131

.

I n t h i s p a p e r w e a r e concerned w i t h c o n t i n u o u s l i n e a r o p e r a t o r s i n LF-spaces and w e s t a t e s u f f i c i e n t c o n d i t i o n s which a s s u r e t h a t t h e a d j o i n t s a r e normalZy s o Z v a b l e . When i n p a r t i c u l a r t h e o p e r a t o r s

in

t h e LF-spaces a r e i n j e c t i v e , o u r c o n d i t i o n s g u a r a n t e e t h e surjectivity of t h e a d j o i n t s . Thus o u r s t a t e m e n t s on normal s o l v a b i l i t y are

gen-

e r a l i z a t i o n s and improvements of t h e above q u o t e d s u r j e c t i v i t y t h e o -

rems of Ssowikowski, P t s k and Retah. Our s i n g u l a r i t y

condition

s i m i l a r t o t h a t of Saowikowski, t h e r e f o r e more a p p r o p r i a t e

for

is ap-

p l i c a t i o n s t h a n t h a t of P t s k and R e t a h , b u t i n a d d i t i o n it i s easier t h a n t h a t of Saowikowski, b e c a u s e it i s a r e q u i r e m e n t f o r c o n t i n u o u s seminorm on

X

and

only

one

Y r e s p e c t i v e l y . The p r o o f s of o u r re-

s u l t s a r e p u r e Z y f u n c t i o n a l a n a l y t i c and d o n o t make u s e o f any m o l o g i c a l methods; c f . a l s o [ 8 ]

.

ho-

W e a p p l y o u r a b s t r a c t r e s u l t s o n o p e r a t o r s i n LF-spaces toLPECs P having v a r i a b l e c o e f f i c i e n t s and a c t i n g i n D'(Q). W e set X = Y = V(R), T = P' and o b t a i n s u f f i c i e n t c o n d i t i o n s which a s s u r e t h a t t h e LPDOs under s t u d y a r e normally s o l v a b l e

o r even s u r j e c t i v e .

c o n d i t i o n s c o n s i s t of openness p r o p e r t i e s , i . e . t h e V'-P-convexity

condition

and a s i n g u l a r i t y c o n d i t i o n of t h e form

a priori

These estimates,

175

NORMAL S O L V A B I L I T Y

Here

(an);

s u b s e t s of

i s a c o v e r i n g of R by nonempty o p e n r e l a t i v e l y compact R w i t h fin C f i n + I J P ' i s t h e a d j o i n t o p e r a t o r o f P w i t h

(a),Cz(R)),

respect t o t h e dual p a i r (V

i s t h e r e s t r i c t i o n of

Pi

P'

d e f i n e d by t h e g r a p h

and

-

Pi

i s t h e c l o s u r e of

Lz(a,i

in

Pi

x

LZiEli.

i s a n LPDO w i t h c o n s t a n t c o e f f i c i e n t s , t'ne p r o p e r t i e s (0.31, ( 0 . 4 ) a r e e q u i v a l e n t t o t h e " c o n v e x i t y " c o n d i t i o n s ( 0 . 1 ) and (0.2) r e s p e c t i v e l y so t h a t i n t h i s c a s e o u r r e s u l t s l e a d t o Hormander's s u r j e c t i v i t y t h e o r e m . F o r LPDOs w i t h v a r i a b l e c o e f f i c i e n t s we s t a t e If

P

s u f f i c i e n t c o n d i t i o n s of g e o m e t r i c t y p e f o r t h e 9 ' - P - c o n v e x i t y e l l i p t i c LPDOs, t h e P l i g o p e r a t o r [ 1 0 1 t h el ess i n t e r e s t i n g

"

,[ 161,

cf. also [ 6 I

t h e " s i n g u l a r P-convexity",

,

[ 171

.

In

and

particular

and t h e v e r y e a s y , b u t never-

r o t a t ion " - o p e r a t o r p = x

z -a xa-l

x

I

-

a axc,

are treated.

1. PRELIMINARIES

families

( E , a ) i s a l o c a l l y convex s p a c e , i t s t o p o l o g y i s d e f i n e d by ra o f c o n t i n u o u s seminorms which w e c a l l b a s e s f o r t h e t o -

pology

a . I'g

If

(E,aJ. For

p

d e n o t e s t h e system E

rn we

set

K

P

of

:= { x

all E

E

continuous : p ( x )

5

seminorms

M i s a s u b s p a c e o f E and d e f i n e a c o n t i n u o u s seminorm dist Ix,M) on ( E , a ) by closed p-unit b a l l i n

E.

If

on

and c a l l i t t h e

13

p E

F a ,

we

P

E p r o v i d e d w i t h t h e s e t o f a l l t h e s e seminorms, b r i e f l y

(E,distr

(

,M)),

a i s a l o c a l l y convex s p a c e which i s n o n - s e p a r a t e d

if

M

# {O}.The s e t

MENN I CKEN AND M ~ L L E R

176

is a basis for the neighborhoods of 0 in ( E , d i s t r

(

,MI).

a

Let I be an arbitrary nonvoid set. The locally

convex

space for

( E , a ) is the inductive limit of the locally convex spaces (Ei,ail

i

E

I, briefly

iff E = span

(1.2)

U

Ei

i€ I and a is the finest locally convex topology on E such that the inis coarser than ai for all i € I,cf. duced topology a E . := a Ei 2 e.g. [ 5 1 , p. 157 ff.

I

If, for i E I, (Ei,ail are locally convex spaces and the vector space E is given by (1.2) then there is a locally convex t o p l o gy a on E such that (E,aI is the inductive limit of the spaces (Ei,ai). (1.3)

( E , a ) = Z $ n ( E i , a i ) , ( F , B ) is an a r b i t r a r y l o c a l l y convex T : E F i s a l i n e a r mapping on E , t h e n T i s (a,B)-con-

I f

s p a c e and

tinuous i f f

TI^^

-+

i s (ai,$)-continuous

for each

i

E I.

The inductive limit ( E , a ) = Z $ n ( E i , a i ) is said to be strict iff = an for all n E n. En En+l and a ~ = n an or an+l I F n

m,

I =

The strict inductive limit is called an LF-space if tnespaces ( E , a n ) n are Frgchet spaces. If

I = (1,2}

we use the notations

and state (1.4)

The s e t

+ K

{ElK Pl

b a s i s o f the neighborhoods o f

a1

A

a2

on

El

ral

: (pl,p21

ra2,

E

> 01

is

a

p2 0

for

the

inductive

Limit

topology

+ E2.

The space ( E I J a l I A f E 2 , a 2 I is isomorphic to a quotient s?ace of the topological product ( E I , a l ) x ( E 2 , a 2 ) , cf. e.g. [ 5 1 , p. 174,

177

NORMAL S O L V A B I L I T Y

whence w e c o n c l u d e : I f (EI,all,

( E 2 , a 2 1 a r e F r g c h e t s p a c e s and the topology ( E 2 , u 2) i s a F r g c h e t s p a c e t o o . i s separated t h e n ( E I J a I I

(1.5)

El

A

a2

T h e f o l l o w i n g f o r m u l a , t h e p r o o f of which i s i m m e d i a t e , w i l l b e

applied several t i m e s : L e t C

E, A

C

and

C

C

(1.6)

E

be a v e c t o r s p a c e , f u r t h e r m o r e A , B ,

C

a v e c t o r s p a c e . Then (A

+

BI n

c =

A

+

(B

n CI.

,

(F, 6 ) a r e l o c a l l y convex s p a c e s and S : E F i s a l i n e a r r e l a t i o n . W e d e n o t e i t s domain by D ( S ) , i t s g r a p h by G l S ) , i t s r a n g e by R ( S I and i t s n u l l s p a c e o r k e r n e l by N(S). The a d j o i n t l i n e a r r e l a t i o n 5'' : F' + E ' i s d e f i n e d by G(S') = I n t h e f o l l o w i n g (E,aI

+

G (-SI

1 (ExF,F'xE')

If

A, B C E

Let

ru

and

and

rg

= B

B + N(S)

then obviously

be b a s e s on ( E , u ) o r ( F , B )

c a l l e d open i f f f o r each

p

E

U

t h e r e is a

q E

respectively. S is

rg

and

P > 0 such

that

The c l o s e d l i n e a r r e l a t i o n

is c a l l e d t h e c l o s u r e of S ' = S', w e d e d u c e :

-

-

S : E + F i s d e f i n e d by G ( S ) : = G(S1 and S. From [ 6 1 (1.11)' (1.121, ( 2 . 1 ) and (2.61,

using

(1.9)

If

For

S

i s open then

A C E

-

S

i s open and

N(sI

= N(S7.

the inclusion

h o l d s , t h e p r o o f of which i s i m m e d i a t e . F i n a l l y , w e a s s e r t

(1.11) L e t

S

denote t h e

a1

b e a Z o c a Z l y c o n v e x t o p o l o g y on al x f 3 - c l o s u r e of S . Assume t h a t

a l C u. Let (uI,BI-open and

E

uith

S

is

MENN I CKEN AND MOLLER

178

Let

PROOF.

r = {(p,p,)

t h e r e i s some

q E I'g

n

R(SI

K~

rz

E

x

ro

such t h a t

c S(K

Pl

: pl

"1

5 p } . For each ( p , p l )

E

r

~ ( 3n) x c S ( K 1.

),

P

4

Using ( 1 . 9 ) and t h e f o r m u l a s ( 1 . 7 ) and ( 1 . 1 0 ) w e c o n c l u d e

T h i s proves (1.11) b e c a u s e of pology

"

x

r

(1.4) since

i s basis f o r

the

to-

.

2 . NORMAL SOLVABILITY I N DUALS OF LF-SPACES Throughout t h i s s e c t i o n w e assume t h a t (X, T ) = Z i m ( X n , -+

(Y,al = Zirn(Yn,an) a r e LF-spaces and t h a t

T

i.e. D(T) = X .

E L(X,Y),

i s a c o n t i n u o u s l i n e a r o p e r a t o r on X t o Y w i t h . (X T . ) -, ( Y n , a n l by the restrictions T i,n * i' 3 G(Ti,nI/ T

i, n

i s a c l o s e d l i n e a r o p e r a t o r from

(2.1) for arbitrary

(2.1')

:= G ( T ) n (Xi

9. (A) J>n A C X.

For

= TIA. n

A = X

R(T$,,)

x.) 3

U,).

x

X

to

i

Yn

with

n yn

t h i s formula y i e l d s

= ~ ( x . 1n Y 3

.

Tn)

that

and

T

W e define

179

NORMAL S O L V A B I L I T Y

(2.2)

R(T) is s e q u e n t i a l l y c l o s e d iff t h e f o l l o w i n g

LEMMA.

proper-

t i e s are f u l f i l l e d :

Suppose t h a t ( 2 . 3 ) and ( 2 . 4 )

PROOF.

sequence i n

R ( T ) converging t o a

3

n

find a

k

2

cf. [ 6 ]

,

some

j

Y.

:

v

E lN) i s bounded i n

E

nV}

{y,

such t h a t

y E R(T

Conversely, l e t

j

2

n

i)

is

n

an

E

E

fl

choose

C

33

RIT).

C

R ( T ) be s e q u e n t i a l l y c l o s e d . Fix an

TIX.) n Y n

such t h a t

3

of t h e F r 6 c h e t s p a c e I R ( T I n Y n , o n l

i s a non

n

meagre

E

W . Since

subspace

from which t h e e q u a l i t y

= TIX.) n yn " n

R ( T ) n yn

(2.5)

(Y,o) t h e r e

y

TIX.). According t o ( 2 . 3 ) w e can 3 i s ( ~ ~ , a ~ l - o p eR n( T. k .I is closed,

R ( T ) n Yn

k,

be a

W e have t o show t h a t

c f . [ 51, p. 1 6 1 . By ( 2 . 4 ) w e

Yn,

C

j such t h a t T k,J( 3 . 1 ) ( B ) . W e conclude

which y i e l d s

there is a

E

{yv : w

R ( l ' l . Since such t h a t

y

m

a r e f u l f i l l e d . L e t (yw)o

3

follows. Now w e going t o show t h a t ( T ( X . I n Y n , u n l a barrel i n I T ( X . 1 n Yn,un).

