Classical and Modern Methods in Summability (Oxford Mathematical Monographs)

OXFORD MATHEMATICAL MONOGRAPHS Series Editors

E. M. FRIEDLANDER I. G. MACDONALD L. NIRENBERG R. PENROSE J. T. STUART

J. M. BALL

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Classical and Modern Methods in

Summability JOHANN BOOS Fachbereich Mathematck Fern Universitdt-Gesamthochschule Hagen, Germany assisted by

PETER CASS Department of Mathematics The University of Western Ontario

London, Ontario, Canada

OXFORD UNIVERSITY PRESS

OXFORD UNrvsRSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dares Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris Sao Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw with associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York Johann Boos, 2000 The moral rights of the authors have been asserted Database right Oxford University Press (maker) First published 2000 All rights reserved. No part of this publication may be reproduced. stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographic rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this book is available from the British library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 850165 X

Typeset by the author using LaTeX Printed in Great Britain on acid-free paper by T.J. International Ltd., Padstow, Cornwall

to

Christiane and to Daniela and Nicolas

Preface In its broadest meaning, summability theory, or in short summability, is the theory of the assignment of limits, which is fundamental in analysis, function theory, topology and functional analysis. For instance, we are interested in the assignment of limits in the case of real or complex sequences (x,,) for the limit process `n -i cc'. series (convergence of series), sequences and series of functions like power series, Fourier series, etc. (pointwise, uniform or compact convergence),

limit of a function at a point (continuity, continuous extension), differentiation of functions, integration of functions, sequences and series in topological (vector) spaces. Except for examples of applications, we deal in this book with summability

in a narrower sense, that is, with the assignment of limits in the case of real or complex sequences by so-called summability methods which are most often defined in this book by an infinite matrix (matrix method) or by a power series (power series method). We aim to give a broad introduction to summability theory and develop some of its most important methodology. We distinguish between classical ('hard') methods and modern ('soft') methods which are essentially based on analytical and functional analytic methods, respectively'. The book is subdivided into three parts which are, roughly speaking, devoted to classical methods (Part I, Chapters 1-5), modern methods (Part

II, Chapters 6-8) and the combination of classical and modern methods (Part III, Chapters 9-11). Concerning summability, the heart of Part I is contained in Chapters 2-4 where we deal mainly with inclusion, ToeplitzSilverman-type theorems, consistency theorems and the like. In Chapters 2 and 3 we discuss matrix methods like Cesaro methods, Hausdorff methods and others, and power series methods like the Abel method and the Borel method. Chapter 4 deals with Tauberian theorems for certain (classes of) summability methods. Some applications are given in Chapter 5. In Part II we investigate the structure of the domains of matrix methods, that is the set of all sequences which are `summable' by the method in ' This distinction comes from the fact that functional analysis is the younger theory and should not be understood in the sense that classical analysis has nothing new to offer.

viii

Preface

question. We apply modern methods, that is functional analytic methods, to these matters in Chapter 8. The essential tool here is FK-space theory which is due mainly to K. Zeller and which we develop in outline in Chapter 7. All functional analytic tools beyond the basics, which are applied

in Part II or III, are discussed in Chapter 6. So in this sense the book is self-contained. To give an impression of the advantages and elegance of functional analytic methods of proof we prove anew the Toeplitz-Silverman theorems that we proved earlier in Part I by classical methods. Whereas the classical methods require detailed and technical arguments, the functional analytic methods shorten these arguments and provide new insights into the meaning of some of the conditions that arise in the theorems being proved. The functional analytic methods, however, are not constructive and this can be a disadvantage in some situations. Both points of view should be appreciated for what each can offer. In Part III we continue certain investigations of Part I and II, for example the `consistency examinations', where we combine classical and modern

methods of proof. Often we can handle broad steps of the proofs with functional analytic methods but we must still perform the fine work by relatively extensive and technical classical methods. Whereas in Chapters 9 and 10 we deal with problems concerning the `consistency' of matrix methods which arise in summability and are solved by using the tools of topological sequence spaces, in Chapter 11 we deal with problems in this theory and solve them by the application of results from summability. So in Parts II and III we show--by way of examples-the connection between summability and topological sequence spaces and demonstrate how each field helps the other. The book is designed as a textbook for a variety of courses on summability and its applications at the graduate level. Its study requires solid knowledge of calculus and linear algebra as well as a basic knowledge of function theory and, concerning the applications, of numerical analysis. The study of Parts II and III requires very little knowledge of topology but basic knowledge of functional analysis concerning metric and normed spaces. As already mentioned, functional analytic tools, beyond the basic ones, are discussed in Chapter 6.

At this time no textbook or monograph is available that presents summability and its applications in the breadth undertaken here. Recent developments and applications to some related topics are discussed and open questions of interest to current research are posed. Thus this book should be of interest to those conducting research in summability or topological sequence spaces. Mathematicians working in areas bordering on summability should find much of interest. This book should also serve well

as a textbook for a variety of graduate courses in summability and topological sequence spaces and provides ample background to books or papers which present the subject in overview, such as, for example, Kangro's paper

Preface

ix

[123] and Zeller and Beekmann's book [267] which, in spite of databases like NfathSciNet from the American Mathematical Society and ZMATH from

Zentralblatt der Mathematik (etc.), retain their value and importance. Hagen April 2000

J. B.

Acknowledgements This book arose from a course for distance learning, entitled `Limitierungstheorie' (Limitierungstheorie is the German term for summability theory) which was written by me with the assistance of Georgia Thurner, typeset by C. Maurer and S. Sikora, and revised by me with the aid of K: E. Spreng. Some parts of Chapter 3 and 4 come from another summability course for distance learning (cf. [227]) which was written by U. StadtmUller. H. Tietz and K. Zeller drew my attention to a recent paper (cf. [240]) in which they give a modification of Wielandt's well-known elegant proof of the Hardy-Littlewood JD-Tauberian theorems for the Abel method. This is an elementary proof and I decided to use the material of this paper for

the important part of Section 4.4. Similarly, W. Kratz gave-for use in this book-an elementary proof of the D-Tauberian theorem for the Borel method which he adapted from a more general situation in a joint paper with U. Stadtmuller (cf. [134]). I use this material in Section 4.5. In the fall of 1998 I stayed for a couple of weeks at the University of Manitoba (Winnipeg, Canada) at the invitation of M. R. Parameswaran. We had fruitful discussions concerning `potent matrices' and I wrote, in this period, Section 2.9 and the corresponding part in Section 3.2. The Summability/Functional Analysis Group in my department, consisting of W. Beekmann, K.-G. Grof3e-Erdmann, K: E. Spreng (and me) discussed many aspects of the work for my book. My doctoral student and colleague D. Seydel spent a lot of time proof reading and gave many hints and valuable suggestions. Sylvia Sikora supported me in typesetting the book in I}TEX2e. She typed some parts of the book and made thousands of corrections and changes with incredible patience and accuracy. I thank all these friends and colleagues for their support. A special word of thanks goes to Peter Cass (The University of Western Ontario in London (Ontario, Canada)). From the beginning he aided the project with discussions and suggestions. He went very carefully through

the whole book and gave a 'nearly uncountable' set of suggestions and corrections concerning both the language and the mathematical contents. Last but not least, thanks to my loving family-my wife Christiane, my daughter Daniela and my son Nicolas-who provided an atmosphere of unerring support and patience to make possible the realization of such a large project as this book. J. B.

Contents PART I CLASSICAL METHODS IN SUMMABILITY AND APPLICATIONS 1

Convergence and divergence 1.1 The early history of summability-the

1.2 1.3 1.4

2

3

devil's in-

vention Summability methods: definition and examples Questions and basic notions Notes on Chapter 1

Matrix methods: basic classical theory Dealing with infinite series Dealing with infinite matrices Conservative matrix methods Coercive and strongly conservative matrix methods Abundance within domains; factor sequences Comparison and consistency theorems Triangles of type M The mean value property Potent matrix methods 2.10 Notes on Chapter 2

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

3

Special summability methods 3.1 3.2 3.3 3.4 3.5 3.6 3.7

4

Cesaro and Holder methods Weighted means, Riesz methods Norlund methods Hausdorff methods Methods of function theoretical type Summability methods defined by power series Notes on Chapter 3

Tauberian theorems 4.1 4.2 4.3

4.4

Tauberian theorems for Cesaro methods Tauberian theorems for Riesz methods Tauberian theorems for power series methods Hardy--Littlewood's D-theorems for the Abel method

3 5 20 25

26 27 34 39 51 61

75

82

87 92 97 99 100 112

126 136 152

157 165 167

168 178 186 191

xii

Contents

4.5

4.6 5

Hardy-Littlewood's ID-theorem method Notes on Chapter 4

Boundary behaviour of power series Analytic continuation Numerical solution of systems of linear equations Fourier effectiveness of matrix methods Notes on Chapter 5

PART II

6.6 6.7 6.8

Topological spaces Semi-metric spaces Semi-normed spaces, Banach spaces Locally convex spaces

Continuous linear maps and the dual space of a locally convex space Dual pairs and compatible topologies Frechet spaces Barrelled spaces

Topological sequence spaces: K- and FK-spaces 7.1

7.2 7.3 7.4 7.5 7.6 7.7

8

FUNCTIONAL ANALYTIC METHODS IN SUMMABILITY

Functional analytic basis 6.1 6.2 6.3 6.4 6.5

7

Borel

Application of matrix methods 5.1 5.2 5.3 5.4 5.5

6

for the

Sequence spaces and their c-duals K-spaces FK-spaces Functional analytic proofs of some ToeplitzSilverman-type theorems The dual of FK-spaces Distinguished subspaces of FK-spaces Notes on Chapter 7

Matrix methods: structure of the domains 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Domains of matrix methods as FK-spaces Distinguished subspaces of domains Replaceability and it-uniqueness of matrices Examples Bounded divergent sequences in the domain Consistency and perfectness Replaceability and invariance Notes on Chapter 8

Contents

xiii

PART III COMBINING CLASSICAL AND FUNCTIONAL ANALYTIC METHODS 9

Consistency of matrix methods 9.1 9.2 9.3 9.4

Consistency and theorems of Mazur-Orlicz type p-bounded sequences and domains p-consistency and p-comparison Singularities of matrices

10 Saks spaces and bounded domains 10.1 Saks spaces and mixed topologies 10.2 The Saks space m fl WE 10.3 A theorem of Mazur-Orlicz type 10.4 b-comparison through quotient representations 10.5 Notes on Chapter 10