By

3

t h e i n t e r i o r of

B in

B

i s nonempty because

T(X

0

With

B

also

a neighborhood of tion

B

( R ( T 1 n Y,,an).

n

T(X.1 3 0 -

i s b a r r e l l e d : L e t B be w e d e n o t e t h e c l o s u r e B a n , by

- 3

.)

3

Yn =

n

Yn

Since B

U rnE fl

i s absorbing

mB.

i s non meagre i n ( R ( T ) n Y n , u n l . 0

i s a b s o l u t e l y convex which y i e l d s 0. As

B

i s closed i n ( T ( X . I 3

0 E

n Yn,on)

-

i.e. B is the

equa-

MENN ICKEN AND MOLLER

180

holds which shows that B is a neighborhood of 0 in ( T ( X . 1 n Y n , a , l . J

Because of (2.1')

is surjective and thus open by Pt6k's open mapping theorem, cf. e.g. I , (3.1) ( B ) . Applying [ 6 ] , (3.1) ( B ) to the mapping T i , n : ( X ~ , T ~ J * ( Y n , u n l we conclude that R I T j , n ) is on-closed. Hence, by (2.5) an3 [ 6

(2.1') I

In the following we assume in addition that there are Banach spaces and ( B 2 , r ) such that the canonical injections

(BI,q)

-

are continuous. ( B Z J r ) is supposed to be reflexive. T closure of -

Tk,k

in the product space ( --4 Xk,q)

x

where X ; f , is the closure of x k in f B 1 , q ) and Y n in I B 2 , r ) , We set q o := q l x and r o : = P l y

k,

([Yz,r)

Pz

denotes the (YkJukl)

the closure

of

.

( 2 . 6 ) THEOREM. A s s u m e t h a t t h e f o l l o w i n g p r o p e r t i e s a r e f u l f i l l e d :

The foregoing theorem is a generalization of S*owikowski's surjectivity statement published in [16]. In S3owikowski's paper l' has

181

NORMAL S O L V A B I L I T Y

to be injective; the conditions filled for systems of norms q ,

B, (iii) and B, (iv) have tobefuland T, forming bases on ( X , T ) or ( Y , o l respectively. These assumptions are obviously stronger than ours, they imply the sequential closedness of R(T), i.e. the conditions B, (i) and B, (ii), cf. lemma ( 2 . 2 ) . In [12], Ptsk and Retah also obtained a surjectivity theorem. They require a different reflexivity property which, to some extent, is stronger than that in the foregoing theorem. In their paper B, is a (iii) is fulfilled a priori since they set q , : = r o T ; q, norm because N(T/ = {Ol. Their singularity condition seems to be a bit artificial as we have already mentioned in the introduction. PROOF OF THEOREM (2.6)

A

we choose

Because

j,

D(T1 = X

n

.

j 2

and

2 j,

= u j 2

E W .

and therefore

T -1 (Y.)

j2j2

B y Baire's theorem there is a

2

n x j2

3

j

According to B, (iv) and

such that

X = T-'(Y)

we have

x

n

We fix an

such that

j,

This leads to X

(2.9)

C

T-'(Y.l

3

j 2

is closed in ( Y , o l . j We fix a k 2 j. According to B, (ii) and B, (iii) there are k > 1 and k 2 2 k l such that

since T is (~,~)-continuousand

k

From (2.lo), (2.1)

and

Xk2 3 Xkl

Y

we deduce that

MENNICKEN A N D MBLLER

182

R(TI n Yk = R(Tk2,kI.

(2.12) According t o ( 2 . 1 1 )

w i t h a suitable

y

I n t e r s e c t i n g b o t h s i d e s by

0.

w i t h t h e a i d of

(2.1).

Hence

t h e r e f o r e a l s o t h e mapping the injection

( ynr’

r)

T T

k2,k

--f

(Xk2,q,)

’

. (B1,q)

k2,k

(Uk,5k)

A

Y T k 2 J k( Kq ,

Kro

R(Tk2,k)

I

+

( Y k , r o ) i s open and

(?&r),

--f

‘

we obtain

Yk

Accordirg to (1.3) conclude

We

(?E,r) i s c o n t i n u o u s .

that

-

Let

T

b e t h e c l o s u r e of

k2,k,n

main s p a c e ( B 1 , q l

T

w i t h r e s p e c t t o t h e do-

k2, k

a n d t h e r a n g e s p a c e (Y:,r)

A

( U k , u k ) . By ( 2 . 1 3 ) and

(1.9) t h e r e l a t i o n

and

N(’T2,k , n I

(2.15)

Since the i n j e c t i o n (Yk,5k)

(Lr,rl n

A

= N(T

(Yk,ak)

k2, k

+

j q

(B2,r)

is continuous

i s s e p a r a t e d and t h u s a F r g c h e t s p a c e by ( 1 . 5 ) .

range theorem for c l o s e d l i n e a r r e l a t i o n s , c f . e.g.

[ 6

1,

The

(Fr r) n’

A

closed

(3.1)

(B)

,

y i e l d s t h a t the range (2.16) The i n c l u s i o n

‘(“k,, C(Tk2,k)

k , nI

c

is closed.

G(Tk

),

t h e c o n t i n u i t y of t h e

in-

2’

-r jection ( y n , ~ iA (Yk’5k) of Xz i n ( B I J q I l e a d t o

+

(Yi,r)

A

(ykZ,ok

and t h e 2

q-closedness

183

NORMAL SOLVABILITY

(2.17) From ( 2 . 7 ) w e o b t a i n

Using ( 1 . 6 ) w e g e t (2.19)

According t o

A w e c h o o s e some

+

(2.20)

The r e l a t i o n s h i p s

(2.17),

The i n j e c t i o n ( X j

1

1

N(TIqO

(2.181,

+ N(TIq,ql

2 k2 C

such t h a t

XI + N ( T I .

( 2 . 1 9 ) and ( 2 . 2 0 ) l e a d t o

A

(Xl,~lI

by ( 1 . 3 ) . Hence, by t h e d e f i n i t i o n of

-+

( B I J q I is

continuous

T

k2, k, n

and ( 2 . 2 1 1 ,

is a closed l i n e a r r e l a t i o n . The domain s p a c e and t h e r a n g e s p a c e i n ( 2 . 2 2 ) a r e F r 6 c h e t spaces a c c o r d i n g t o ( 1 . 5 ) . Now w e a p p l y t h e c l o s e d r a n g e t h e o r e m f o r c l o s e d l i n e a r r e l a t i o n s , c f . e.g.

[ 6

1 , (3.1)

( B ) , and o b t a i n t h a t

i s open. Next w e p r o v e t h a t ( 2 . 2 3 ) h o l d s t r u e w i t h

T

k Z , k , n'

For t h i s p u r p o s e w e make u s e o f

Tk z , k

(1.11). W e s e t

instead

of

MENNICKEN AND MULLER

184

and -

S := T k 2 , k

s = ? k; 2 , k , n (2.23).

S

From the definition of *

is

(al,B)-open by ( 2 . 1 3 )

T k2, k, n

and

'

and ( 2 . 2 1 )

we get

3 is (a,B)-open

by

Thus the assumptions of (1.11) are fulfilled and we conclude

that

Here we made use of the relationship

which is immediate from ( 1 . 3 ) .

Now we show that the canonical inclusion

is continuous. According to (1.1) we have to prove that for every there is a p 2 E I7 and an E > 0 such that 0

pl

E

rCr

NORMAL

185

SOLVABILITY

W i t h o u t l o s s o f g e n e r a l i t y w e assume

K

C Kr

p2

i s e a s i l y c h e c k e d . I n t e r s e c t i n g b o t h s i d e s by

w i t h t h e a i d of

(1.6).

.

The i n c l u s i o n

Yk

we obtain

0

T h i s i n c l u s i o n , t o g e t h e r w i t h (2.12) and (2.28),

yields

D ( T k Z J k ) C Xl

From ( 2 . 8 1 ,

implies t h a t

(2.12),

whence ( 2 . 2 6 )

(2.9)

and (2.27) w e f o l l o w t h e r e l a t i o n s h i p s

i s proved w i t h

E

=

ri 2Y

*

W e h a v e p r o v e d t h a t . t h e p r o p e r t y ( V I ) of Theorem (1) i n o u r pa-

per [ 7 ]

is r e f l e x i v e , c f . e . g . Theorem (1) i n [ 7 ]

[ 5 ]

,

.

-

R := R ( T ) and r RIT) is (?i,r) i s r e f l e x i v e s i n c e ( B 2 , r )

i s f u l f i l l e d with respect t o

s e q u e n t i a l l y c l o s e d b y Lemma ( 2 . 2 )

p . 2 2 9 and p . 2 7 2 . Hence p a r t ( i i )

yields that

R ( T ) i s well-located,

i.e.

of

186

MENN I C K E N AND MOLLER

where

The canonical injection

is continuous by ( 1 . 3 ) . Since R ( T I is well-located the adjoint operator i ‘ maps Y ’ into ( R ( T I , U l ’ and is surjective by Hahn-Banach‘s theorem. Let

To

be defined by the following diagram

We show that P o is continuous. Fix a k E LV and choose j 2 k according to (2.9) such that T ( X k I C R ( T ) n Y We conclude that the j‘ restriction

is continuous. The canonical injection (R(T)

Yi,o.I

3

c-+

lR(Tl,o

R(T’i)

is continuous whence the composition

is continuous. Since

R ( T I is sequentially closed ( R I T I n Y n , o n )

is a

Frschet

) an LF-space. Thus To space for each n E LV whence ( R ( T ) , D ~ ‘ ~ ) is is a surjective continuous linear operator acting between LF-spaces. According to a theorem of Dieudonn6 and Schwartz [ l ] , p. I 2 T o is

I a7

NORMAL S O L V A B I L I T Y

[ 6

o p e n . By B a n a c h ' s c l o s e d r a n g e t h e o r e m , c f . e . g .

= N ( T o )1

R(TL

(2.29

T' = T' o i '

W e have

.

R ( T ' ) = R ( T L ) since

and t h e r e f o r e

j e c t i v e . Furthermore N ( T o ) = N ( 9 )

i' i i s i n j e c t i v e whence

as

R(9') = N(T)

h o l d s by

I , (3.1), t h e ad-

i s normally s o l v a b l e , i . e .

TA

j o i n t operator

is sur-

1

(2.29).

3 . NORMAL SOLVABILITY O F LPDOS I N

o'(R)

T h r o u g h o u t t h i s s e c t i o n w e s h a l l u s e t h e n o t a t i o n s of H6rmander [ 4

I

and H o r v s t h [ 5 d

Let

a sequence

E

W \

(Rn o

.

I

10)

and

Q

C

D d b e a n open nonempty s e t .

Choose

R

o f nonempty open r e l a t i v e l y compact s u b s e t s of

such t h a t m

-

(3.1)

'n

D(Bn/

Since

1

5

p

J

u

cf. e.g.

[ 5

an.

n=l

i s a F r G c h e t s p a c e of e a c h

i s a n LF-space,

K

'n+1

R =

n E W

the space

1 , p . 165.

d e n o t e s t h e s e t of t e m p e r a t e w e i g h t f u n c t i o n s on

5

m

and

i s d e n o t e d by

W

d

k E K . The norm of t h e w e i g h t e d S o b o l e v s p a c e I , c f . [ 4 1 , p . 36. With p = 2 and

Ip,k

ks(EI = ( 1 +

151 2 s/2

(6

E

IRd,

w e have t h e u s u a l S o b o l e v s p a c e s

I Is)

(H,,

F o r an a r b i t r a r y s u b s e t

A

B'

P =k

C

= ( B2 , k s J ' lRd

(A) = B

12,k

w e set

P >k

n &!(A).

''

s

rn)

.

Let B

PJ k

188

M E N N I C K E N AND MULLER

The f o l l o w i n g c a n o n i c a l i n j e c t i o n s

D ' ( 5 2 ) i s e q u i p p e d w i t h t h e weak t o p o l o g y w i t h

a r e continuous where

If see [ 4 1

p
1

( 3 . 8 ) REMARK.

Theorem (3.7)

tion ( 3 . 5 ) , i.e. by P ' 1,n

3.1.

according t o (3.6)

I

(ii) such

is

that

By ( 1 . 9 )

r e m a i n s v a l i d i f w e weaken t h e

t h e s t r o n g P - c o n v e r x i t y of

condi-

R, by s u b s t i t u t i n g

LPDOS WITH CONSTANT COEFFICIENTS Let

P be a n o n t r i v i a l LPDO w i t h c o n s t a n t c o e f f i c i e n t s :

( 3 . 9 ) PROPOSITION.

For each

1 E i7V

t h e r e is a

Cl > 0

such t h a t

-

Pi

NORMAL SOLVABILITY

E E BZoc b e a f u n d a m e n t a l s o l u t i o n of P’. c f . m,P’ Theorem 3 . 1 . 1 . Then E * P ’ u = u f o r u E Czfnl). If $ E Cz

4

Let

PROOF.

see [ 4 ]

,

1,

then

$ which i s i d e n t i c a l l y J on a n e i g h b o r h o o d of t h e a l g e b r a i c d i f f e r e n c e R,+, - Rl We fix

Theorem 1 . 6 . 5 .

We choose a f u n c t i o n

.

a

J

9 E Czfzl+,j which i s i d e n t i c a l l y

on

a

neighborhood

of

RI.

(3.10) yields

which i m p l i e s

9f(l

- $)E * P’u) = u

W e apply [ 4 ]

,

Theorem 2 . 2 . 5

C Z t R ) and

u

(ii) u E H : ( R I

Then PROOF.

P

:

D’(Rl

and +

Hence

= pu = p f $ E

*P‘ul.

and Theorem 2 . 2 . 6

(3.11) THEOREM (Hormander). L e t there i s a j E W such t h a t (i)

0.

s E

s u p p P’u c

IN. Assume t h a t for e a c h n

an

s i n g s u p p ~ ’ uc

0 (Rl

:= ( H s J I

implies

an

i s surjective.

W e a p p l y Theorem ( 3 . 7 ) and d e f i n e

(Bl,ql

and o b t a i n

E

2Q

s u p p u c s2 j’

implies

sing suppu c

i.

192

R

MENNICKEN AND MULLER

D'-P-convex by assumption (i). P'

is

n, n

( p = 2,

tion ( 3 . 9 )

k = k s l . Thus 62 is strongly P-convex if we show

that ( 3 . 5 ) is fulfilled: Fix assumption. Let

1

is iq,rl-openbyProposi-

2

j + 1

n

and choose j according to the

E W

and

It follows that there is a u E H Z ( S l ) such that P i u = v . The defiand thus nition of the singular support yields s i n g s u p p v C m sing s u p p u C 5 by assumption (ii). We choose a P E C o ( Q j + l ) which j is identically 1 in a neighborhood of R If follows that

an

j'

By ( 3 . 9 ) the operator

is open for

m

6 W.

Thus the operator

Im

: rn is open because of ( 3 . 3 ) and since { I seminorms on V ( 2 n 1 , cf. e.g. I4 1 , p. 45.

P'

E

nV}

is

a

basis of

is injective by ( 3 . 9 ) . Therefore the theorem is proved.

3 . 2 . LPDOS WITH VARIABLE COEFFICIENTS

Let P : D r ( n ) * D'(R) be an LPDO with Cm-coefficients and is an open subset of R. sume that R '

as-

P' is said to fulfil the u n i q u e n e s s of t h e Cauchy p r o b l e m (UCP) i n Q' if for each x E R ' and each closed p-ball 2P ( y l c Q r with x E a K (yl and for each open neighborhood U C R ' of x there is an P open neighborhood V C U of x such that for all u E C i ( R r ) the following is true: if -

P'ulu = 0 and supp u I u c U \ K ( y l

P

then

uIv = 0 .

P' is said to fulfil the u n i q u e n e s s of t h e Cauchy problem for the

193

NORMAL S O L V A B I L I T Y

-

R'

if for each x

and for each K ( y ) C f i r P with x E aK f y ) and for each open neighborhood U C 0' of x there P is an open neighborhood V C U of x such that for all u E -%'(fit) the following is true: singuZ.arities

(UCPS) i n

Assume t h a t

(3.12) PROPOSITION.

ulRr = 0

R'

is c o n n e c t e d a n d t h a t

R'

for a l l

P' f u Z -

R'.

fiZs t h e p r o p e r t y UCP i n Then

E 0'

suppu

or

3

u E Cm(fi/

with

P'uIfi,

= 0. m

PROOF.

9 {@,R'}. Choose

Fix some -

K

R'

Since

ZP'

u E

C

(2')

R'

C

(a) with

P ' u l f i r = 0. Assume s u p p u n 0' =: A

is connected A has a boundary point x' in f i r . and y E f R ' \ A ) n K p r ( x r l . Since dist(y,Al =

d i s t f y , A n KZprfx')l there is an

= d i s t ( y , A ) =: p

-

> 0. Let

x

E A

n K Z p r ( x ' ) such that

V := KZp,fxr). Choose

9

Iy - x i

such

E C:(Q')

-

that 9 is identically 1 on U. Then P'IpullU = P ' u I u = 0 and s u p p 9 u I C U \ K p (y) . By the property UCP there is a neighborhood

of x such that

V C U

( 3 . 1 3 ) PROPOSITION.

uI

=

= 0. This contradicts x E A

pull,

Assume t h a t

C

supp u.

-

R'

is c o n n e c t e d and t h a t m

f i l s t h e p r o p e r t y UCPS i n a'. T h e n u i Q I E C f R ' ) f o r aZZ u E V'(R) w i t h P ' u l R I E C m ( R ' l .

or

P' fd-

singsuppu 3 f i '

The proof is similar to the foregoing one and thus omitted. Let B be a subset of Q which is closed with

R. Let

induced topology on We set (PI

@

: = (Q' E Q, :

R'

is unbounded or

and

-

B R := B

V

lJ

a'

R' E Q 2 where

-

a'

respect

to

the

be the set of all components of R \ B .

denotes the closure of

R'

in

IRa .

-

R' P a},

MENNI CKEN AND MOLLER

194

w i t h r e s p e c t t o t h e inducedto-

is a subset of R which is closed

A

Q. F u r t h e r m o r e , i f B i s a compact s u b s e t of

pology on

51 s o is BR.

W e observe t h a t

el

(3.14)

i s t h e s e t of t h e components of

Assume t h a t

(3.15) LEMMA.

gS2 is

that

Then R

f u ~ f i l l st h e p r o p e r t y UCP i n Q \ B and

P'

com-

il.

n

c;(EQ).

NIP') c

is D'-P-convex and

F i x some

s e t K of

.

t h e u n i o n o f a ZocaZly f i n i t e f a m i Zy o f d i s j o i n t

pact subsets of

PROOF.

(62 \ B ) ;

A t f i r s t w e prove:

6 IN.

There i s a compact sub-

Q such t h a t

(3.16)

Let

B b e a l o c a l l y f i n i t e f a m i l y of d i s j o i n t compact s u b s e t s of

Q

such t h a t

-

BQ

Since B

-

u

B k := (B'

is l o c a l l y f i n i t e t h e s e t

f i n i t e f o r each

k

E

.--

Zn

j

E 8 : 3'

n S k # 01

is

Thus

IN.

Bn

i s compact. Choose a

B'.

B ' E €3

E

Bn =

A'

B'

such t h a t B'

U

B'E

u B'E Bn

'

=

B \ Bn

B n C il

(gR \ R . )

'

U

i-

Then U

B'

B'E B.\Bn 3

which p r o v e s t h a t \Bn i s c l o s e d w i t h r e s p e c t t o Q. T h e s e t KO:= i s compact and K O n (En \ B n l = 0. Hence t h e r e i s an E > 0 f i n U Bn such t h a t

W e define

of

R

\

li,

(zn

U

:= K O + K E ( O ) . B l . W e set

Let

Qo

d e n o t e t h e s e t of a l l c m p o n e n t s

NORMAL SOLVAB I L I TY

@ 032

@

Q,

033

-

:= {R'

E

a0

:= { a '

E

@o,2 : 52'

:= I n ' E

0,4

R'

:

:

Q,o,3

C

R'

195

R

and

R'

n

5in # 81,

e

UE)

i s bounded},

and assert t h a t t h e s e t (3.18)

Q,

TO p r o v e t h i s l e t

is finite.

0,4

e 0 , 4.

Rf E

R' n

Thus

UE

# n ' . From

E @

0,3

we infer

R' n

Hence

UE

#0.

The d e f i n i t i o n of

x

fl' t

E

and

UE

KE(xJ

T h i s and

R'

Since

i s . c o n n e c t e d t h e r e i s an x

C

(KO + K , E ( O l )

C

R \ (En

QO

U

all

E

n a'.

En)

\KO C

Q \ (B

U

In).

implies t h a t

whence ( 3 . 1 8 ) i s p r o v e d s i n c e KZE(O/

E

(3.17) y i e l d .

d o e s n o t depend on

E

R'

and

KO

+

i s bounded.

By ( 3 . 1 8 ) t h e r e i s a

u

(3.19)

k

2 n

such t h a t

R f C U E U

QfE

033

u

"'@0,4

R' c E k .

W e assert t h a t @

(3.20)

For t h i s l e t t h e set

R'

R'

E

0,2

\

\Q,

0,3

.

C O 2 .

Since

i s closed with respect t o

R' R\

i s a component of R

(En

U

B l , i.e.

(3 u B )

MENN I CKEN AND MULLER

196

\ B ) because of " @0,3 O2 follows from R' E @ o , 2 . We set

proved. R'

K

'

-

a ' = R' n /R

We infer

and

is

52' E @

is a compact set and

by (3.19) and (3.20). Hence (3.16) is proved. Let

8 ' = { B ' E 8 : B ' n K # 01. As above we conclude:

is compact, \ L is closed with respect to R and there bounded neighborhood U of K U L such that U C R \ (En \ L ) . some j E W such that U C We assert that j-

Let

v

E RIP')

n C:(En)

v . We fix a component

R'

is a Fix

and fix some u E C z ( R ) such that of R \ ( E n U B ) ; , i.e. R' E a0 \ 0

P'u=

0,2

by

( 3 . 1 4 ) . We infer s u p p u 2 R' since s u p p u isa compact subset of R. By assumption P' fulfil.1~the property UCP in R \ B and hence also in the set R' C ,r2 \ U B l . Thus we obtain ulR, = 0 from P ' u l R l = 0 with the aid of Proposition (3.12). This proves

(zn

Let

ip is identically 1 in a neighborhood In V we have u = q u and thus P ' u = P ' ( i p o u / . From (3.22) and the definition of U we infer

V

of

K

ip E

U

L.

C z ( U ) such that

197

NORMAL S O L V A B I L I T Y

Ipu = 0 and thus P ' ( 9 u l = 0 in R \ (K U L l . This proves P'u = P ' ( I p u l since s u p p P ' u C B,,,C K. Finally $7 E C z ( U l C C Z ( 2 . ) 3 imp1ies

which yields

2,

For N(P'I

C

= P'u = P ' ( l p u l

E P'tc;tn.,i.

3

the assertion (3.22)

u E NIP')

K = g.

holds with

Hence

C;(BRl.

Let

(3.23) LEMMA.

K

b e a compact s u b s e t o f

R

f i l s t h e p r o p e r t y UCPS i n

\ K. T h e n

P'

R. Assume t h a t

R i s singularly

ful-

P-convex with

respect t o (q,r).

PROOF. Fix some W such that

n

(an U

holds for all -

V

1

E

(an u K); -W . - Since . We prove

3

v

j. Let

(Be

E

PZJk2

-

= P'U = P ' u

is compact there is a j e that

K I R C "-I

for some

(sn)+

( a l ) and

u E Be

Pl' kl in (3.22) we obtain by using (3.13) that 1

-

.