11 Some aspects of topological sequence spaces 11.1 An inclusion theorem 11.2 Gliding hump and oscillating properties 11.3 Theorems of Toeplitz-Silverman type via sectional convergence and ... 11.4 Barrelled K-spaces 11.5 The sequences of zeros and ones in a sequence space 11.6 Notes on Chapters 9 and 11

459 460 475 483 500 515 516 522 527 529 536 538 538 540 546 550 557 561

Bibliography

563

Index

575

Part I Classical methods in summability and applications In Part I we develop basic theory in summability by applying exclusively classical (analytical) methods of proof. In Parts II and III we will re-prove some of the results of Part I by using modern (functional analytic) methods. Chapter 1 has an introductory character. After making some remarks on the early history of summability we introduce the basic notation and formulate the main problems in summability. In Chapter 2 we present some of the most important results in summability by applying exclusively classical methods of proof; thus the spectrum of the results presented reach from very classical theorems like the wellknown Toeplitz-Silverman theorem to recent results generalizing a wellknown theorem of Hahn on matrix methods summing all sequences of zeros and ones. Chapter 3 is devoted to the consideration of examples of (classes of) matrix methods and power series methods; in fact, we present them in consideration of the theory developed in Chapter 2. Whereas in Chapter 2 the problem of characterizing matrix methods, which sum given classes of sequences, is the main priority, we deal in Chap-

ter 4 with the converse problem. For a given (class of) regular matrix method(s) or power series method(s) we seek for sufficient conditions such that each sequence, summable by the method under consideration, is con-

vergent. Such conditions are called Tauberian conditions and the corresponding theorems are called Tauberian theorems. In closing Part I we consider in Chapter 5 some applications of matrix methods to other fields of mathematics. The reader should note that Section

5.5 contains hints and references to further applications of summability methods.

1

Convergence and divergence This first chapter has an introductory character. We start in Section 1.1 with some remarks about the early history of summability, that is about the history from the time when the essential evolution of summability started until the end of the nineteenth century. Based on both Cauchy's and Abel's theorems (cf. 1.2.7 and 1.2.9), in Section 1.2, we agree what we should understand by a summability method. By way of examples we then develop further this agreement, which appears at first to be artificial and abstract. The notion of a summability method and the examples, considered in Section 1.2, give rise to natural questions and problems. We list some of them in Section 1.3. Some of these problems may only be understandable when we deal with them seriously in the subsequent chapters.

1.1

The early history of summability-the devil's invention

Tracing the origins and development of the ideas that crystallized towards the end of the nineteenth century to form the basis of the theory of summability is a complex undertaking. A full discussion of the historical issues involved is beyond the scope of this book. An admirable and full treatment is given in chapters 1 and 2 of Hardy's book Divergent Series (cf. [108]) to which we enthusiastically refer the reader. To help motivate and introduce the first ideas of summability we offer some comments which borrow from Hardy's discussion. Hardy writes Newton and Leibniz, the first mathematicians to use infinite series systematically, had little temptation to use divergent series (though Leibniz played with them occasionally). The temptation became greater as analysis widened, and it was soon found that they were useful, and that operations performed on them uncritically often led to important results which could be verified independently. The formula, for example,

4

Convergence and divergence

1-1+1-1+ can be arrived at as follows. Let

s=1-1+1-1+ Then,

1-s.

Thus, s = 2. Hardy carries out various manipulations arriving eventually at the formula

and notes that

(1-1+1-1+ )(1-1+1-1+ ) = 1.1-(1.1+1.1)+(1.1+1.1+1.1)- , so that equation (1.1:2) can be `obtained' by squaring equation (1.1:1) using Cauchy's rule. Later Hardy describes a valid process of arriving at a `sum' of a divergent series that yields the bizarre result

1+2+4+8+

=-1.

Hardy states that It is plain that the first step towards such an interpretation [of his various calculations] must be some definition, or definitions, of the `sum' of an infinite series, more widely applicable than the classical definition of Cauchy. In 1828 Abel wrote `Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever.' Such was

the authority of Cauchy, Abel and their successors that divergent series were, in Hardy's words, `gradually banished from analysis'. It was not until 1890 when Cesaro published a paper about the multiplication of series that, according to Hardy, `for the first time a "theory of divergent series" is formulated explicitly'. Indeed, Cesaro's idea was quickly picked up and applied fruitfully to Fourier series. One of the early successes was a beautiful theorem of Fejer's which we state after first giving the simplest form of Cesaro's idea. The limit of a sequence (x,,) according to Cesaro can be defined to be

limy,, where (y,,) is the sequence of Cesaro means of the sequence (x,,) given by

n

yn

Exi i=O

Fejer's theorem states that if f is in LP(0, 21r), 1 < p < oo, then the Cesaro means yn of the partial sums of the Fourier series of f converge

Summability methods: definition and examples

5

to f in the LP-norm. If, in addition, f is continuous and f (7r) = f (-ir), then the y converge uniformly to f (cf. 5.4.15). Bibliography: [204]; [267]

Summability methods: definition and examples

1.2

The essential content of this section is the agreement on what is meant by a summability method, that is a method generating a limit map, and the consideration of examples of summability methods. With the aim to encompass all well-established summability methods, we will give a very general definition and we will take the bad with the good that the definition provides. In the next two chapters we will consider almost exclusively matrix methods. Before we discuss the notion of a summability method, we introduce some simplifying notation. We will introduce further notation in the course of the book.

General assumption 1.2.1. In the sequel let K := C or lid := R be the field of all complex or real numbers, respectively. Further, let N be the set of all positive integers (natural numbers) and N° := N U {0}. L\ Throughout the book sequence spaces will be in the foreground; we will understand the following:

Definition 1.2.2 (sequence space). If w := KN° := IT = (Xk)kEN° j x : No -* K, k --+ Xk := x(k)} 1

denotes the set of all sequences, then w together with coordinatewise addition and scalar multiplication defined by

((xk), (yk)) -+ (xk + Yk) and

(.\, (xk)) -} (Axk),

respectively, is a linear space (over K). Each linear subspace of w (with the induced addition and scalar multiplication) is called a sequence space. If x = (xk) E w, then we also use the notation [x]k for the kth coefficient xk of X.

Examples and Notation 1.2.3 (sequence spaces). Besides w the following subsets of w are obviously sequence spaces:

m

l

x = (xk) E w 1 Ilxlk := sup IxkI < 00 kEN°

1 In the sequel we specify the domain of the running index of a sequence only if it is

different from NO. Furthermore, we use the notation (xk), (xk)k and (xk),n where

n E N. Analogously we make use of limk' and `Ek' instead of limk...,,o' and `E0k0=°', respectively.

Convergence and divergence

6

(the set of all bounded sequences).

m -- R, x -4 IIxII is called the supremum norm.

The map rII

c :_

{ x = (Xk) E w f (xk) converges2, that is lim Xk exists k-oo I 111

(the set of all convergent sequences). Note, lim : c -+ K, x = (xk) -4 limk xk is a linear functional. { x = (xk) E c I lir xk = 0 }

co

{x=(xk)Ew

bs

(the set of all null sequences)

IxIIbs:=suPxk
I

n

k=O

I

)JJ

(the set of all sequences with bounded partial sums). cs

l

(Xk)

x = (Xk) E w I

E

cI

(the set of all summable3 sequences).

e = el :_ {x=(xk)E

I

Ixkl <

Ilxlli

}

k=a

(the set of all absolutely summable sequences).

The maps 11 llbs:bs-->R, x -+IIxIIbs and 1111, :£-+R, x --lxlll

are called bs-norm and 11-norm, respectively.

[x = (xk) E w I IIxIIbv := Ixol + E Ixk - xk+i I < oo }

by

k

)

(the set of all sequences with bounded variation). bvo

:= by fl co.

The map II cp

Ilbv

: by -+ R, x -4 IIxIIbv is called the by-norm.

:= {x=(xk)Ew13NE`l° dk>N:xk=O} (the set of all finitely non-zero sequences) ({enInENa})

4 .

where 2 (in the ordinary sense)

-

3 in the sense that the series Z. xk converges 4If M is a (non-empty) subset of a linear space V, then we use the notation (M) for

the span of M; for a E V we put (a) := ({a}). Furthermore, if U and W are linear subspaces of V, then the sum U+W = (UUW) is called a direct sum, if UnW = {0} holds; in such a case we use the notation U EB W.

Summability methods: definition and examples

e' :_ (snk)kENo

with

6,,k :=

1

ifk=n

0

if k o n

7

(k, n E ll°)

denotes the n'' unit vector (in w). It is an easy exercise to check

p CPCcsCcoCc=c:°®(e)CmCw

(1.2:1)

eCbv°Cbv=by°®(e)Cc,

(1.2:2)

and

where e := (1,1.... ).

A

Throughout the book the so-called thin sequences and the sequences of zeros and ones will play an important role, so we introduce here some notation for the corresponding sets and their spans.

Notation and Remarks 1.2.4. We put X

{x=(xk)EwI dkE No : xkE{0,1}} (the set of all sequences of zeros and ones),

m°

(X) _ {x = (xk) E w I {xk I k E No } is a finite set} .

By definition, a sequence x = (xk) E X is called thin, if there exists an index sequence

with

k - oo (v -* oo) and with Xk = 1 if

k = k (v E N°) and Xk = 0 otherwise. Further,

T := {x E w I x is thin}

(the set of all thin sequences).

Obviously, cp C T C X C (X) = m°. (Note, each e' is representable as the difference of two thin sequences.]

We are going to find and motivate a usable notion of a summability method. We base our decision on the following. Problem: Let D C w with D 0 0 be given. We ask for a rule which gives

for each x E D a `generalized limit' in 1K, that is a map D -> 1K (for example, D := c and lim : c - 1K, (Xk) -+ limk Xk ). To accomplish this, we often start with two subspaces D C w and E C w and a map ip : D -+ E. The map zb is often required to be linear in such cases. We then consider those sequences x E D for which Vi(x) E c and define the ip-limit of x to be fi(x) = lim'(x). We thus base our notion of a generalized limit on the ordinary notion of limit. In the case that c C D and z/i(c) C c we sometimes require that lim '(x) = lim x for all x E c. Such summability methods are called regular. This `natural' requirement is, however, not always appropriate and much would be lost from our theory were we always to insist on it. Similar ideas can be formulated in other contexts. For example, we may work in situations where

E is a double sequence space or a function space instead of a sequence space. We now give some examples of summability methods to bring this

8

Convergence and divergence

broad outline into focus. The first example may appear trivial but arises naturally when one considers the possibility of addition and subtraction of different summability methods.