Choose

borhood of

-

Cz(?i.l

Ip E

3

such that

(3.24) PROPOSITION. R.

n

R(P;I.

singsuppP'u

C

(Sin

Hence

an.

As

K); C is identically I in a neighsingsuppu

C

U

Then

Let

B

f u l f i l s t h e p r o p e r t y UCP i n UCPS i n

lro

c;(R))

Then

dimN(P'1

b e a compact s u b s e t o f

0 \B

and t h a t

-

P'

R. Assume that P'

f u l f i l s the property

m.

PROOF. Fix some 1 E W such that 8, C El. Let u E c % ' ( R ) and P ' u = 0. Proposition (3.13) implies that s i n g supp u = 0. Hence, by (3.15), N ( P ' ) C C:(sl) and N l P ' l = N I P ; ) is a closed subspace of DiGl) and

I I

1 . Both topologies coincide on I V ( P ' ) by (3.3) P l 4 and the open mapping theorem, cf. e.g. [ 6 I , (3.1) ( B ) . V ( c l ) is a Schwartz space, cf. e.g. [ 5 1 , p. 282. This shows that N i p ' ) is a normed Schwartz space. By the definition of Schwartz spaces its unit

of ( B e (Ql1, P l 4

MENN I CKEN AND MOLLER

198

ball is precompact. Hence of F. Riesz, cf. e.g. [ 5 1 (3.25) PROPOSITION. L e t

N(P’) is finite dimensional by a theorem

,

p. 147.

n, s , s ‘ , t

s > s’.

IN,

E

N ( P ’ ) n ~ z ( ~c i cZ(3,) ~ i

(3.26)

-

PROOF.

Assume

Let

P;

be the closure of

P’

n,n -

-

in (H:(3n)J

I

ls)X(H;(Zn),

I It).

PA is a restriction of the mapping PI : D ’ ( Q ) --* D’(i-2) and thus operator. Since I 5 I I s the estimation (3.27) implies

an

Is,

prove that

It)

IIUII,

D, :=

(D(T),

11 I1 ,l

:= luls + l’;ult is complete as

(u E

-

PA

is

open.

D f F p .

is closed. The identity

from D I onto D with the T-norm is bounded because of From (3.27’) we conclude that

IIUII, 5 ZC( The canonical injection the embedding (H:

(En), 1

map

I Is, 5 I Is.

199

NORMAL SOLVABILITY

,

Th. 2 . 2 . 3 ,

whence t h e i d e n t i t y map from

D1

pact. Therefore

R ( P h 1 i s closed i n (HZ(EnIJ I

Itl

14 ]

above c i t e d theorem. (3.1) ( B ) , y i e l d s Let

there is a

y

C

the [ 6

1 ,

It).

(3.28)

that

we obtain

G(PA).

(3.26) i m p l i e s t h a t

which p r o v e s t h e a s s e r t i o n of ( 3 . 2 9 ) THEOREM.

com-

such t h a t

0

GI,?;,,)

to

(3.28).

I n t e r s e c t i n g b o t h s i d e s by

because

is

X

Banach's c l o s e d r a n g e theorem, c f . e . g .

d e n o t e t h e u n i t b a l l i n (HZ1En), I

Kt

into

according

Let

(3.25).

P b e a n e l z i p t i c LPDO o n R , L e t

B

b e a cZosed

s u b s e t o f R s u c h t h a t E n is t h e union of a ZocaZZy f i n i t e f a m i Z y o f d i s j o i n t c o m p a c t s u b s e t s of R . Assume t h a t t h e c o e f f i c i e n t s o f P are anaZytic f u n c t i o n s in

R

\ B.

* D ' ( R I i s normaZZy s o Z v a b Z e . I f P i s s u r j e c t i v e i f B i s empty.

We a s s e r t : T h e o p e r a t o r B

is c o m p a c t t h e n

P : P'(il)

c o d i m R(PI
0

of a l l

tained i n

s u c h t h a t t h e b a l l of r a d i u s r c e n t e r e d a t x i s conU and f i s bounded on i t . W e h a v e (see 1141 8 7 , propo-

sition 2):

If E and F a r e Banach s p a c e s and n E I N , then there e x i s t s P E PclnE;FI a u n i q u e e x t e n s i o n P” E P c , ( n E 1 f ; F 1 . The operat o r d e f i n e d b y T n P := P” h a s t h e f o l l o w i n g p r o p e r t i e s :

LEMMA 1.

for e v e r y

= 1

(1)

llTnll

(2)

T~

(3)

f o r every

f o r a12

n E fl;

i s an isomorphism from 2,

m

E .TJ,

k

i k i T m P l l y ) = T k d-k P l y )

5

P ~ ( ~ E ; F oI n t o

m

and

P

for all

P ~ , ( ~ E ~ ~ ; F I ;

E

P~PE;FI,

y

E

E.

I t i s enough t o remember t h a t , by d e f i n i t i o n ,

PROOF.

P ~ ( ~ E ; F:=I P

(

f

n 8~ F , ~

P ~ , I ~ E ~ ~ ,:=F IP * ( n ~ r f ) Q F ,

f

closure i n

P ( ~ E , F ) ~ ,

closure i n

P ( ~ E ~ , , F I ~ ,

a n d u s e [ 1 3 ], Lemma 1 and Remark 2 . LEMMA 2 .

I f

E i s a normed s p a c e t h e n t h e r e i s a n u n i q u e

such t h a t

PROOF.

F o r (1) and ( 2 ) , see 1131

,

Corollary 5.

isomorphisrn

MORAES

208

(3) For each rn E Bv and P E P u u ( r n E l we have by the first part of this lemma that TrnP E Pw*u(m.Crr) C H I E " / . S o , for each y E E there exists a neighbourhood U y C E" where the Taylor series expansion of PmP at y converges uniformly to TrnP and so we have rn T , P ( ~ ~ ~=J 2 (I /~!);~TC(T~P) ( y ) ( x ~-" y )

for every

xlr E

u

k=O

Y

C E".

By (1) we have T r n P ( x ) = P ( x l for every x E U n E, where U n E is Y Y a neighbourhood of y .in E . So, there exists a neighbourhood V of Y y in E such that rn

Z (1/k!)2kp(yl(x - y/

P(x) =

k=O

for every

x

E

V

Y,

and we have for every

5 m:

k

( l / k ! ) $ ( T r n P ) ( y ) ( x - yl = ( l / k ! ) d k P ( y ) ( z - y )

(*)

for every

x

E

V

Y

As ( 1 / k ! ) z k ( T m P I I y ) and ( 1 / k l l ~ k P ( y l are holomorphic functions El

(*)

. on

implies

(I / k!)ak(5!'mP) l y l = ( 1 / k ! ) z k P ( y ) Now, if we prove that

the bounded subsets of

Zk ( Tk ~! P (Y) )

is

0( E ,

E

1 )

on

E.

-uniformly continuous on

E , we have

k!

k!

Let B be a bounded subset of E" . Without loss of generality, we can suppose B a balanced set (take the balanced hull, if necessary). We known that ( T r n P ) / i y + B I is o ( E f l ,El)-uniformly continuous and so, given E > 0 there exists 6 > 0 and ( p 1 , . . . , ( p P E E' such that

such that I(pi(x - z / I < 6 for i = I,. . . , p . Now, if we take x, z E B such that l ' p i l x - z l I < 6 we have y + Ax, y + Xz E y + B for all X such that ( AI = 1 (as B is balanced) and l i P i ( h i - Az) I = I V i ( s - 2) 1 < 6

A

and non-empty

L e t E and F and .f E f f , ( U ; F I

and a unique

7E

PROPOSITION 3 .

take

b e Banach s p a c e s . I f U C E is o p e n t h e n t h e r e e x i s t s an o p e n s e t W C Err

such t h a t

Uc.IW;FI

209

U

C

W

and

7/U

= f.

We

may

t o be t h e s e t

W

Furthermore,

t h e r e i s an isomorphism T : ucb(E;F)

such t h a t ( T f ) IE = f PROOF.

Let

for all

f

E

y E U

For each

y

E

pC,i

W

=

Y

rciy;fl

= B

2

f E ffeblE;Fl.

r(

E

a

Pc( kE ; F ) unique

E";FI.

U U and YEU Y

m

T f(x)

H c *b ( E " ; F )

U, let

E

I t i s clear t h a t

for a l l

+

f f c l U ; F ) . W e h a v e by d e f i n i t i o n t h a t d-k f l y ) a n d k E ill. S o , by Lemma 1, t h e r e e x i s t s

k T ~ C ?f l y )

extension

since

THEOREM

HAHN-BANACH EXTENSION

k=O

tre

can d e f i n e

1

( T k ~ !a k f ( y ) ) ( x- yi

rb(y;f)

r. T h e r e f o r e

Tyf

for

is

Tyf

:

x

well

E

Uy

+

u

Y

F

by

.

defined

and

21 0

MORAES

holomorphic on

U

Y

Tyf E He* (U ;PI.

and we have

Y

Now suppose:

is satisfied. Then Tf : W F may be defined by T f / U y :='yf Tf E He*(W;F). Furthermore the mapping T is clearly linear. +

and

Now we are going to show that (A) is satisfied. We define:

Iz E E "

u 3 :=

Y

:

IIZ

-

rb(Y;f)

y II
0 e x i s t s a s e q u e n c e of p o s i t i v e r e a l numbers and

IIfII


1

Hence there exists

and so U is a 0-neighborhood in

D. such that

E.

1.12. PROPOSITION. L e t E b e a b - F - b a r r e l l e d precompact t h e n H i s e q u i c o n t i n u o u s in E ' .

s p a c e s . If

H

5 EA

PROOF. If B E A and E > 0, there exists X E F , 0 < Ihj < m such that there exists P 5 E d , , with c a r d ( P I

H C - C(P) + x E hPo

But if

Hence for

x'

and

E HI

= f + X g and I x ' ( x 1 I < E continuous for each B E A

1.13. PROPOSITION. Ei.

Let

E

and

E

XBO.

then

f E GIPI

there exist

is

f E C I P I , g E Bo

such that

if x E V n B . Thus we obtain HB and then H is equicontinuous. be b-F-barrelzed

and

H

x'

equi-

be precompact

in

Then:

ii)

H

(ii) I f

i s r e l a t i v e l y weakly c-compact.

F i s a ZocaZ f i e l d t h e n

H

i s r e Z a t i v e Z y weakly compact.

265

NONARCHIMEDEAN gDF-SPACES

PROOF.

If H

Eb

is precompact in

r(H)

then H and

(the F-convex

m/'

hull of H ) are equicontinuous. By Van Tie1 [ 8 1 , is weakly is weakly compact whenc-compact, and hence (i) Furthermore,r ( H 1 '

.

ever F is a local field, and hence (ii). 1.14. PROPOSITION.

is a

E

b'-F-space

if and o n l y if

hh

is com-

pZete.

fI

PROOF. If f E E* and B is continuous for every B E A , there exists an F-convex 0-neighborhood U such that f E (U n B I o . Let h E F , Ihl > 1, V = X-'U and D = A - l B , then f

E

XVo + ADo.

Hence f is a cluster point of somecauchy sequence in E A . Then

f E El.

Conversely, let F be a Cauchy filter in E L . If CI E F and x E E, then there exists M E F such that M - M a { x } O . Then tbere is a Cauchy sequence of the form { f n ( x l 1 in F. We define g

ists

€

Let

E' D

l i m f n ( x ) , f E E*.

B E A

and

flg-'(D)

and then

1.15. COROLLARY.

then there ex-

g-'(D)

B

is a O-neigh-

5

D. Hence f B is continuous for f E E ' , because E; is a b'-F- space. fl

B)

F. Then

If E is c o m p l e t e t h e n ( E ' , T ~ ) i s a b ' - F - s p a c e .

1.16. COROLLARY. If F is a l o c a l f i e l d and ( E ' , T ) i s a b'-F-space.

52

B E A

If

g E f + Bo.

be the unit ball in

borhood in B every

f(x) =

such that

E

is

complete

then

NONARCHIMEDEAN gDF-SPACES

Now we give an Nouredinne [ 5 1 and

extension of the classical definition of [ 6 1 for gDF-spaces.