Example 1.2.5 (zero map). We take as a basis the notion of ordinary convergence of sequences and consider for D := w the map V : w -+ c, x -* (0, 0, ...) as well as the `limit functional' lim oV : w -4 K which is obviously the null functional. Nevertheless, in this way, we get for the sequence x of the partial sum of the series 1 - 1 + 1 - ... considered by Euler, that is for x = (1, 0,1.... ), the limit (lim oV) (x) = 0. A If one had data in the form of a set of measurements from an experiment, it may make sense to consider one of the following `notions of convergence':

Examples 1.2.6. Again, we take as a basis the ordinary limit of sequences, that is lim : c --- 1K.

(a) Let D := w and xre_1

Z1:w->w, x=(xk)--+ C

(with x_1 := 0)

+xn)

2

_1

czi

{x E w I Z, X E c} = Z (c).

Then we obtain for every x E

czj

a (generalized) limit by the setting

Z -lim := limz := lim oZj. :

cz

-i K, x -+ lim Zix.

Obviously,

cC

cz#

and

limz1 x = lim x (x E c).

(1.2:3)

Also 1

and limza (1, 0,1, ...) = cz 2* Thus we see that Z2 assigns limits to some divergent sequences.

(1,0,1,0 ....) E

(b) Using the arithmetic mean, we obtain a further notion of a limit. Let D := w and

/C1 : w -f W, x = (xk) -+ (l TL + l 1

Also let

"

xk

k=0

)n-

_1

CC, :_ {x E W I C1x E c} = C1(c).

Then for each x E cc, a (generalized) limit is defined by

C1- lim := limo, := lira o C1 : cc, --- K, x -* lim Cl x

Summability methods: definition and examples

9

and, as we may easily check, we have

y := (1, 0,1.... ) E cc,

with

lime, y =

2

and

x := (1, 0, -1,1, 0, -1, ...) E cc, Note, however, that x

with

limci x = 0.

0

CZ .

In (1.2:3) we saw that limz i is defined on c and extends lim from c to ; that is, the `new' notion of convergence is defined for each converczj gent sequence and each convergent sequence is assigned the `right' limit. That these observations are valid also in the case of limc1 is confirmed by the following theorem which-as we will see-follows as a special case of Theorem 2.3.711.

Theorem 1.2.7 (Cauchy's limit theorem). c C cc, and limc, x = lim x for every x E c. -1

Proof. First we note that C1 : w -* w is linear and that cc, = C1(c) is consequently a sequence space and limc, is linear. Because e = Cle and c = co ® (e), to prove the statements in the theorem, it is sufficient to show

co C cc, and limci y = 0 (y E co). For that let y = (Yk) E co be given and let

n

i tn._ nEYk (nEN°). k=0

Then we have to verify t := (tn) E co. For an arbitrarily given E > 0 we may choose a ko E No with Iyk I <

E

2

for each k > ko

[since y E co ]

and then choose no E No such that ko < no

and

I.-

I

< no+i Iykl k=0

E

2

This gives for all n > no the inequalities n

ko

Yk + Y Yk k=0 k

<

°

k=0

Hence t E co.

k=ko+1

n-ko e n

E E < 2+2 = E.

10

Convergence and divergence

Thus, in Example 1.2.6 we have encountered two (generalized) `notions

of convergence' which are reasonable-in the sense of our broad outline discussed above. That is, the limit functional is based on a proved `notion of convergence' (here on lim) and a suitable map (here it is Zj and C1, respectively) and the new `notion of convergence' extends the functional lim beyond c. To motivate further the notion of a `summability method', we consider in the next example a `notion of convergence' which is based on the `limit of a function at a given point'.

Example 1.2.8. Let D1 := {z E K I izi < 11, C :=

CA,

If: D1 -+K I

t->1-l tmD f (t) exists } ,

{x = (Xk) E w I g(t)

xktk converges for Its < 1, k

and f E C is given by f (t) := (1 - t)g(t) } and

Al : CA, -* C, x -+ f with f (t) := (1 - t)g(t). Then a (generalized) limit is defined on CA, by

Al-lim : CA, -* K, x = (xk) -+

lira t->1-,tE1

(1 - t) E xktk. k

It is an easy exercise to verify cp C CA,

and A1- lim x = 0 (x E W),

e E CA,

(1, 0,1, 0, ...) E CA,

(1.2:4) (1.2:5)

and A,-lime=l, and

Al-lim (1, 0,1....) =

1

2'

Beyond the statements in (1.2:4) and (1.2:5) we also have c C CA,

and

.41- lim x = lim x for every x E c,

(1.2:6)

as we will deduce from the following theorem.

A

Theorem 1.2.9 (Abel's theorem). Let (ak) E w and Ek akrk be convergent for a given r > 0. Then Ek aktk converges for all Itl < r and

ark = k

holds.

lim

t-r-, tER

7 k

aktk

Summability methods: definition and examples

11

Proof of (1.2:6) in Example 1.2.8. Let x = (Xk) E c. Then F_k ak converges where ak := xk - xk-1 (k E No and x_1 := 0). Because the power series >k xktk converges for each ItI < 1 since x E c C m, we obtain for every Iti < 1 the equalities

E aktk = E(xk - xk_t)tk = (1 - t) E xktk. k

k

(1.2:7)

k

Therefore, by Abel's theorem 1.2.9 (in the case r = 1), the statement

lira Xk = E ak = k

k

aktk =

lim

t->1-, tER

k

(1 - t)

lim

t-+ 1-, tER

xkt k k

holds. This shows in particular that x E CA, .

Proof of 1.2.9. Without loss of generality we may assume r = 1 since otherwise we may consider the sequence (bk) with bk := akrk (k E NO). v0 a (k E N°), then we Thus, let (ak) E cs be given. If we put xk obviously have x = (xk) E c and Ek xktk converges for each ItJ < 1. In particular, we retrieve (1.2:7). Now, if we put

f(t):=(i-t)Y' xktk (It{<1)

and

a:=Ea,, v

k

as well as

00

rk:= 57

xk

(kEN°),

v=k+1

we obtain for all Iti < 1 the identities

a - f (t) = (1 - t) E(a - xk)tk = (1 - t) E rktk.

(1.2:8)

k

k

The remaining part of the proof is quite similar to the corresponding part of the proof of Cauchy's limit theorem. For a given r > 0 we may choose for (rk) E co a k0 E N° with Irkd < i (k > k0). If we now put 5 := min {1, 2{TI+1i } with M := Fkk.° 0 Irkl, then we get, by (1.2:8), for 1 - 8 < t < 1 the inequality 00

ko

Ia - f (t)I

<

(1-t)E1rkl+(1-t) 1 Irkltk k=0

k=ko+1 00

<

(1-t)M+(1-t)26 1: tk < 2+2 = e. k=ko+1

Consequently, a =

lim

t-+1 -, tCR

f (t). Thus Abel's theorem is proved.

12

Convergence and divergence

What is common among the examples of `new notions of convergence' considered in 1.2.5, 1.2.6 and 1.2.8 are three important principles. the choice of a `standard' notion of convergence, on which the new notion of convergence will be based;

the definition of the map, by which one connects to the `standard' notion of convergence; the determination of the set of all sequences, on which the new notion of convergence is applicable. On the basis of these observations we now formulate a general notion of

a summability method. (It is not really a definition since we do not make precise exactly what we mean by a 'standard' notion of convergence.)

Formulation 1.2.10 (summability method). A summability method V is a triple (V,Nv, V-lim) consisting of a map V : Dv -* M, where DV C w and (roughly speaking) M is a set such that at least on a suitable subset N, 0 $ N C M, there exist a (standard) limit functional f : N - K, -2

the domain Nv := V(N) of V and the summability functional V-lim := f a V INv : Nv -+ K5 Each x E Nv is called summable (by V to the value V-limx) or V-summable to V-lim x. A series Fv a, is called summable (by V to the value a), if (xk) with xk :_ FY=°a (k E N°) is summable (and A a = V-limx holds). It may appear at first unnatural to consider the map V on Dv instead of directly on Nv, but examples and also the theory which we develop subsequently will prove that it is more sensible to do it this way, since often

we may know Dv precisely but not be able to determine Nv exactly. Now, we give some examples of summability methods. The first series of examples is based on the ordinary notion of convergence. In particular, we will verify that the examples in 1.2.5 and 1.2.6 are summability methods in the sense of 1.2.10.

Examples and Notation 1.2.11. Let M := w, N := c and f := lira (in 1.2.10).

(a) Putting V := id,,, : w ---* w, x -+ x we get cv = c and V- lim = lim; therefore (id,,,, c, lim) is a summability method which is just the ordinary notion of convergence.

(b) If V : w -t w is the zero map (cf. 1.2.5), then cv = w and V-lim = lira o V is the zero functional. In accordance with our formulation in 1.2.10 (V, w, V- lim) is a summability method that sums each sequence to zero. (c) If Z I., cZi and limZ3 are defined as in 1.2.6(a), then (Z , cza , limz, } 2

2

2

2

3

I In the following we will not distinguish between the notation for maps and their restrictions to subsets of their domains.

Summability methods: definition and examples

13

is a summability method which is called the Zweier method6 (of order 1) 2

(d) If C1, cc, and limc, are defined as in 1.2.6(b), then the summability

method (C1 i cc, , limc,) is called the Cesaro method (of order 1) or briefly the Cl-method, A The last examples of summability methods have in common that the defining map in the definition of the method is representable by a matrix. We emphasize that property of such summability methods by the following definition.

Definition 1.2.12 (matrix method). Let aoo

aol

a02

aok

alo

all

a12

alk

ano

and

arz2

A = (ank) = t`

...

...

ank T

E- nth row 11

kthcolumn

with [A]nk := ank E K (n, k E N°) an infinite matrix and let wA

{x = (xk) E W I Ax := (1: ankxk)

exists, n

that is all series 57, ankxk converge

}

J

k

(application domain of A), A : WA -> W, x -+ Ax (matrix map induced by the matrix A),

(domain of A), CA :_ {x E w I Ax E c} = A(c) limA := A-lim := lim oA : CA -+ K, x -+ lim Ax. Then the summability method (A, CA, lima), A for short, is called a ma-

trix method. (Note, we use the letter A for the matrix A, the matrix map A : WA -4 w and the summability method (A, CA, limA).) Moreover, coA :_ {x E cA I limA x = 0} = A(co)

and rn fl cA

are called the null domain and the bounded domain of A, respectively. 6The name `Zweier method' comes from the German word Zweier-Verfahren where zwei means `two' and Verfahren means `method'.