K.

W. Ruess

2.1. DEFINITION. A locally F-convex space gDF-space (n.a. gDF-space) if

E

is a n o n a r c h i m e d e a n

(a)

There exists a fundamental sequence of bounded sets, and

(b)

E

is a b-F-barrelled space.

266

NAVARRO AND SEGUEL

If E i s a n . a .

2.2. PROPOSITION.

complete then the strong dual o f

PROOF.

If

gDF-space B

and

F

is

spherically

i s a n,a. Frgchet space.

is a fundamental sequence of bounded sets then the

{Bn}

polar sets { B Z } form a countable basis of 0-neighborhoods for E L . Now we will prove (a) that every Cauchy sequence in E b is u-converqent and (b) every Cauchy sequence in o-convergent is B-conEb verqent. lxn}

Let

be a Cauchy sequence in

E i . Then the F-convex

hull

of { x n } is an F-convex and precompact subset of E L . Hence by 1.13, H is a-c-compact and 0-closed. Thus Cx,} is weaklyconvergent.

H

Now let be the basis of 0-neighborhoods in 8 ( E ' , E I consisting of the a-F-barrels. If $ is the elementary filter associated to { x n } and V E v , then there exists M E $ such that M - M 5 V . Then, as 4 converges to x , x belongs to -a M 5 y + V for each y E M.

M 5 x + V , and thus $ is convergent to x in

Hence 2.3.

COROLLARY.

gDF-space i s a

b'-F-space.

It is a consequence of 1.14.

PROOF. 2.4.

n.a.

A

Eb.

COROLLARY.

PROOF.

n.a.

A

gDF-space i s a

b-F-space.

It is a consequence of 1.10.

COROLLARY. E i s a n . a . gDF-space i f a n d o n l y if i t s t o p o l o g y i s L o c a l i z e d i n a f u n d a m e n t a l s e q u e n c e of b o u n d e d F - c o n v e x and closed set. 2.5.

2.6.

Every n.a.

PROPOSITION.

DF-space

i s a n.a.

gDF-space.

PROOF. Let E be a n.a. DF-space. We assume that { B n } is an increasing fundamental sequence of bounded F-convex and closed sets. If each

n

W

5E

E ZV,

We define

verifies

W n Bn

is a

0-neighborhood

then there exists a 0-neighborhood

V = n V

n

and obtain

V

5

W.

Vn

in

Bn

such that

for

N O N A R C H I M E D E A N gDF-SPACES

On the other hand, for each such that

aBn

5

Vn

for every i n n ( a ( . But

=

vn

n

CY

v k i n B~

267

n B~

F-convex we obtain

I n

vk)

n

( C X B ~ = )

n 0, n E W , n 2 m such that

and hence If(km)I = 1 for every m E W . Similarly, we obtain f l k ) = 0, this contradiction implies that K is finite. Now as X is W-compact, X must be finite, then (f) implies (9). That (9) implies (a) follow from the invariance under of the property of being c-Monte1 and n.a. gDF-space.

products

Now we will consider the vectorial case. Let {A,} be anincreasing fundamental sequence of F-convex bounded sets of E. For each sequence (Un} of 0-neighborhood of E we denote by r(An n Unl the F-convex hull of

{An n UnIn.

If follows from Garling [ 2 ]

that

the

family

IrfA, n Un)l. where {Un) is a sequence of 0-neighborhoods is a filter basis of F-convex 0-neighborhood for the b-topology of E. Here we call b-topology the finest locally F-convex topology which agrees with the topology of E on the sets An, for every n E iB. 3 . 3 . PROPOSITION. The family { n (Ai + V i l j , where {Vi) is a sequence of F-convex 0-neighborhoods of E , is a filter baseof F-eonvex 0-neighborhoods f o r the b-topoZogy.

PROOF. It is sufficient to show for each sequence {Ui} of F-convex #-neighborhoods of E there exists a sequence {Vi} of F-convex O-neighborhoods of E such that

271

NONARCHIMEDEAN g D F - S P A C E S

We claim that { V . ) verifies the inclusion required. Let x be an element of f' ( A i + V . ) . Then x = a i + v i where a i E A i , vi E Vi.

vi

- u

i-1

, bi = a i

bl = al

If we call

if

bi

bi E A i

then

Ai

and

b.

n Ui

for

i = I,.

E

i > 1

if

and

u 1 = v l J ui =

i > 1 , we obtain that

2

E '

i

i B

x =

But

- ai-1

j=l

UiJ

b j i

since

. .,j

B u j=] i .

and

Ai

also

ui

E

Ui

are F-convex, hence

Ai n Ui

for

i =I

.,... ,j

x E ri(Ai n U i l . The following is an extension of Hollstein's results

[3]

for

the vectorial case. 3.4. PROPOSITION. L e t X b e a n u l t r a r e g u l a r t o p o l o g i c a l s p a c e . T h e n ClX) a n d E a r e n . a . gDF-spaces if and onZy if C(X;Sl is a n.a. gDF-space. PROOF. If C ( X ; E ) is a n.a. gDF-space, then, since C(X) and E can be considered as complemented subspaces of C(X;E) and the n.a. gDF-spaces are invariant under separated quotients, it follows that C(Xl and E are n.a. gDF-spaces. Conversely we assume that C ( X l and E are n.a. gDF-spaces. By Navarro [ 4 I , it is obtained that {M(X,Anl 1 is a fundamental sequence of bounded sets for C ( X ; E l whenever { A n } is a fundamental sequence of bounded sets in E . We can assume each M(X;A I as F-convex and the sequence can be assumed increasing. If A is the sequence

IM(xx;A~)I n

then will show that

T~

is the compact-open

3.3 topology. Let U be a TA-neighborhood of zero. By Proposition in X and a sequence there exists a sequence of compact sets CK,} {V,} of F-convex 0-neighborhood of E such that

By Navarro [ 4 ] , 2.1, it follows that n M(Kn;Vn n

As

+ An)

5

U.

C(Xl is a n-a. gDF-space then X is W-compact, then there exists

NAVARRO A N D SEGUEL

272

a compact s e t K such that

If w e denote gDF-space

V =

n (V + n n

U Kn

n

A I

5

then

K

then V is a 0-neighborhood of then.a.

n

E.

Finally we obtain M ( K , V l the compact open topology for

- U. C

Then U is a 0-neighborhood

in

ClX; E l .

REFERENCES [

1]

N. DE GRANDE - DE KIMPE, c-compactness i n l o c a l l y K - c o n v e x spaces. Indag. Math. 3 3 (1Y71), 176-180.

[

21

D. J. H. GARLING, A g e n e r a z i z e d form of i n d u c t i v e - l i m i t topology for v e c t o r s p a c e s . Proc. London Math. SOC. 14 (1964), 1-28.

[ 3

1 R. HOLLSTEIN,

C (X; E l . Manuscripts Math.

Permanence P r o p e r t i e s of

38 (19821, 41- 58. [ 4 1 S. NAVARRO, N o n a r c h i m e d e a n D F - s p a c e s , [ 5

[

to appear.

1 K. NOUREDDINE, Note sur l e s espaces Db. Math.

Ann.

219 (1976), 97- 103.

6 1 W. RUESS, The s t r i c t topology and D F - s p a c e s . Functional Analysis; Surveys and Recent Results, pp. 105 -118. North-Holland Math. Studies, Vol. 27. North-Holland, Amsterdam, 1977.

[ 7

1 J. SEGUEL, E s p a c i o s gDF-no a r q u i m e d i a n o s . Tesis de Magister en Ciencia, Universidad de Santiago, 1982.

[

8 1 T. A. SPRINGER, Une n o t i o n d e cornpacite dam l a t h e o r i e des e s p a c e s v e c t o r i e l s t o p o Z o g i q u e s . Indag. Math. 27 (1965), 182- 189.

[ 9

1 J. VAN TIEL, E s p a c e s l o c a l m e n t K - c o n u e x e s , I, 11, III. Indag. Math. 27 (19651, 249 -289.

[lo] S. WARNER, The t o p o l o g y

of

compact c o n v e r g e n c e

on

f u n c t i o n s p a c e s . Duke Math. J. 25 (1958), 265

continuous

- 282.

COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

273

PSEUDO-CONVEXITYl u-CONVEXITY AND DOMAINS

OF u-HOLOMORPHY 0. W. Paques and M. C. Zaine

Instituto de Matemstica Universidade Estadual de Campinas 13.100 - Campinas, SP Brazil

By using the concept of uniform holomorphy ibtroduced by L. Nachbin in [ 3 ] , we define domains of u-holomorphy and u-holomorph-

ically convex domains. Relationships between these concepts and those of domain of holomorphy and pseudo-convex domains will be obtained. We would like tothank Professor Mzrio C. Matos and Jorge Mujica for many helpful discussions on the subject of this paper.

1. NOTATIONS AND PRELIMINAIRES

Throughout this paper, E will denote locally convex Hausdorff space over d', c s i E l is the set of all continuous seminorms on E. X i U I will denote the space of all holomorphic functions from U (an open subset of E ) to d'. If a E c s ( E l , will denote by ( E , a I the space F endowed with the topology generated by ci and by E , the normed space associated to ( E , a l . Let ia : E E , denote the canonical surjection. +

DEFINITION 1.

Let U be a non-void open subset of E . U is u n i f o m Z y a E c s l E ) such that U is open in ( E , a i . Let I be

o p e n if there is

the set of such a . We remark that I is a directed subset of c s ( E l and generates the topology of E . We refer to Nachbin [ 3 ] for examples of uniformly open sets. REMARKS. U n V.

(I)

If U and V are uniformly open, then so is

(2)

If U is uniformly open, then so is each connected U.

ponent of

From now on, U will denote a connected uniforly open of

E.

com-

subset

274

PAQUES AND Z A l N E

DEFINITION 2 . A function f E K ( U l is UniformZy h o l o m o r p h i c if there are a 6 I and f, E J C f U , ) such that f = f, o i a ,where Ua = i a ( U I . X u f U l will denote the space of all uniformly holomorphic functions on U. EXAMPLES (1) If E = F A , where F if isa separable Frdchet space, that is, the dual of F endowed with the topology of compact convergence, then every open subset U of E is uniformly open and Ku(Ul = KfUI (see Mujica [ 2 1 ) . E = JC(gI

with the compact-open, topology Nachbin in f ( 9 I = 9 ( 9 ( 0 l l , for q E E, is holomorphic, but not uniformly holomorphic. (2)

If

[ 3 I shows that the function defined by

DEFINITION 3. U is a domain of u-holomorphy if there are no con@ f U 2 C U n Ul nected uniformly open sets U l and U 2 in E , with and U 1 $2 U, and such that for each f E J C u ( U I , there exists g E Xu(Ull with f = g on U 2 s the domain o f u - e x i s t e n c e of a function f E DEFINITION 4 . U J c u ( U I if there are no connected uniformly open sets U l and U g in such that 9 # U 2 C U n U I J Ul g U E l and a function g E X U ( U l , U is a domain of u-existence if U is the doand g = f on U 2 main of u-existence of some function f € J e u t U l .

Clearly every domain of u-existence is adomain of u-holomorphy. For definitions and properties of holomorphically convex dcsnains, polynomially convex domains, pseudo-convex domains, Runge domans, domain of holomorphy and domain of existence, see Noverraz 111. If E = F E , as in example (1), then U is a domain of holomorphy if and only if U is a domain of u-holomorphy. Later on we will see that if E is a nuclear space, then U is a domain of holomorphy if and only if U is a domain of u-holomorphy. PROPOSITION 5. of

f

PROOF.

Let

i f and onZy i f

f

E

U

JCufUl.

Then

U i s t h e domain o f u - e x i s t e n c e

i s t h e domain o f e x i s t e n c e of

To prove the non-trivial implication, suppose

f.

U

is not the

275

DOMAINS OF U-HOLOMORPHY

domain of existence of f. Then there are connected open sets U z and U 2 in E , and g E K ( U l ) , such that U l U, 0 # U 2 C U ill and g = f on U 2 . Without loss of generality we may assume that U 2 is a connected component of U n U l . Take a point a E U I n a U 11 ail, (see Mujica [ 2 ] ) , and choose a E I and r > 0 such that B a ( a ; r l and g is bounded on B " ( a ; r l . Let b E U 2 n B a ( a ; r l , and choose C Ul B E I and s > 0 such that B R ( b ; s l C U 2 n B a ( a ; r l . Then Vl = B B " ( a ; r l and V 2 = B ( b ; s l are uniformly open and g E J C u ( V , l . Since V I I F u, @ # U 2 C U n V 1 and g = f on U 2 , U is not the domain

of u-existence of PROPOSITION 6 . A

f

f-

Let

=

{U

f

E

xu(U) a n d

I; f = f o i

E

u

u'

f,

with

E J€(U,)},

U is the d o m a i n of existence of f if a n d o n l y if Ua is the dom a i n of e x i s t e n c e o f f,, f o r every u E A Then

f'

Firstly suppose that there is u in Af such that U, isnot a domain of existence of fa. Then there are connected open subsets U i and il: in E, and in JCClJil such that

PROOF.

fi

U:

0#

UaJ

2

U, C U, n V i

and

1

f, = f,

We remark that U = i-'(U,) and Ui = i-a ' (U,), U open sets in ( E , u ) . Furthermore, Uz

U,

@

# U2

C

U n U1

fi

where f1 = ia and f = f, o i,. not the domain of existence of f.

and

2

U,

on

.

i = 1 , 2 , are connected

f1 = f

on

U2,

If follows from this, that U is

Conversely, if U is not the domain of existence of f, then there are connected uniformly open subsets U', U 2 in E and f1 in 1 Ku(U l such that U1

U,

0 #

U 2 C U n U1

It is possible to find a in (E,uI, f = f, oi, and JC(UiI.Then

U, = i , ( U l

E

and

and

f = f, on

Af such that U, f1 = f ,1o i , , with

Ud = i , ( U i ) ,

U1 f,

i = 1,2,

U2

and E

KIU,)

. U2

are and

open

fi

E

are connected open

PAQUES A N D ZAINE

sets in

E,

and

f = f, o i a' Then, we can conclude that

where

of

Ua

is not the domain of

existence

f.

DEFINITION 7.

U

is s e q u e n t i a l l y u-ho l o m o r p h i c a l l y convex (Sequentially U , which converges to a point x € all, there exists a function f e J c u ( U / (K(Ul) with s u p I f ( x n l I = + m . h o l o m o r p h i c a l l y c o n v e x ) if for each sequence ( x n ) in

It is clear that if U is sequentially u-holomorphically convex, then U is sequentially holomorphically convex. i s s e q u m t i a l l y hoZomorphically PROPOSITION 8. I f for some a E I , U u convex, then U i s s e q u e n t i a l l y u-holomorphically convex.

PROOF. Let ( xn ) be a sequence in U which converges to a point xo in a l l . Then ( i a ( x n l i is a sequence in U,, converging to i a ( x o i in aU,. Therefore there exists f, in WU,) such that supIf,,oi,(x,)l =+m. If we take f = f, o i , , we have that U is sequentially u-holomorphically convex. PROPOSITION 9. If U i s s e q u e n t i a l l y u - h o l o m o r p h i c a l l y U i s a domain of u-hoZomorphy.

convex,

then

PROOF. Suppose that there are connected uniformly open sets U l and U 2 in E, with U l U, @ # U 2 C U n U l , and such that to each f E . K u t U l there corresponds a function f l in J c u ( U I ) , with fl = f on U2. Without l o s s of generality we may assume that U 2 is a connected component of U n Ul. Then we can find a sequence ( x n l in U 2 which converges to a point x in aU n U l n aU2 (see Mujica [ 2 1 ) . By hypothesis there exists f 3 C u ( U I with s u p l f ( x n l I = + m . Then, on one hand the sequence I f l ( x n ) l converges to f l l x o / and on the other hand ( f l f x n ) ) = ( f ( x n ) ) is unbounded. This is impossible.

e

Let K be a compact subset of

U. We will denote by:

277

DOMAINS OF u-HOLOMORPHY

is u-hoZomorphically c o n v e x if for every U there is a 0-neighborhood V in E such

DEFINITION 10. U pact subset K of

comthat

ii+vcu. a i n I , s u c h t h a t Uu U i s u-holomorphically convex.

I f there i s

PROPOSITION 11.

p h i c a l l y convex, then

is

holornor-

Let ir' be a compact subset of U. Then i u ( K 1 = K u pact in Uu. By hypothesis, there is an open ball B , ( 0 , 6 / such that

is comin E,,

PROOF.

Then Since

( z a ) + B ' ( O , 6 1 U , where B' denotes the open ball in (E,a/. 2 C i a l ( z u ) , we have the result.

-1

i,

C

PROPOSITION 12. If U i s a d o m a i n 07 u - h o l o m o r p h y , l o m o r p h i c a l ly c o n v e x .

then

U i s u-ha-

The proof of this proposition is similar to the proof of rem 3.5 in Noverraz [ 5 ] .

U

DEFINITION 13.

mials on

El

tipology

T

is u-Runge if the space of the continuous pOlyn0-

P ( E J , is dense in ~

Theo-

Jcu(UI

endowed with the compact-open

.

If for each

ci

I,

Ua

is Runqe, then U is u-Runge.

If U is a Runge uniformly open subset of E, then xu(U) is obviously .r0-dense in X ( U ) . We don't know if this is true ingeneral.

The following result, whose proof we omit, parallels of Aron and Schottenloher [ 1 I . PROPOSITION 14.

Let

E

b e a ZocaZly c o n v e x s p a c e w i t h t h e

m a t i o n p r o p e r t y . If U i s u - h o l o m o r p h i c a l Z y c o n v e x , conditions are equivalent:

i)

U

i s u-Runge;

ii)

U

is Runge;

iii)

U

i s polynomially convex.

a

result

approxi-

t h e n t h e following

PAQUES AND Z A l N E

278 2 . O W ' S THEOREM

Let

OKA'S THEOREM. mental system J,

J

E

be a l o c a l l y ccnvex space which

o f c o n t i n u o u s seminorms,

has an e q u i c o n t i n u o u s S c h a u d e r

Ea

p s e u d o - c o n v e x u n i f o r m l y o p e n s u b s e t of

such t h a t , f o r each

basis. E,

has a fundaCL

in

Then e v e r y connected

is s e q u e n t i a l l y u-holomor-

p h i c a l l y convex.

PROOF. Let U be a pseudo-convex, connected uniformly opensubset of is pseudo-convex. E. Then there is a. in I n J such that uaO

By Levi-Oka's Theorem (see Noverraz [ 5 1 ) , holomorphically convex. Then by Proposition u-holomorphically convex.

is U,O

6,

U is

sequentially sequentially

COROLLARY. L e t E b e a l o c a l l y c o n v e x s p a c e w h i c h s a t i s f i e s t h e hyp o t h e s i s i n O k a ' s Theorem. L e t U b e a c o n n e c t e d u n i f o r m l y open subset of

E.

Then t h e f o l l o w i n g c o n d i t i o n s a r e e q u i v a l e n t :

(1)

U

i s a domain o f u-hclomorphy;

(2)

U

i s a damain of h o Z o m o r p h y ;

(3)

U

i s s e q u e n t i a l l y hoZomorphically convex;

(4)

U

i s s e q u e n t i a Z Z y u-hoZornorphicaZ Zy c o n v e x ;

(5)

U

i s pseudo convex;

(6)

U

i s hoZomorphically convex;

(7)

U

i s u-hoZomorphicalZy

convex.

Observe that every nuclear space satisfies the hypothesis Oka's Theorem.

in

REFERENCES

[ I ] R. M. ARON and M. SCHOTTENLOHER, Compact h o l o m o r p h i c mappings on J. Funct. Banach s p a c e s and t h e a p p r o x i m a t i o n p r o p e r t y . Anal. 21 (1976), 7 - 30.

DOMAINS OF U-HOLOMORPHY

[ 2 ]

279

J. MUJICA, Domains o f hoLornorphy i n DFC-spaces. Functional Analysis, Holomorphy and Approximation Theory, pp.500-533.

Lectures Notes in Math. Vol. 843. Springer, Berlin, 1981. [ 3

1 L. NACHBIN, U n i f o r m i t ; d ' h o l o m o r p h i e e t t y p e e x p o n e n t i a l . Seminaire Pierre Lelong 1970, pp. 2 1 6 - 224. Lectures Notes in Mat., Vol. 205. Springer, Berlin, 1971.

[4 ]

L. NACHBIN, R e c e n t d e v e l o p m e n t s i n i n f i n i t e d i m e n s i o n a l holomorp h y . Bul1:Amer.

[ 5

I

Math. SOC.

P. NOVERRAZ, P s e u d o - C o n v e x i t ; ,

79 (1973), 625-640.

C o n v e x i t ; P o l y n o m i a l e e t Domaines

d ' H o Z o m o r p h i e e n D i m e n s i o n I n f i n i e . North-Holland

Studies, V o l .

3. North-Holland, Amsterdam, 1973.

Math.

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COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) 0 Elsevier Science Publishers B.V. (North-Holland). 1986

281

THE PROOF OF THE INVERSION MAPPING THEOREM IN A BANACH SCALE Domingos Pisanelli PontifIcia Universidade Cat6lica de S a o Paulo, Brazil

Sao

Paulo

INTRODUCTION In a previous paper 1 2 1 we proved an inversion mapping theorem in a Banach space using power series. This theorem cannot beextended, ipsis litteris, in a Banach Scale. We gave a counter example in 1 3 1 . In [ 4 J we enunciated that, using our generalized Frobenius theorem, one can obtain a "right" inversion theorem [ 5 1 . We want now to give complete proofs and prove that one can obtain an inversion theorem ("right" and "left"). The classical inversion theorem in a Banach Space is a particular case of such situation. For some applications see [ 1 1 . DEFINITION.

A Banach S c a l e is a complex vector space

X =

XsJ

U O < S 0

with

such that

f'(xl E

0 < p' < p such that

i.e.

REFERENCES [ 1 ] L. NIRENBERG, Topics in Non Linear FunctionaZ Analysis. Courant

Institute of Mathematical Sciences, New York 1973 - 1974. [

2]

University,

D. PISANELLI, SuZZ'invertibiZit~ degZi operatori anaZitici negli spazi di Banach. B o l l . Un. Mat. Ital. (3) 19(1964), 110-113.

[ 3 ] D. PISANELLI, An extension of the exponential of a matrix and a counter example to the inversion theorem of a holomorphic mapping in a space H ( K ) . Rend. Mat. (6) 9(1976), 465- 475. [

4]

D. PISANELLI, Thiorzmes d'ovcyannicov, Frobenius, d 'inversion et groupes de Lie Zocaux duns Zes ichelles d'espaces de Banach.

T H E INVERSION MAPPING THEOREM

285

C. R. Acad. Sci. Paris 277 (1973), 9 4 3 - 9 4 6 . [ 5 ]

D. PISANELLI, T h e p r o o f of F r o b e n i u s t h e o r e m i n a B a n a c h S c a l e . Functional Analysis, Holomorphy, and Approximation Theory (Guido Zapata, ed.) 1983.

, pp. 379 - 389. Marcel Dekker, New York,

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COMPLEX ANALYSIS, FUNCTIONAL ANALYSIS AND APPROXIMATION THEORY, J. Mujica (Editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1986

ON CERTAIN METRIZABLE LOCALLY CONVEX SPACES

M.