1 4

Convergence and divergence

Examples 1.2.13 (matrix methods). (a) If

A:= I:=

1

0

10

1

0 0

... ...

0

0

1

...

is the identity matrix, then the corresponding matrix method yields obviously the summability method (id,,, c, lim) in 1.2.11(a). (b) The zero matrix (all coefficients are equal to zero) corresponds to the summability method in 1.2.11(b).

(c) The Zweier matrix

and the Cesaro matrix (of order 1) 1

1

2

2

a

C1 :=

[

nn

1

1

n+1

n+1

1

...

n+1

+l repeated (n + 1)-times in the nth row] yield the Zweier method in

1.2.11(c) and the Cesaro method in 1.2.11(d), respectively7.

(d) If Bi

then the matrix Bi and the matrix method Bi = (B1y' , CBi , limBi) are called the Borel matrix and the discrete (e-n k,`)

,

Borel method, respectively.

0

In the course of this book we will encounter many examples of (classes of) matrix methods and we will subsequently deal almost exclusively with

matrix methods. However, to demonstrate the scope of the notion of a summability method introduced in 1.2.10, we identify first the example in 1.2.8 as a summability method. After that we will examine some summability methods which are not based on the notion of ordinary convergence of sequences. 'The big `0' indicates that in the corresponding domain, that is over the main diagonal, the coefficients are equal to zero.

Summability methods: definition and examples

15

Examples 1.2.14 (Abel and Borel methods). (a) Let M := N:= C. Referring to Example 1.2.8, let

F : C -* K, f -3 F(f) :=

lim

t-+1-, tE?t

f (t).

Then (A1,CA1,A1-lim), Al for short, is a summability method and is called the Abel method. (Here A1, CA, and Al-lim are defined as in 1.2.8, and we have Al-lim = F o A1. ) (b) Let Jim a-t f (t) exists I f : Ilk -4 K I j:= If: t-'+°° 11

CB,

B1 : CB, -+ j, (xk) -* f with f : K -+ K, t --- f (t)

l

tk

k

and

B1-lim : CB, -> K, (xk) -3 Then

a-t 1: Lk tk.

lim

lit a t-r+oo

k

B1 for short, is a summability method which is

called the Borel method. We have, as the reader should prove in Exercise 1.3.11,

cc CB,

and B1- lim x = lim x (x E c).

Further, as in the case of the Abel method, we have

x :_ (1,0, 1,...) E CB,

with

B1-limx =

2

because

.f (t) _

k

and

lim erat->+oo

xk tk = kl et

t2k 1..

+e-t) 2

= et + e-t

(tEK)

2

F2 k-)-!

lim

H2at-4+oo

1

2

+

e=zt 2

2*

We will return to the Abel and Borel methods in Section 3.6 where we will study more generally the so-called power series methods. The last class of examples of summability methods which we consider in this section is based on the uniform convergence of double sequences. For that we introduce some notation. Note that other authors (cf., for example. [267]) use the notation B0 instead of B1 and Bo instead of B,*. Our notation is consistent with that in 3.6.10(b).

16

Convergence and divergence

Notation and Remarks 1.2.15 (double sequences). Let {y = (ynp) I y : No x No -) K, (n, p)

Yap := y((n,p))}

(the set of all double sequences),

{yE11 3aEK: ynp-4a (n -oo uniformly for pENo )} {y=(ynp)Efl I 3aEK VE>O 3n°EN°

C

bn>n° VpEN° : (ynp-al <e} (the set of all (uniformly) convergent double sequences) and

Lim:C--+K, y -- Limy:=a (where a is chosen in accordance with the definition of C). Obviously, 11 is a linear space (over K), where the addition and the (scalar) multiplication are defined coordinatewise, C is a linear subspace

of 1 and Lim is a linear functional on C. Moreover, we remark that (uniformly) convergent double sequences are not necessarily (uniformly)

bounded. That is, there exists a y = (ynp) E C with sup,, , l ynpl = oo; for example, y = (ynp) with ynp := p, if n = 0 and p E ld, and ynp := O otherwise, is an unbounded member of C. However, as an immediate consequence of the definition of C we have

VyEC 3noEN0:

sup n> no, p£N°

Iynpl
First we consider some trivial examples of summability methods based on the notion of convergence in 1.2.15.

Examples 1.2.16. (a) Let V : w -a fl, x = (xk) --+ (ynp) with Yap :

-1 xn (n, p E No). Then Cv = V (C) = c and V- lim = lim holds obviously; that is, the summability method (V, c, lim), as that in 1.2.11(a), reduces to the notion of ordinary convergence of sequences.

(b) If V : w --+ 1 is the zero map, then we see that Cv = V(C) = w, and V- lim = Limo V is the zero functional. Thus the summability method (V, w, V- lim) provides the same (trivial) notion of convergence as that in 1.2.11(b).

(c) If A = (ank) is an infinite matrix and

V : wA -* Il, (xk) 4 (ynp) with ynp

ankxk

(n,p E N°),

k

-1

then, since Yap does not depend on p, we obtain Cv = V(C) = CA and V-lim = lima . Thus the summability method (V, cA, lima) provides the same notion of convergence as the matrix method A. L1

Summability methods: definition and examples

17

So, there is not yet anything new in the examples in 1.2.16 and the question arises whether there exist examples of summability methods which are based on the (uniform) convergence of double sequences and are not

identical to a matrix method. As we will prove subsequently the notion of almost convergence, introduced in 1948 by G. G. Lorentz (cf. [148]), provides such an example.

Example and Definition 1.2.17 (almost convergence). Let P+n

F:w-+f2, x_ (xk) ---+F(X) :=

n+l k=p Xk

be given. Then -1

f := CF = F(C) P+n {x=(xk)EIaEK:Exkna(uniformlyforP) 1

k=p

is a sequence space, F-lim := Lim o F is a linear functional and (F, f , F- lim) is a summability method. The notion of convergence generated in this way is called almost convergence, and the elements of f are called almost convergent (sequences). A We now state some important properties of almost convergence.

Theorem 1.2.18 (almost convergence). (a) c C f C m and F-lim I, = lim. (b) If A is a matrix method, then f 54 CA (that is, almost convergence is not representable by a matrix method) and, moreover, f ,-£ m fl CA.

Proof. (b) Later we will see from 2.5.8 that there is no matrix A with f = CA. We omit the proof that f 54 m fl CA and refer the reader to [148, Theorem 11].

(a) Let x = (xk) E c and a := lima. For a given e > 0 we choose an

NEN° with Ixk-aI<2 (k>N)

(1.2:9)

and for N an no E No

such that no > N and

2(N no

+Ilxll < 2

We then obtain for p E No and n > no the inequalities 1

n+F

n+1 k=P

n+p

n+p

xk-a

1

1: (xk - a) n + I

k=p

<

1: jxk - a,

n + 1 k=p

(1.2:10)

18

Convergence and divergence

n+1(n+1)2, ifp> N <

N

1

<

1

al+

L1 fxk ;T+-1 k=p

n+ l

[cf. (1.2:9)] n+p

Ixk-al, ifp
k=N+1

2 1ifp>N

ifp
(N-p+1)2I1xII.+n+p-NF

n+1 <

n+1

2

[cf. (1.2:9)]

[by (1.2:10) and the choice of N and no ].

e

Thus x E f and F- lim x = a, that is c C f and F- lim l c = lim . Now, we verify x = (xk) : _ (1, 0,1, ...) E f and F- lira x = 1; hence, c

f. Obviously, we have n+p n21 -1 < Exk < n21 +1

(n,pE No),

k=p therefore

n+ 1

1

1

1

2

n+1

n+1

C

2

-l \t< n+1xk 1

k=p

(n21+1)

1

n+p

=

2+n+1

(n,pENO),

which implies n+p

1

(n -3 oo uniformly for p E N° ),

1E'rk--42

k=p

that is x E f . We next prove f C rn. Let x = (xk) E f, a E K and no E N° with 1

n+1

p+n (xk - a)

<1

(n>noandpe °)

k=p

be given. Then we have for each p E N° the inequalities p+no+l

p+no+1

k=p

k=p+1

E xk - E xk

p+no+1

p+no+1

(xk - a) k=p

L.1 k=p+1

(xk - a) + a

(1.2:11)

Summability methods: definition and examples

< no+2+no+1+lal =

19

[cf. (1.2:11)]

2no+3+lal.

That is, x E in. It remains to show f C m. For that we put x = (xk)

0,0,

. .

te=:

.,0,

1, ...}

(n + 1)-times (n + 1)-times and n-1

p0 := 0, pi := 2, A2 := 6, ... , An := 2 J(v + 1) = n(n + 1), ... . v-0

Note, Itn is the index at which the (n + 1)th block of ones begins. Thus {h.+2n+1

An +n

1

E xk=1 and n+1 E xk=0 (nEN°) n+1 k=kn k=un+n+1 0

f. Obviously, x E m. This gives (a).

and therefore x

As a generalization of almost convergence G. G. Lorentz (cf. [148]), later G. M. Petersen (cf. [193]), and later still M. Stieglitz (cf. [230]) along with other mathematicians have considered summability methods defined by sequences of infinite matrices.

Examples and Notation 1.2.19 (SM-methods). Let A := (Atp))p be a sequence of matrices A(P) = A:

f

l

let

WA(P) --4 it , x = (xk) '-} U=0 ank

xkp= n,pEN°

and

-1

CA := A(C) and LimA := Lim o A : CA --+ K The summability method (A, CA, LimA), A for short, generated by the sequence A of matrices, is called an SM-method. Special cases: (a) Let A be an (infinite) matrix and Abp} := A (p E N°). Then we obtain CA = CA and LimA = IimA for the SIN-method A, that is A and A generate the same notion of convergence. (b) For p E No let AW = (ank) be defined by ask

I`l

n+1> 0

if p:5 k < n + p otherwise

(n kEN°).

Then F = A, f = CA, and F-lim = LimA (cf. 1.2.17), that is the methods (F, f , F- lim) and (A, CA, LimA) coincide.

A

20

Convergence and divergence

We will encounter and discuss almost convergence several times in the sequel, but we will not examine SM-methods in detail. In this connection it is obvious to require for the existence of SM-methods whose domains are

different from f and which are (as (F, f,F-lim)) not representable by a matrix method. G. G. Lorentz (cf. [1481) and subsequently other authors (see for instance [230], [229), [31] and [1781) have dealt with this question. For the moment, we just present these examples of summability meth-

ods. In the course of the book, particularly in Chapter 3, we will study many other examples of (classes of) summability methods.

Exercise 1.2.20. Verify the (strict) inclusions in (1.2:1) and (1.2:2).

Exercise 1.2.21. Give an example of a bijective map T : e --i by with pTxjjb = jjxjji (x E f) and determine its inverse map.

Exercise 1.2.22. Prove C cc, and limZ# x = lime, x (x E cz,) for cZi the matrix methods ZI and Cl (cf. 1.2.6 and 1.2.13). Exercise 1.2.23. Verify the following facts for almost convergence:

(a) x

((-1)k) E f and F-limx = 0.

(b) x = (Xk) := (1,0,-1,1,0,0,-1,1,-1,0,0,0,1,-1,1,-1,0,...) E f and F-limx = 0, but jxj := (jxkj) V f. (c) f C m fl cc,.

(d) T C fo := {x E f I F-limx = 0} . Hint: Show first T C coc,. Bibliography: [267], [196], [108], [216], [148]; [193], [230], [178]

1.3 Questions and basic notions Throughout this section let two summability methods V = (V, Nv, V-lim) and W = (W, Nw, W-lim) according to 1.2.10 be given (where they may be based on different notions of convergence). From the definition it is evident that the essential `entries' of a summa-

bility method (V, Nv, V-lim) are the domain NV and the summability map V- lim and that (for a given notion of convergence) these are essentially determined by the map V, which, in the case of matrix methods A, is the defining matrix A. Thus it may be fruitful and natural to ask for necessary and sufficient conditions on V (or the defining matrix A) that give a minimal or maximal size of the domain or also for the summability of a certain class of sequences. Moreover, it is interesting to compare two given summability methods, that is to compare their domains and summability functionals, respectively. Now, we define these questions more precisely, and introduce corre-

sponding notation and definitions, and-as far as possible-we shed light upon the questions and definitions by the consideration of examples.

Questions and basic notions

21

1.3.1. Inclusion theorems Question: Does D C Nv hold for a fixed D C w? Definitions: V = (V, NV, V-lim) is called conservative for null sequences if co C NV,

conservative if c c NV, strongly conservative if f C Nv, and coercive if m c NV. In the case of matrix methods we call the (defining) matrix conservative

for null sequences, conservative, strongly conservative and coercive, if the matrix method has the corresponding property.

Remark: Because co C c C f C m the notion of `conservative for null sequences' is the weakest condition and `coercive' is the strongest condition.

Examples: (a) All the examples considered in Section 1.2 are conservative, and thus conservative for null sequences (cf. 1.2.13, 1.2.14, 1.2.16 and 1.2.17).

(b) Almost convergence, that is (F, f, F-lim), Ci and Ai are strongly conservative since f C cc, C CA, (cf. Exercise 1.3.10 for the latter inclusion). I and ZZ are not strongly conservative and consequently not coercive. [To prove the last statement note that cj = c g f (cf. 1.2.18(a)) and that f ¢ czi holds since x (1, 0, -1,1, 0, 0, -1,1, -1, 0, ...) E f (I,1,-i0,.1,0,-2,0,0,-2,...) c.] (cf. 1.2.23(b)) and Z,x= 2 2 5 (c) The zero matrix is coercive (cf. 1.2.17(b)); Ci thus (F, f, F-lim) is not coercive. [To prove the last statement it is sufficient to verify m 5t cc, since f C cc,. For that we consider the sequence

k=4'

k=40

x=(xk):=(0,

1

k=42

1,1,1,0,...,0,

,0,0,

41-times

...,1,0,... 42-times

This is obviously bounded, but because n-1

[Cix]4°-l = 4n 1: 4k = 3 4n k=0

(n -+ oo)

3

and 1

[Cix]2.4°-i =

4n

*L

k

4n+i _ 1

E 4 = 3 . 2.4n

2

3

(n --+ oo)

k=0

it is not Ci-summable.]

0

In Chapter 2 we will characterize matrices which are conservative for null sequences, conservative, strongly conservative and coercive, respectively.

22

Convergence and divergence

1.3.2. Tauberian theorems Wanted: (a) Necessary and sufficient conditions for the V-summability (non-summability) of sequences.

()0) Determination of certain subsets L of w having the following property: x E L fl Nv x E c. A theorem of this type is called a Tauberian

theorem, and a condition, determining L, is called a Tauberian condition. (-y) More generally, as in (;Q), in the case of two summability methods V and W which satisfy Nw C Nv, we like to determine sets L of sequences with the following property: x E L fl Nv x E Nw. A theorem of this type is also called a Tauberian theorem, and a condition which determines L is called a Tauberian condition. 0 We will consider (a) in Chapter 2 in connection with factor sequences, and (0) in Chapter 4. As we have already observed and will state subsequently, two summability methods (V,NV,V-lim) and (IT',Nw,IV-lim) may be different in all three components or only partially. In the following we will focus our attention on the comparison of their domains and we will also compare their summability functionals pointwise. In the case of equal domains we see from Example 1.2.19(a) that, in general, the summability functionals are different. We now define precisely the questions connected with the comparison of summability methods, which we will examine later in the case of special as well as general (classes of) summability methods.

1.3.3. Comparison of domains Questions: (a) Are (the domains of) V and VV comparable; that is, does Nv C NW or Nw C Nv hold? (J3) Let L C w be given. Are (the domains of) V and W comparable relative to L; that is, does L fl Nv C Nw or L fl Nw C Nv hold? Definitions: If 0 $ L C w, then V is called stronger than W relative to L (and W is called weaker than V relative to L), if L fl Nw C Nv holds. By definition, V and W are equivalent relative to L, if Lf1Nv = L fl Nw. Instead of `equivalent' the notion equipotent is also used. In the case of L = w we speak simply about stronger, weaker and equivalent

(equipotent) whereas in the case L = m we speak of b-stronger, bweaker and b-equivalent. Further, V is called equiconvergent, if Nv = C.

A

Subsequently to 1.3.5 we will consider examples pertaining to 1.3.3 (and also to 1.3.4 and 1.3.5).

Questions and basic notions

23

1.3.4. Consistency of summability methods Questions: (a) Does V- lim x = W- lim x hold for each x E Nv n Nw?

(,3) Is V-limx = W-limx true for a given L C w and for every x E LnNvnNw? Definitions: Let 0 4 L C w be given. Then V and W are called consistent relative to L, if V- lim x = W- lim x for each x E L n Nv n Nw is valid. In the case of L = w we speak simply about consistency whereas, if L = m, we speak of b-consistency. 0 We emphasize a question which is already contained in 1.3.1, 1.3.3 and 1.3.4. Namely, from the fact that a summability method generates a notion of convergence, one may expect that the minimal size of the domain is c and that the method coincides on c with the `ordinary' notion of convergence.

1.3.5. Regularity Question: Do c C Nv and V- lim . = lim or at least co C Nv and V- lim x = 0 (x E co) hold?

Definitions: V is called regular for null sequences, if co c NV and V- lim x = 0 (x E co), and regular, if c C Nv and V- lim x = lim x

(x E c) are valid. d Before we consider some examples pertaining to 1.3.3-1.3.5, we point out that in applications of summability methods, non-regular, and even non-conservative, methods play an important role.

Examples 1.3.6. (a) Except for the zero matrix, which is regular only for null sequences, all examples considered so far are regular (cf. 1.2.13(a), (c) and (d), 1.2.14, 1.2.16(a) and (c) as well as 1.2.17 and 1.3.1).

(b) Zi is weaker than and consistent with C1 (cf. Exercise 1.2.22), but Z! and C1 are not equivalent (cf. 1.2.6(b)). (c) Almost convergence, that is the method (F, f, F-lim) in 1.2.17, is neither equivalent nor b-equivalent to a matrix method (cf. 1.2.18(b)). (d) Al is stronger than and consistent with C1 (cf. Exercise 1.3.10). (e) The matrix

/1

A:=

(-1

1

-1

1

a is conservative and regular for null sequences, but it is not regular. [Note

that Ax = (xk - xk-1) with x-1 := 0 holds for all x = (xk) E w. Thus c C CA and IimA x = 0 (x E c).] Moreover, A and Zi are not comparable [since x := ((-1)k) E cZ, \ CA and y :_ (k) E CA \ CZ, J. 3

E

In connection with the comparison and the question of consistency of summability methods (and the examination of other questions) it is very

24

Convergence and divergence

important to study as far as possible the structure of their domains. This study will be taken up in Part II.

1.3.7, Structure of domains Let a summability method of a distinguished type, for example a 'conservative matrix method', be given.

Questions: (a) Algebraic structure of the domain? (;0) Topological structure of the domain (with respect to various topologies)? A In general we will consider `linear' methods, that is the domain is a linear

space and the limit functional is linear. For instance, matrix methods are linear in this sense. So, the algebraic structure is clear. We will start in Chapter 6 with an examination of the topological structure of the domains of matrix methods. The results will prove to be very useful, for example, to obtain (under certain assumptions) general statements on the consistency of matrix methods.

1.3.8. Applications Summability originated in other fields of mathematics like function theory and applied mathematics, so it is important to point out applications of summability methods in other branches of mathematics; to stimulate the development of other fields in mathematics by using tools from summability.

Mainly in Chapter 2, but also in other chapters, for example in Chapter 11, we will deal with applications of summability. For further information the reader should particularly note the corresponding remarks in Sections 5.5, 10.5 and 11.6.

We will leave the incomplete list of questions at that and turn to the examination of matrix methods and other methods like the Abel and Borel methods.

Exercise 1.3.9. Let (ak) E w with ak _> 0 (k E N°) be given, and assume that the limit limt._41_, tER Ek aktk =: a exists. Show that then the converse of the statement in Abel's theorem holds. That is, Ek ak converges and Ek ak = a.

Exercise 1.3.10. Prove that the Cesaro method Cl is weaker than and consistent with the Abel method Al.

Exercise 1.3.11. Verify the regularity of the Borel method B1. (In particular, the Borel matrix B*t is also regular.) Bibliography: [267]; [196], [128]

Notes on Chapter 1

25

1.4 Notes on Chapter 1 More detailed historical overviews of the development of summability are given in various books. Here we draw attention to the book [216, Chapterl] of Shawyer and Watson and also to Hardy's book [108, I. Introduction]. At the end of Chapter 2 in the survey book [267] of Zeller and Beekmann there are references to papers and books containing historical material. Generally, the interested reader may find historical remarks in Zeller and Beekmann's book which gives a general survey of methods, problems and results in summability along with some history of the subject. It covers very well the period 1880-1968. Later Kangro extended this survey to cover the period 1969-1976 in his paper [123]. Besides the examples of (classes of) summability methods, introduced

in Section 1.2, there are many other types of summability methods. In Section 7.2 we will touch on strong summability which was introduced in the case of the Cesaro method in connection with Fourier series by Hardy and Littlewood [109] and Fekete [81]. Almost all papers concerning strong

summability are connected with Fourier series or orthogonal series9 (cf. [267, Chapter 6] and [123]).