Valdivia

F a c u l t a d de Matemgticas Universidad d e v a l e n c i a D r . Moliner 50 Bur j a s o t , V a l e n c i a Spain

The l i n e a r s p a c e s w e s h a l l u s e a r e d e f i n e d over t h e f i e l d

iK of loi s i t s t o p o l o g i c a l d u a l and

r e a l o r complex numbers. The word " s p a c e " w i l l mean c a l l y convex s p a c e " . I f o(E',E),

logics on

of A

,'A

is

E'

B

i s a space,

E

A

1

E'

a r e t h e weak, Mackey and s t r o n g topoI f A i s a s u b s e t of E , A i s the closure

B(E',E)

respectively.

and

tively. If A

and

v(E',E)

"Hausdorff

a r e t h e p o l a r and o r t h o g o n a l sets of A i n E' respec-

i s a s u b s e t of

E l , B"

i s t h e p o l a r s e t of B i n E .

an a b s o l u t e l y convex bounded s u b s e t of

s p a c e on t h e l i n e a r h u l l of

A

t h e s e t of p o s i t i v e i n t e g e r s and

En

E,F

),

we set

t h e normed

IN

is

i s the topological product

of

an i n f i n i t e c o u n t a b l e f a m i l y of c o p i e s of Given a d u a l p a i r

is

E , EA

whose norm i s t h e gauge of

If

- ,-

A.

E. )

to

denote t h e

asso-

c i a t e d b i l i n e a r form. of c o n t i n u o u s seminorms on

W e say t h a t a set A

a space

is

E

a I and a 2 i n A t h e r e a r e a i n A and A > 0 such t h a t a I 5 X a , i = 1 , 2 . I f 6 i s a c o n t i n u o u s seminorm on E , w e d e n o t e by E B = ( E , B I t h e s p a c e E endowed w i t h t h e unique

d i r e c t e d whenever, g i v e n

B and by

seminorm

E

/B

EB

/ B-'(O)

t h e normed s p a c e a s s o c i a t e d t o

E6 . Following Nachbin, w e d e n o t e by seminorms

c1

i s open: and such t h a t

on a s p a c e CC SlEl

E / a

C O S ( E ) t h e s e t of a l l c o n t i n u o u s

such t h a t t h e c a n o n i c a l mapping

E

t h e s e t of a l l c o n t i n u o u s seminorms

E - E / a

a

on

E

i s complete, [ 5 1.

L e t U b e a n e i g h b o u r h o o d of t h e o r i g i n in a space E. If A i s a b a r r e 2 in E w h i c h i s n o t a n e i g h b o u r h o o d of t h e origin and F is a c Z o s e d s u b s p a c e of f i n i t e c o d i m e n s i o n i n E ' [ o l E ' , E I I , t h e n

PROPOSITION 1.

Uo r-? F

does n o t c o n t a i n

PROOF.

F n E' A"

A'

n F.

i s a f i n i t e codimensional closed

subspace

of

the

VALD I V I A

288

normed space AD X(Ao n F I

, and t h e r e f o r e w e can f i n d a f i n i t e s u b s e t

A0

A > 0

and

E'

E'

s u c h t h a t t h e a b s o l u t e l y convex h u l l

contains

W

J of

of

J

U

A.

Uo n F c o n t a i n 'A n F . Then A' n F is equii s a l s o e q u i c o n t i n u o u s . S i n c e Wo i s c o n t a i n e d A , i t f o l l o w s t h a t A i s a neighbourhood of t h e o r i g i n , a conSuppose t h a t

c o n t i n u o u s hence

W

in tradiction. THEOREM 1.

Let

E

be a metrizabZe space. I f

t h e r e i s n c l o s e d subspace

F

E

of

i s n o t barrelled,

E

such t h a t

F

and

are

E / F

then not

b a r r e 1Z e d . L e t { U n : n = 1 , 2 , ...Ibe a fundamental system of neighbourhoods of t h e o r i g i n i n E . L e t A be a b a r r e l i n E which is n o t

PROOF.

a neighbourhood of t h e o r i g i n . W e t a k e

If

Ml

i s t h e subspace of

{xl}, w e

orthogonal t o

E'

apply

the

former p r o p o s i t i o n t o o b t a i n

Then

Proceeding by r e c u r r e n c e , w e suppose t h a t f o r a p o s i t i v e i n t e g e r w e have found (1) If

XI E

Ln

uIJ xi g

A, u . 3

3

i s t h e s u b s p a c e of

immediate t h a t A n L , Therefore there i s

If

9 u;, u .

Mn+l

E

E AO,

(x.,u.) z 3

= 0,

orthogonal t o

i, j = 1,2,. ..,n.

{ u l J u2 J . . . J u n l ,

it

i s n o t a neighbourhood of t h e o r i g i n i n

is t h e subspace of

E'

n

is

Ln.

orthogonal t o ~ x l , x 2 J . . . J x n J x n + l ~ J

w e a p p l y t h e former p r o p o s i t i o n t o o b t a i n

289

METRIZABLE LOCALLY CONVEX SPACES

Un+l

9 u:+1

u n+l

',+I>

E

A'

.

n M,+]

Then

F b e t h e c l o s e d l i n e a r h u l l of

Let

F n A

that

{xl,x 2,...,xnJ...}.Suppose

i s a n e i g h b o u r h o o d of t h e o r i g i n i n

tive integer p

F.

There i s

posi-

such t h a t

u

P

~

F

C

A

~

F

A c c o r d i n g t o (1) w e o b t a i n t h a t x

P

E U

P

which i s a c o n t r a d i c t i o n . Thus

"F, F

x

P

FA

is not barrelled.

E / F i s b a r r e l l e d . The t o p o l o g i c a l dual 1 i s i d e n t i f i e d i n t h e u s u a l way w i t h F Since

Now l e t u s s u p p o s e t h a t

of

E / F

is a

ofF

1

,E/FI-bounded

c o n t i n u o u s on

E

.

F , it follows t h a t

s u b s e t of

/ F , and h e n c e o n

E.

B

is

equi-

T h e r e f o r e t h e p o l a r set

Bo

of

i s a n e i g h b o u r h o o d of t h e o r i g i n i n E , whence t h e r e i s a p o s i t i v e i n t e g e r q s u c h t h a t U C Bo. C o n s e q u e n t l y B C Uo On t h e 9 4' o t h e r h a n d , a c c o r d i n g t o (1) w e h a v e t h a t B

in E

which i s a c o n t r a d i c t i o n . Thus

E / F

is not barrelled.

L e t E b e a non-barrelled subspace o f a Frichet space G. T h e n t h e r e is a c l o s e d s u b s p a c e F o f E s u c h t h a t F # F and E + $ is n o t b a r r e l l e d .

COROLLARY.

PROOF.

W e a p p l y Theorem 1 t o o b t a i n a c l o s e d s u b s p a c e F of

that F

and

follows t h a t morphic t o

E / F F E +

are n o t b a r r e l l e d . -

Since

F

E

such

is a Frgchet space it

i s d i s t i n c t from

F . O n t h e o t h e r hand, E / F is i s o -

F/F, from where

it

follows

that

E +

F is

not

VALD I V I A

7-90

barrelled. Let

write

q.e.d be t h e c l o s e d u n i t b a l l

B

t o d e n o t e t h e e l e m e n t of

en

the

of

Hilbert

whose

E2

space

coordinates

e x c e p t t h e n-th c o o r d i n a t e which i s e q u a l t o o n e , n = l , 2 ,

E l := { ( a n ) E := { ( a , )

E~

Clearly

El

and

2

(1

E k2

a2n = 0,

:

naZn = a2n-l,

For every p o s i t i v e i n t e g e r Therefore dense i n

eZn k2.

The s e q u e n c e (-

E2

e 2 n - l l of

E El

+ E2.

vanish

.We

set

.. I .

n = 1,2,.

a r e c l o s e d s u b s p a c e s of

E2

...

...I

n = 1,2,

:

We

k2.

and E l

k2

n , e2n-l E El

e 2n

and

From where it f o l l o w s t h a t

El

lo).

n E2 = +

e ~ w - E~~ * +

E2

is

W e can w r i t e

(-

El

1

e Z n l c o n v e r g e s t o t h e o r i g i n i n E and t h e s e q u e n c e which is t h e p r o j e c t i o n of

( 1n e2,1

does n o t converge t o t h e o r i g i n . Therefore

El

on

El

+ E 2 # k2.

P r o c e e d i n g by r e c u r r e n c e s u p p o s e t h a t f o r p o s i t i v e i n t e g e r

w e h a v e found c l o s e d s u b s p a c e s

El,E2,.

:= E l + E 2 +

Ln-l

...

..,En

of

along

R2

n > 1

such t h a t

+ E n # k2.

W e have t h a t B~

:= E~ n B + E~ n B +

...

+

E~

n 3

i s a n a b s o l u t e l y convex weakly compact subset of

k2 contained i n i s d e n s e i n k 2 a n d d i s t i n c t from E2, it f o l l o w s i s n o t b a r r e l l e d . W e a p p l y t h e C o r o l l a r y t o o b t a i n a closed t h a t Ln+l s u c h t h a t Fn+l # Fn+l and Ln-l + Fn+l i s s u b s p a c e Fn+l o f L%-l n o t b a r r e l l e d . I f w e s e t En+l . -- F n + l , we obtain that

Ln-l

.As

Ln-l

Ln

PROPOSITION 2 .

:=

El

+ E2 +

...

+ En+1 # k 2 .

For e v e r y p o s i t i v e i n t e g e r

n’

‘Bn

i s isomorphic t o

29 1

M E T R I Z A B L E L O C A L L Y CONVEX SPACES

L2

and

PROOF.

f,

Let

b e t h e mapping from

El

x

.

L2

is a w e a k l y c o m p a c t s u b s e t of

Bn

Bn

E2

...

x

x

into

E~

R'

d e f i n e d by ".,X

fniXZ'X2'

B,,

E2 x

. ..

...

+ x .

( B n E 2 ) x . . . x ( B n E n ) u n d e r f n coincides and t h a r e f o r e f, i s a l i n e a r homeomorphism from El X

The image of with

+ x2 +

1 = x

x

(B

Ell

fl

onto

En

x

R2

Bn

.

This implies t h e conclusion.

q.e.d. 2

) . If we n ( A I i s a s e q u e n c e o f a b s o l u t e l y convex weakly com-

Now w e w r i t e E t o d e n o t e t h e i n d u c t i v e l i m i t o f (L, set

An

.-

nBnJ

p a c t s u b s e t s of

c o v e r i n g i t . S i n c e B i s b a r r e l l e d , e v e r y bounded

E

s u b s e t of E is c o n t a i n e d i n some A

n'

[7,Theorem 6 1 . Therefore u(E',E)

and B ( E ' , E l

c o i n c i d e on

THEOREM 2 .

There i s a closed subspace

d e f i n e t h e t o p o l o g y of PROOF.

W e set

G:= E'

E'.

F

and

F

of

(L21N

such t h a t C O S ( F l

C O S I F I is n o t d i r e c t e d .

1 I - r ( E ' , E I ] , where E i s t h e l i n e a r s p a c e

con-

s t r u c t e d above. L e t

be a n e t i n

such t h a t

G

l i m Ia(xi) : i

f o r every

a

W e have t h a t

of

G /an

in

an

E

C O S I G I . For e v e r y p o s i t i v e i n t e g e r

i s a c o n t i n u o u s seminorm on

c a n be i d e n t i f i e d w i t h

T h e r e f o r e , G / an

I, > } = o

i s isomorphic t o

En,

G

n , w e set

and t h e s t r o n g d u a l

which i s i s o m o r p h i c

R2.

Consequently

a,

to

Lz.

E C 0 S(GI.

From where i t f o l l o w s t h a t

( 2 ) converges t o t h e o r i g i n uniformly on m = l , Z , ..., a n 6 t h u s ( 2 ) c o n v e r g e s t o t h e o r i g i n G , and C 0 S ( G I d e f i n e s t h e t o p o l o g y of G.

every subset in

Since

B,,

Ln-l

i s d i s t i n c t from

Ln

,

n = 2,3,.