Another regular summability method introduced by H. Fast (cf. [77]) and which is not equivalent to any regular matrix method is called statistical convergence. During the last decades statistical convergence played an important role in the literature and was investigated by J. Fridy (cf. [85]) and in joint papers with H. I. Miller (cf. [86]) and C. Orhan (cf. [87]), by J. Connor (cf. [67], [60], [61]) and in joint papers with J. Fridy and J. Kline (cf. [64],[65]), by J. Maddox (cf. [160], [161]) and by E. Kolk (cf. [129], [130], [131]). Using regular matrices with non-negative entries the

concept of statistical convergence leads to T-statistical convergence. An overview of the theory of statistical convergence is given by J. Connor in [63]. For further remarks concerning T-statistical convergence see Section 10.5.

Other keywords for (classes of) summability methods are, for instance,

absolute summability, intersection method and union method (cf. [267, Chapters 6 and 7]). Because we restrict our interest to sequences in K we do not deal with summability methods for multiple sequences. In the case of double sequences these are very often matrix methods defined by four-dimensional matrices and based, respectively, on the notion of convergence due to Pringsheim and to Hardy (cf. [267, Chapter 8]). For information on other questions, problems and applications we refer the reader to the books [267], [123] and [108].

9To get an overview in the database ibfathSci use the keywords strong summability, or the German keywords starke Limitierbarkeit and starke Summierbarkeit.

2

Matrix methods: basic classical theory As we have already mentioned, in this and the following chapters we will deal mainly with matrix methods, in particular with their domains and the notion of convergence generated by them. Corresponding to the questions formulated in Section 1.3 we aim in Chapter 2 to give the most basic inclusion, comparison and consistency theorems. We will make use exclusively of classical (analytical) methods; some of these results we will re-prove in Parts II and III by functional analytic methods. First of all, in Sections 2.1 and 2.2 we develop useful tools for dealing with series and infinite matrices.

After that we turn our attention to matrix methods and present inclusion theorems in Sections 2.3 and 2.4. In 2.3 we characterize, in the Toeplitz-Silverman-Kojima-Schur theorem, matrix methods which are conservative for null sequences and those which are conservative. In addition, we characterize matrix methods which are regular for null sequences or which are regular. Section 2.4 includes the characterization of matrix methods which are coercive (Schur theorem) or which are strongly conservative. All these characterizations have in common that the corresponding inclusion is characterized entirely by conditions in terms of the defining matrix and that in each case the non-trivial part of the proof is based on the `gliding hump method' which is an essential tool in summability, functional analysis and other fields of mathematics using analytic methods. In Section 2.5 we prove-again by applying the gliding hump method and by the consideration of `factor sequences'-that the domain of a matrix method contains a extensive `parcel' of sequences, when the domain contains sequences having certain properties. In this connection we speak about `abundance within domains'. As an immediate corollary of these considerations we find that a conservative matrix method sums an unbounded sequence if it sums at least one bounded divergent sequence. As a further corollary we get in Section 2.6 the bounded consistency theorem, due to Mazur, Orlicz and Brudno, which tells us that regular matrices A and B are b-consistent, when B is b-stronger than A. Also we prove in 2.6 some simple, but very useful, results on the comparison and consistency of matrix

Dealing with infinite series

27

methods. We continue the consistency considerations in 2.7. Motivated, for instance, by the bounded consistency theorem, we look for a class of regular triangles A such that each matrix, which is stronger than A, is consistent

with A. This leads us to the class of all matrices of `type M' which we investigate in this section. In Section 2.8 we deal with the `mean value property' of matrices which turn out to be sufficient for the property of type M in the case of regular

triangles. Since, in the case of special matrices, it is not easy to verify directly the definition of the mean value property, we give necessary and sufficient conditions for the mean value property.

Hahn's theorem (cf. 2.4.5) tells us that a matrix which sums all sequences of zeros and ones sums all bounded sequences. In this connection

the question arises whether for a given matrix A another matrix B has to be b-stronger than A, if B sums all sequences of zeros and ones which are summable by A. Such matrices A are called `potent'. In Section 2.9 we give necessary and sufficient conditions for matrices to be potent. We will continue in Chapter 3 the study of the potency of special (classes of) matrices.

2.1 Dealing with infinite series In the sequel, for example in connection with matrix methods, we have to deal with infinite series. For that purpose, in this section we present Abel's partial summation formula and some convergence tests which follow from it. Furthermore, we discuss the major rearrangement theorem which contains as a special case Cauchy's theorem for double series. First of all, we turn to Abel's partial summation formula and Abel's test.

Theorem 2.1.1 (Abel's partial summation formula). Let (av), (bv) E w. xn := _V_° av (n E 1`V° } and x_1 := 0. Then the equality ntk

n+k

E avbv = E xvbv - by+1) - xn-i bn + xn+kbn+k+1 v=n

(2.1:1)

v=n

(Abel's partial summation formula) holds for all n, k E N. If (xvbv+l) E c, the series F_v a,bv converges if and only if the series >v x, (bv - by+i) does, that is (avbv) E cs

(xv(bv - by+i)) E cs

(Abel's convergence test).

Proof. Because av = xv - xv_1 (v E N°) we obtain (v E NO), avbv = xvbv - by+1) - xv-lbv +xbv+l which implies (2.1:1) by summation. Abel's test is an immediate consequence of (2.1:1) if we note that (xvbv+l) E c and consider the case n := 0 0 and let k -* oo.

28

Matrix methods: basic classical theory

As special cases of Abel's test we get some convergence tests which are sufficient for series of the type F,v avb,,. (Of course, we may write every series in the form Ev avbv, but the applicability of the tests depends on the clever choice of the factors av and bv.)

Theorem 2.1.2 (Abel's test). If (av) E cs and (bv) is bounded and monotone', then (avbv) E cs.

Proof. By assumption

E c where x := L.v-oav (n E N°). More-

over, (bv) E c since (bv) is assumed to be monotone and bounded. Hence (xvbv+i) E c and, because Fv-o(bv-br}1) = bo-b,,+1 (n E N°) and (bv) by}1) converges absolutely. is monotone and bounded, the series 0 converges, and (avbv) E cs by 2.1.1. Thus, E, xv(bv -

Theorem 2.1.3 (Dirichlet's test). If (av) E bs and (bv) is a monotone null sequence, then (avbv) E cs.

Proof. Clearly (xvbv+l) E co where xn := Ev=0av (n E N°), and the convergence of Ev xv (bv - by+1) follows as in the proof of 2.1.2. Now 2.1.1 0 again implies (avbv) E cs.

Theorem 2.1.4 (Du Bois-Reymond's/Dedekind's test). (a) If (bv) E by and (av) E cs, then (avbv) E cs. (b) If (bv) E bvo and (av) E bs, then (avbv) E cs.

Proof. Let xn :=

v=0 av (n E NO). As in the above proofs it is easy to deduce the convergence of Ev xv(bv - by+1) and of (xvbv}1) from the 0 hypothesis of (a) and (b). Now (2.1:1) yields the result. One of the aims of this section is to prove Cauchy's theorem for double series (cf. Theorem 2.1.11). We begin by discussing the effect of rearranging the terms of a series.

Definition 2.1.5 (rearrangement). If v : NO -* NO is a bijective map, then for a given series Ev av the series Ell a,(v) is called a rearrangement of it. Theorem 2.1.6. If Ev av is absolutely convergent with sum s := Ev av, then each rearrangement Ev a,(v) is absolutely convergent with sum E. aa(v) = S.

Proof. Let x and y, be the ntt' partial sums of E, av and Ev a,(v), respectively, and let xn and y be those of F_v (avI and Ev la,(v)I. First is monowe prove (a,(,)) E t which is equivalent to (yrz) E m since i v E 1"12, we see that tonically increasing. Putting N := max {o (v) 0 < yn G 2N. Hence E m because (i,_) E m. Thus E. a,(v) converges (absolutely) with sum t say. It remains to show that t = s. To 1 The monotonicity of 2 NO

includes b E R.

Dealing with infinite series

29

that end, for given n we set no = min I k E No I C {o (O), ... , o(k) } } . Obviously n < no. Thus the finite sum yno - xn contains only, terms a I

with v > n. Hence jyno - xnj < F_'.+, la,j, so jS - tj

<

18-xnj+jxnYnoj+jyno-tj

<

js - xnj+jyno -tj+

lad -+ 0 as n -aoo. v=n+1

0

This gives s = t.

As a consequence (cf. Corollary 2.1.8) of our next theorem we will see that E,, a, converges absolutely if each rearrangement Ev a0.(v) converges (not a priori absolutely).

Theorem 2.1.7. Let (av) be a real sequence with E cs \ 2. (a) For each s E IR there exists a rearrangement > a,(,,) of EV a, such that Ev a,(,,) = S.

(b) For s = +oo and s = -oo there are rearrangements 57,

and

which diverge definitely to +oo and -oo respectively. Combining Theorems 2.1.7(b) and 2.1.6 we obtain the following corollary, which holds for av complex as well as real. The proof is left to the reader.

Corollary 2.1.8. A series converges absolutely if and only if each rearrangement of it is convergent.

Proof of 2.1.7. For

E cs \ e we build the subsequences (cv) from the non-negative terms and the negative terms of

and respec-

tively. The series J: by and F_v cv diverge definitely to +oo and -00 respectively since otherwise we would have sumption.

E P contradicting our as-

Now, let s E R be given. Because F_ b = +oo and F-v c = -oo we can define

n-1

nl=min n>2 j and then

( 2.1:2)

Eb,, >s

nl-1

n-1

_°

v=o

kl=min n>2I

(2.1:3)

We define inductively index sequences (nr) and (kr) with no = ko = 0, n1 and k1 given by (2.1:2) and (2.1:3) respectively, and assuming n, and k, to be defined, we set r

n-1

nr+1=min n>nr I E E bv+ µ=1

c

E v=n,.

30

Matrix methods: basic classical theory

s

nr + kr_1 - I

nr + k,. - I

n,.+1 + kr - 1

nr+1 + kr+1 - I

Fig. 2.1.1: Illustration of the choice of the index sequences and k,-1

n,,-1

kr+l = min

n > kr

E µ-1

v-kY_

v=np_1 nr+1-1

1

/

n-1

+ Y' by + v=nr

cv < s

v=kr

))1!!1!

See Figure 2.1.1 for an illustration of the choice of these index sequences. On the vertical axis the sum s is marked, and we indicate how EnV=0 dv varies for nr + kr_1 - 1 < n < nr+1 + k,.+1 - 1 where (dv) := (bo,...,bn,-1,CO,...,Ck1-1,bnl,...,bnz-1,Ck1.... ).

We now prove that the series Fv d,,, which is obviously a rearrangement of F_v a,,, converges to s. Using dn =

Jby withnr_1 :5 v0 cv with k,._1 < v < kr if do < 0,

where r is suitably chosen, we obtain for the partial sums sn :_ J:u=o di., (n E N°) the estimation ((cf. Figure 2.1.1)

max{lC-, -11. Is - snI < 4 m ax{bnkr_1, Ick

if d

if do < 0

and therefore sn -+ s since (a,,) E co; thus (be,) E co and (cv) E co. Altogether we have proved the theorem in the case of s E lit

Dealing with infinite series

31

If s :_ +oo we modify the choice of do as follows. First take terms bo, ... , bn, until the sum bo + ... + b,, exceeds 1 for the first time. Then take co. Next take terms bn,+i,... ,bn2 if necessary until the sum bo + ... + bn, + co + bn,+1 +... + bn2 exceeds 2 for the first time. Then take cl continuing in this way and using the fact that E., b, = oo and that (cu)

is a null sequence, we obtain a rearrangement E1 a,(v) of F, a with

E

oo. The case s = -oo is similar.

As a further corollary of 2.1.6 we get a theorem on the convergence of the product of two series.

Theorem 2.1.9 (product of series). Let E a and E. b be absolutely convergent series with sum a and b, respectively. If (c,) is any arrangement of the products anbk (n, k E N°) into a sequence, then the series F_,, c, is absolute convergent with the limit ab. In particular, if we arrange `in diagonals', then

(av) (b)

_

C

E akbn-k)

where the absolute convergence of the series on the right hand side is as-

sured and the sum is ab. This series is called the Cauchy product of

Ev a and E b,,.

Proof. Let E e, F_, la,I =: a. r b, and let (cv) be an arbitrary arrangement of anbk (n, k E N°) into a sequence. First we prove (cu) E e. To that end let p E N° be given and let A denote the largest of the indexes n or k in the products anbk determining co, ... , cµ. Obviously,

E Ic-I <

l(EIb')

< ab;

thus (cv) E f. Now, we have ab = F c,, from Theorem 2.1.6. Therefore we can choose a special arrangement of anbk (n, k E N°) into a sequence. Let do := aobo , d1 := a,bo , d2 := aob1 , d3 := a,bi,

that is (ao + al)(bo + bl) = do + dl + d2 + d3.

Continuing in this way we get

=do+...+d(n+1)2-1 Thus ab d if n tends to oo. In particular, we obtain the equality (Ek=o akbn_k) = ab for the arrangement `in diagonals'. En

32

Matrix methods: basic classical theory

In the last'part of this section we deal with a major rearrangement theorem which contains as special case Cauchy's theorem for double series. The latter gives us sufficient conditions for the convergence of double series =o a,,

and tells us that-under suitable assumptions-convergence

does not depend on the order of summation. This means for example that we get the same convergence behaviour and the same sum if we write the coefficients of the double series as an infinite matrix and `sum' first the coefficients in the rows and then add the row sums, or if we `sum' first the coefficients in the columns and then add the column sums. That is, under suitable assumptions, we have E E avµ = E E avµ)

which means that the double series are both convergent and their limits are equal. First of all, we give a very general rearrangement theorem.

Theorem 2.1.10 (major rearrangement theorem). Let I be an arbitrary countable index set and let (Io,1i, I2....) be a partition of I (that

v#µand v,pEN°).

is, I=UvIv, I"0- O (vEN°) and

Let (ai i E I) be a family of numbers in K such that !

sup tiEJ E jail

i J C I , J is finite } < oo,

(2.1:4)

J

and for each v E No let (cvo, CO, cv2, ...) be any arrangement of all ai (i E Iv)

(2.1:5)

into a (finite or infinite) sequence. Then the sums3 cv := >u cv, exist for v E N°, and the series Ev c,, converges absolutely. Moreover, the sum s := EL, c,, = F_,, . cvu, is independent of the chosen partition I and the arrangement in (2.1:5). is an arrangement Note, if each I is an one-element set, then of ai (i E I) so that we have the situation of Theorem 2.1.6.

Proof of 2.1.10. Let (bk)kENo be an arrangement of (ai

C

i E I) into a

sequence. Because of (2.1:4) the series Ek bk is absolutely. convergent and

its limit r := Ek bk is independent of the chosen arrangement of ai by Theorem 2.1.6. Thus s, defined in the theorem, is proved to be independent of the chosen partition of I (and from the arrangement in (2.1:5) if we can verify r = s). To that end we consider an arbitrary partition (I,, I v E NO) of I and for every v E N° an arrangement of all ai (i E Iv) into a (finite or 3 As a finite sum or limit of the series under consideration which is absolutely convergent because of (2.1:4).

Dealing with infinite series

33

infinite) sequence (coo, cvli ...) as formulated in the theorem. Then >µ evµ is for each v E No either a finite sum or, by (2.1:4), an absolutely convergent series, and in both cases we denote the value (sum or limit) by ev. To prove

the convergence of Ev cv and r = s := F_v cv let e > 0 be arbitrarily given. Since (bk) E f we may choose a K E No with Ek K+1 (bkj < e and then an N E No such that bo,... , bK are contained in the families (at i E Iv) with v E N. Since the series F,µ evµ (v E N°) and Ek bk I

are absolutely convergent we obtain for each n > N that n

r -

Cv L,=o

k

CO

bk - E Cl, v=°

< E JbkJ < 6k=K+1

Thus Fv cv converges to r, so r = s. Moreover, the convergence of Fv cv is absolute since, as we may easily show, _v =o icv4 < Ek JbkJ < oo holds 0 for each n E N°. As already mentioned above, our next theorem, Cauchy's theorem for double series, is a special case of the major rearrangement theorem and we will apply it several times throughout the book.

Theorem 2.1.11 (Cauchy's theorem for double series). Let avµ E K (v, µ E N°) be given such that supra Eµ =o v=o Iav,I < oo. Then all of the series E. avµ (v E N°) and Ev avµ (p E N°) converge absolutely. If av = E. avµ and bµ = F. avµ, then F-v av and Eµ bµ converge absolutely and we have

av=EEavµ = v

µ

bµ

avt' _ µ

v

µ

for the limits.

Proof. In Theorem 2.1.10 we take I := No x No as our countable index set and consider the partitions (Iv I v _E N°) and (Iv 1 v E N°) of I defined by Iv := { (v, n) I n E No } and Iv := { (n, v) it E No } (v E NO), respectively, with the natural ordering. Now, it is an easy exercise to check I

that Theorem 2.1.10 is applicable to the present situation and that the desired results follow.

0

Exercise 2.1.12. The well-known Leibniz test tells us that ((-1)"bo) E cs if (bv) is a monotonically decreasing null sequence. Give a proof of that theorem by applying one of the convergence tests 2.1.2-2.1.4. Exercise 2.1.13. Use the convergence tests in the present section to prove the following statements:

(a) If (av) E cs, then (a,,+ ) E cs. (b) If (a,,) E cs, then the power series Eo avty converges for t E [0, 1]. (c) Let (av) E w and to E ]0, oo[ be given. If E. avv-t0 converges, then

34

Matrix methods: basic classical theory

Ev avv-t converges for every (real) t > to. Series of the form F_ are examples of Dirichlet series.

Exercise 2.1.14. Let (bv) E w,

E cs and z

avv-t

ak (v E No).

Apply Abel's partial summation formula to prove that n+k

n+k

E avbv = E zv(bv -

znbn_1 - zn+k+lbn+k

(n, k E N°).

v=n

v=n

Bibliography: [128]

Dealing with infinite matrices

2.2

As we mentioned above, matrix methods will play an essential role through-

out. We must therefore understand how to work with infinite matrices. Calculations with infinite matrices are quite different from those with finite matrices. We use this section to introduce some of the most basic ideas.

Throughout this section let A = (ank)n,kENo, B = (bnk)n.kENo and C _ (cnk)n,kENo be infinite matrices with ank, bnk, cnk E 1K (n, k E N°).

Definition 2.2.1 (addition and multiplication by scalars). In the case of infinite matrices addition and multiplication by scalars are defined componentwise analogously to the case of sequences (cf. 1.2.2). More precisely

A + B := (ank + bnk)n,kENo and AA := (Aank)n,kENo

(A E K).

With these definitions of addition and multiplication by scalars, the set of all infinite matrices is a linear space over K IL Formally the product of infinite matrices is defined similarly as in the case of finite matrices. However, because the formal definition of the product of infinite matrices involves infinite series, the existence of the product

is not assured for arbitrary matrices A and B. The same issue arises in the case of the product of a matrix and a sequence. We make the following definition.

Definition 2.2.2 (products). Let x = (xk), y = (Yk) E w and A = (ank) and B = (bnk) be infinite matrices. We define

yx E ykxk

(scalar product of sequences),

(2.2:1)

k

Ax :=

(ankxk) k

and yB := (Ynbnk) nE FI°

n

(2.2:2) kENo

(product of a matrix and a sequence),

Dealing with infinite matrices

AB :_ (Cnk) where cnk

35

(2.2:3)

an.,,bvk (n,k E N0)

(product of matrices), provided all of the series appearing in (2.2:1), (2.2:2) and (2.2:3) converge.

In such a case we say that yx, Ax, yB and AB exist or are defined.

The following remarks apply to products of matrices. There are, of course, corresponding results for the other products defined in 2.2.2.

Remarks 2.2.3. (a) If AC and BC exist, then so does (A + B)C and (A+B)C = AC+BC holds. The corresponding statement holds for A(B+

C) when AB and AC exist (distributive laws). (b) Just as with finite matrices, the product of infinite matrices is not, in general, commutative. That is, there exist matrices A and B with AB # BA. Moreover, we can choose A and B so that AB exists and BA does not. An example of this is obtained by taking A and B to be the matrices B and C respectively that are given in the following part (c). Then AB does not exist while BA does exist. (c) In contrast with finite matrices the associative law is not true in general as the following example proves. If

A:=

0

0

0

0

0

,

B:=

and

then AB =

1

0

1

0

0

0

0 x

1

0

and (AB)C = 0. However, BC, and thus A(BC), does not exist (whereas CB does), as we may easily verify. As we will see in 8.5.7, it is also possible

to find matrices A. . B and C for which all of the products AB, BC, (AB)C and A(BC) exist but still (AB)C A(BC) holds. d Since associativity plays an important role, we give in the next three theorems broadly applicable sufficient conditions that ensure associativity for various types of products that arise.