. . , it f o l l o w s

that

VALD IV I A

292

i s a non-normable F r e c h e t s p a c e . I t i s immediate t h a t El + E2 is E , and t h e r e f o r e i f c1 i s a c o n t i n u o u s seminorm on G such t h a t a 1 5 a, a 2 5 a i t f o l l o w s t h a t a- 1 ( 0 1 = {O). Since G / a i s G

dense i n

G - G / c1 i s n o t open, from a d o e s n o t b e l o n g t o C O S t G l and t h u s C O S ( G l

a normed s p a c e t h a t c a n o n i c a l mapping where i t f o l l o w s t h a t

is n o t d i r e c t e d . F i n a l l y , if

i s t h e s t r o n g d u a l of

Hn

w i t h a c l o s e d s u b s p a c e of isomorphic lI { H n : n

.Z

to

Hn.

1,2

,... }

Let

lI { H n

be

ip

onto

incides with

.

= 1,2,.

a

(!L2ID.

t h e d e s i r e d s u b s p a c e of t e 2I w

FINAL REMARKS.

: n

k2 Bn

. . 1.

G can be i d e n t i f i e d

W e have t h a t

topological I f w e set

isomorphism

F : = v ( G l , then

1). Observe t h a t i n o u r former theorem of

F

C 0 S(Fl

seminorms

and i t i s n o t d i r e c t e d .

F

is from

is

q.e.d.

C C S ( F l . T h e r e f o r e t h e system

d e f i n e s t h e t o p o l o g y of

k2

Our

co-

CCS(F)

Theorem

2

answer n e g a t i v e l y t h e q u e s t i o n posed by Nachbin t h a t i f t h e t o p o l o g y of a l o c a l l y convex s p a c e

i s d e f i n e d by

E

CUS(El

then

COS(El

is directed. 2 ) The i d e a t h a t

of

E

C O S I E l i s d i r e c t e d and d e f i n e s t h e

was i n t r o d u c e d i n ( 3 ]

,

topology

where some i n t e r e s t i n g examples of cur-

r e n t s p a c e s s a t i s f y i n g such a c o n d i t i o n are g i v e n . I n [ 3 l t h e r o l e of t h i s c l a s s of s p a c e s

E

,[ 4 l ,

15 ]

i n s t u d i n g uniform holomorphyand

holomorphic f a c t o r i z a t i o n i s t r e a t e d . F , it i s e a s y t o chek t h a t t h e p r o p e r t y

3 ) For a F r g c h e t s p a c e s

of

COS(F)

b e i n g d i r e c t e d and d e f i n i n g t h e t o p o l o g y of

l e n t t o t h e " r e l a t i v e l y complete" p r o p e r t y of W.

F is equiva-

Slowikowski

and W.

Zawadowski [ 6 ] . I t a l s o c o i n c i d e s w i t h p r o p e r t i e s " s t r i c l y r e g u l a r " N. Zarnadze, I 8 1 (see a l s o [ l ]). T h i s class of F r g c h e t s p a c e s w a s also c o n s i d e r e d by Grothendieck i n [ 2 1 , 11, 5 4 , NQ 1 L e m m a 11.

and " c o m p l e t e l y r e g u l a r ' ' of D.

REFERENCES [ 1]

A.

GROTHENDIECK, P r o d u i t s t e n s o r i e l s t o p o l o g i q u e s e t e s p a c e s nu-

c Z e a i r e s . M e m . Amer. Math. SOC. 1 6 ( 1 9 5 5 ) . [ 2 ]

S . DIEXOLF and D.

N.

ZARNADZE, A n o t e o n s t r i c t l y r c g u Z a r

s p a c e s . Arch. Math. 4 2 (1984) , 549

- 556,

Frgchet

293

METRIZABLE LOCALLY CONVEX SPACES

[ 3

1 L. NACHBIN, U n i f o r m i t ; d ' h o l o m o r p h i e e t t y p e e x p o n e n t i e l , Ssminaire Pierre Lelong (Analyse), Annse 1 9 7 0 , 2 1 6 - 2 2 4 . Lecture Notes in Math. 2 0 5 . Springer, Berlin, 1 9 7 1 .

4 1

L. NACHBIN, R e c e n t d e v e l o p m e n t s i n i n f i n i t e d i m e n s i o n a l m o r p h y . Bull. Amcr. Math. SOC. 7 9 ( 1 9 7 3 ) , 6 2 5 - 6 4 0 .

51 L. NACHBIN, A g l a n c e a t h o l o m o r p h i c f a c t o r i z a t i o n

holo-

uniform

and

h o l o m o r p h y . In: Complex Analysis, Functional Analysis and

Approximation Theory, ( E d .

J. Mujica),

pp.

2 2 1 - 245.

North-Holland, Amsterdam, 1 9 8 6 . [ 6

1

[71

W.

SLOWIKOWSKI, and W. ZAWADOWSKI, N o t e on r e l a t i v e l y €?,-space. Studia Math. 1 5 ( 1 9 5 6 1 , 2 6 7 - 2 7 2 .

complete

M. VALDIVIA, A b s o l u t e l y conz.'ex s e t s i n b a r r e l l e d s p a c e s .

Ann.

Inst. Fourier, 2 1 ( 2 ) ( I 9 7 1 ) , 3 - 1 3 . [81

D. N. ZARNADZE, On c e r t a i n l o c a l l y c o n v e x s p a c e s o f c o n t i n u o u s f u n c t i o n s a n d Radon m e a s u r e s . Tr. Wychisl. Tsentr. Akad. Nauk Cruz. S. BR 1 9 ( 1 9 7 9 ) , 29 - 4 0 .

This Page Intentionally Left Blank

295

INDEX

absolute neighborhood retract, 116

complete regularity property, 292

absolute retract, 1 1 6 additivity property, 1 1 7 almost strict inductive limit, 248

analytic set, 2 analytic singularity, 9 6

complete quotient condition,240 condensing map, 1 1 2 cone, 1 2 6 continuous selection, 4 7 contractive projection, 6 8

asymptotic trajectory, 1 1 6 autonomous process, 111

convolution operator, 1 3 8 convolution operator of type zero, 155 convolution product, 1 4 3 , 148

Baire-like space, 2 5 0

COS(F) , 2 8 7

Baire space, 2 5 0 basic projective limit representation, 2 2 7 basic system, 2 4 3 Berqman space, 3 4 b-F-barrelled space, 2 6 1 b-F-space, 2 6 2 b'-F-space, 2 6 2 boundary value, 3 9 Cauchy problem, 9 5 Cauchy-Riemann function, 7 3 Cauchy-Riemann manifold, 7 3 Cauchy-Riemann submanifold, 7 4 c-compact set, 2 6 8

(db)-space, 2 5 0 defining sequence, 2 4 8 dilation, 3 7 domain of u-existence, 2 7 4 domain of u-holomorphy, 2 7 4 D'-P-convex open set, 1 8 8 D-space, 8 8 Dn-space, 8 8 egress point, 1 1 6 embedding, 1 2 equiamply bounded set, 2 2 2 equilocally bounded set, 2 2 2 ( E l -space, 8 3

C C S ( F ), 2 8 7

Chebyshev subspace, 4 7 closed vector sublattice, 49, 50,

51

commutativity property, 117

factorization lemma, 17 F-barrel, 2 6 1 F-convex, 2 6 1 finite dimensional analytic set, 2

296

finitely defined analytic set, 2 Fourier-Bore1 transform, 134 Fr6chet space, 247 Fredholm map, 3 frequency cone, 96 (F)-space, 83 functional Hilbert space, 26 generalized function, 57 generic submanifold, 74 graph theorem, 21 Grothendieck condition, 235 Hardy class, 34 Helly's theorem, 90 Hermitian operator, 67 holomorphic factorization, 224, 225, 231 homotopy property, 117 hyperbolic polynomial, 97 xcb ( E ) I xc*b ( E f f )I 206 xwub(B) r jcw*ub(E") I 206 Kc(U) xc*(u) 206 JCwu ( u ) Jew*, ( u ) r 206 immersion, 12 inductive limit, 176, 248 inductive sequence, 248 infinite dimensional distribution, 58 infinite dimensional holomorphy, 62 integral of a process, 115 interpolation manifold, 40 JB* triple system, 67 k-set contraction, 111 (LB)-space, 248 Leray-Schauder degree, 117, 122

INDEX

(LF)-analytic mapping, 282 (LF)-space, 176, 248

(LF)l-space, 252 (LF)2-space, 252 (LF)3-space, 252 Lindelof condition, 235 local condensing map, 112 local reflexivity principle, 90 local semidynamical system, 110 local strict set contraction, 112 locally F-convex space, 201 locally Holder continuous map, 114 measure of noncompactness, 111 metric projection, 47 microhyperbolic function, 98 minimal embedding codimension, 7 multiplication ofdistributions, 57 multiplication operator, 32 multiplier, 32 neutral equation, 112 nonarchimedean gDF-space, 265 nonarchimedean valued field , 261 nontangential manifold, 40 normalization property, 117 normally solvableoperator, 180 open linear relation, 177 openness condition, 232, 2 3 3 openness criterion, 18 open projective limit representation, 243 orbit of a process, 114 order of an entire function, 133 Pick-Nevalinna condition, 35 Pick-Nevalinna interpolation theorem, 35 positive-definite kernel, 26

297

INDEX

process,

111

p r o p e r mapping t h e o r e m , 1 3 p r o x i m i n a l s u b s p a c e , 47

s t r i c t (LB) - s p a c e ,

248

s t r i c t ( L F ) - s p a c e , 248 s t r i c t r e g u l a r i t y p r o p e r t y , 292 s t r i c t l y strict inductive l i m i t , 248 s t r o n g l y P-convex open s e t , 189 s u b o r d i n a t i o n of k e r n e l s , 28

q-capacity, q-energy,

38

surjective l i m i t representat i o n , 243

37

q u a s i - B a i r e s p a c e , 250

s y m m e t r i c domain, 68

r a d i u s of b o u n d e d n e s s , 207

tangent space, 6

r a d i u s of c o n v e r g e n c e , 206

t o t a l l y r e a l submanifold, 74

r e l a t i v e completeness p r o p e r t y ,

t r a j e c t o r y of a p r o c e s s , 1 1 4

292

t y p e of a n e n t i r e f u n c t i o n , 1 3 3

r e l a t i v e p r o j e c t i o n c o n s t a n t , 83 reproducing kernel, 26 restriction operator, 42 R-property, (R-R) (R

79

p r o p e r t y , 250

- T - Y)

p r o p e r t y , 250

u-holomorphically

convex

open

s e t , 227 u l t r a f i l t e r , 67 u l t r a p o w e r , 68 u l t r a r e g u l a r s p a c e , 268 u n c o n d i t i o n a l b a s i s , 257

s e c t o r i a l o p e r a t o r , 113 s e m i - c o n t i n u i t y of t h e f i b e r dimension, 1 7 semi-dynamical s y s t e m , 111 semi-Fredholm map, 1 0

u n i f o r m holomorphy,

225, 226,

231 u n i f o r m l y holomorphic f u n c t i o n , 274 u n i f o r m l y open s e t , 227, 273

semi-Fredholm o p e r a t o r , 2

uniformly s t a b l e operator I 1 1 2

s e p a r a b l e q u o t i e n t p r o b l e m , 254

u n o r d e r e d B a i r e - l i k e s p a c e , 250

s e q u e n t i a l l y u-holomorphically

u-Riunge,

convex o p e n s e t , 276 s e t of u n i q u e n e s s , 3 1

v a r i e t y , 251

277

S F - a n a l y t i c s e t , 3 , 11 s i n g u l a r s e t o f a F r e d h o l m map, 1 9

wave f r o n t s e t , 9 6

s i n g u l a r l y P-convex open s e t , 189

wedge, 126

s i n g u l a r i t y c o n d i t i o n , 173, 1 7 4

w e i g h t e d Bergman s p a c e , 34

smallest v a r i e t y , 251

well-located s u b s p a c e , 185

s p l i t t i n g p r o b l e m , 256

W-compact,

strict egress point, 116

Wilansky-Klee c o n j e c t u r e , 250

268

s t r i c t inductive l i m i t , 176 s t r i c t i n d u c t i v e s e q u e n c e , 248

Zymund c l a s s , 39

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