Theorem 2.2.4 (associativity of t(Bx) ). Let B be an infinite matrix and x = (xk),t = (tk) E w. If (i) x E wn and t e ca or

36

Matrix methods: basic classical theory

(ii) t E 1, x E m and IIBII := sup,, E,, Ib,,,l < oo

is valid, then t(Bx) and (tB)x exist and t(Bx) = (tB)x holds. Proof. (i) Now t(Bx) = F_u t, F_Y b,,,x,, = Eµ Ell tIAb,,vx exists since it is, with t E ,p, a finite sum of convergent series. Thus we may interchange the order of summation to obtain

t(Bx) = E E µ

E x E t,b, = (tB)x.

V

V

(2.2:4)

µ

(ii) The product Bx exists because N

E IbuYxvl <_ Iixll. IIBII < 00

(µ, N E N°),

v=o

and tB exists since N

It,bkvl <_ IIBII IIthII < 00

(v,N ENO).

IA_°

Furthermore, for all N E N° we have

NN E I tb,,,,xI <_ IIxII. IIBIIIithI; < oo. p=O V =O

Thus all series which appear in the (formal) product t(Bx) converge absolutely. Now Cauchy's theorem for double series (cf. 2.1.11) shows that 0 (tB)x exists and the identities in (2.2:4) hold. As an immediate consequence of 2.2.4, we get in Theorems 2.2.5 and 2.2.6 corresponding results for products A(Bx) and A(BC).

Theorem 2.2.5 (associativity of A(Bx) ). Let A and B be infinite matrices and x = (xk) E w. If (i) x E WB and A = (ank) is row-finite (that is, (ank)k E cp (n E N°)) or {ii) x E m, IIBII < oo and (ank)le E 1 for each n E No holds, then A(Bx) and (AB)x exist and A(Bx) = (AB)x. Proof. This is an immediate consequence of 2.2.4 since the nth coefficient 0 (A(Bx)}n of A(Bx) is t(Bx) where t is the nth row of A.

Theorem 2.2.6 (associativity of A(BC) ). Let A, B and C be infinite matrices. If

(i) BC is defined and A is row-finite or (ii) IIBII < oo, (c k) E m (k E N°) and

E t (n E N°)

Dealing with infinite matrices 37

holds, then A(BC) and (AB)C exist and A(BC) = (AB)C. Proof. This theorem is also a corollary of Theorem 2.2.4 since the coefficient {A(BC))nk of the matrix A(BC) is t(Bx) where t and x denote 0 the nth row of A and kth column of C respectively.

We close this section with a discussion of right and left inverses of infinite matrices.

Definition 2.2.7 (inverse). Let A and B be infinite matrices. If AB exists and AB = I, then A is called a left inverse of B, and B is called a right inverse of A. If in addition BA exists and AB = BA = I holds, then the matrix B is called a bi-inverse or simply inverse of A. The inverse of A, if it exists, is denoted by A-'. For a large class of infinite matrices the existence and uniqueness of the right inverse are assured as the following theorem shows.

Definition 2.2.8 (triangular matrix, triangle). A matrix A = (ank) is called a (lower) triangular matrix if ank = 0 (k, n E NO with k > n). A lower triangular matrix A = (ank) with ann # 0 (n E N°) is called a

triangle or normal matrix. Theorem 2.2.9 (triangle). If A is a triangle, then the following statements hold: (a) For each y E w there exists a unique solution of the system of equations Ax = y.

(b) There exists a unique right inverse B of A. Moreover, B is also a triangle and a left inverse. So A-' exists. (c) The matrix A may have more than one left inverse, but there is exactly one that is also a triangle, namely A-'. Proof. We sketch the proof and leave the mechanical details to the reader.

(a) Let A = (ank) be a triangle and y = (yk) E w be given. There is a solution x` = (x *n) E w of the system Ax = y given by the recursion formula n-1

x0 =

YO

a00

and

xn =ann- yn -

ankxk

for each n > 1 (2.2:5)

k=0

which determines uniquely a solution x*. (b) For j E N° consider the system of equations Ax = ej where ej = ((Sjn) is defined as in 1.2.3. (Note, ej is the jth column of I.) By part (a), there is a unique x(j) = (xnj)) E w with Ax(j) = ej. Let X := (x(nj))n,jENo be the

matrix obtained by taking x(j) as its jth column. It follows immediately (see (2.2:5)) that X is a triangle. Therefore the products X A, AX, A(XA)

and (AX)A exist and A(XA) = (AX)A = A holds. By part (a) this

38

Matrix methods: basic classical theory

implies XA = I because Aek = (a,,k)fENo for every k ENO. Therefore, X is also a left inverse of A. (c) Let X be the left inverse of A determined in part (b), and let Y be a left inverse that is also a triangle. Then Y = Y(AX) = (YA)X = X (note, all products exist since all of the matrices A, X and Y are triangles). On the other hand, it is easy to check that the matrices

and C :_

E :_

0

-1

0

0

-1 -1

0

0

0

-1 -1 -1

(2.2:6)

are left inverses of the triangle

/1 -1

A:= 1

1

-1

1

(2.2:7)

.l

t0

and that for each A E [0, 1] the matrix AE + (1 - A)C is also a left inverse

0 of A. Note that, of course, AE = I, but AC # I. Remark and Notation 2.2.10. Let the matrices A and E be given by (2.2:6) and (2.2:7). Then E is called a summation matrix. Since E is the unique (triangular) inverse of A, we fix for the sequel the notation E-1 := A for the matrix A. d Exercise 2.2.11. Examine whether in the following examples the products t(Bx) and (tB)x exist and whether they are equal. Furthermore, determine the value of t(Bx) in such a case. (a) t:= (1, -1,1, -1, ...) , x := (1,0,2,03,0,...), 0

1 1

B

0 0

1 1

0 0

... ...

0

1

0

...

1

0

...

0

...

0 (b) t

(0, 1, 0, 14 0, 91 0,

x := /1- '-2 1- 1,1 - 14 3

.

/2-1 0

B :_

2-2

2-3

2-4

0

0

0 3-4

...

3-1

3-2

3-3

4-1

o

0 4-2

4-3

4-4

0

0

0

0

0

0

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

.

Conservative matrix methods

39

Exercise 2.2.12. Determine the right inverse of the matrices Ci and Z1. Bibliography: [267], [68]

2.3 Conservative matrix methods Throughout this section let A = (ank) be an infinite matrix with coefficients in K, and let WA be the application domain and CA the domain of A. We intend now to characterize, in terms of the matrix coefficients, those matrices which are conservative, those which are conservative just for null sequences, those which are. regular, and those which are regular just for null sequences. Concerning the notation, we follow our customary `abuse of terminology' (see Definition 1.2.12) by identifying the matrix A with the `matrix method' it defines. Obviously, C C WA is necessary for A to be conservative, that is the series F_k ankxk (n E N°) are necessarily convergent for each x = (xk) E c,

or-as one says-for each n E No the sequence (ank)k is a convergence factor sequence of c (in cs).

Definition 2.3.1 (convergence factor sequence, 3-dual). For X C w with X $ 0 we define4

Xa:={t=(tk)EW Vx=(xk)EX: tx:=(tkxk)Ecs}. Then Xa is called the 3-dual of X or the set of all convergence factor sequences of X (in cs). For y E w, we write ya instead of {y}a. Remarks 2.3.2. Let 0 # X C w and A = (ank) be an infinite matrix. Then the following statements are valid:

(a) X3 is a sequence space with p < Xa < w.5 (b) X C Xaa := (Xa)d. (d) = w and wa = (p. (e) X C WA i d n E l ` 1 ° X X

converges.

(ank)k E X°. Proof. We will just show wa = gyp. The rest is an immediate consequence of Definition 2.3.1 and is left to the reader. Using (a) we need only show V n E No

:

if ws C V. So let t = (tn) E w3. Let y = (yn) be given by y, to # 0 (n E N°) and yn := 0 otherwise. Clearly En tnyn = En.a #o I.

Thus ty E cs implies t E V. In particular, the statement (e) tells us that co C WA, C C WA and m C WA are characterized in terms of the matrix A if the 0-duals of co, c and m, respectively, are determined. 4 Note, we use the notation tx for both the sequence (tkxk) and the sum Ek tkxk.

We call (tkxk) the coordinatewise product of t and x. 5'<' stands for `is a linear subspace of'.

40

Matrix methods: basic classical theory

Theorem 2.3.3. (a) co = co = m'3 = P. (b) PF = m.

(c) T'3=X0=mo =P. Proof. (a) Obviously mQ C c'3 C co/3 by 2.3.2(b). So (a) is proved if we show t C m'3 and c0'3 C P. Now, let t = (tk) E P and x = (xk) E m be given. Then n

(n E N°),

E Itkxkl < IIxII.IItIIi < oo k=0

which shows tx E P C cs. Thus t E m'3 so P C m'3. On the other hand, for a given t = (tk) E w \ P we prove the existence of an x E co with tx cs, which implies t 0 co ; thus, altogether, co' c P. in NO with Because t P we may choose an index sequence6

no=0 and

Itkl>v (iEN). 1

If we define x = (xk) E co by7

k < n and v E N),

xk := 1 sgn tk v

then we get n -i

tkXk

1

E N);

k=n,_1

therefore tx cs and thus t co . This proves (a). (b) To show 113 = m, we note from 2.3.2(c) that m C (m')Q = Pa since

m,3 = P. Now assume the existence of a t = (tn) E Pa \ in. Since t

is

unbounded there exists a subsequence (tn,,) of (tn) with for each k E No.

Itn, I > (k + 1)2

The sequence (xn), defined by xn := k+i) sgn tn,, if n = nk (n, k E No) and 0 otherwise, is obviously a member of P. But

tnxn = Hence t

(k +)2 >

1 = 00.

0, which contradicts our assumption.

6 An index sequence in NO is a strictly increasing sequence in NO. Recall that if z E C then sgn z := F -I if z ¢ 0 where T is the conjugate of z. 0

ifz=0

Conservative matrix methods

41

(c) Using (a) and the fact that T C X C m° C m we get e = m'3C ma C Xa C TO. The following proof of the converse inclusion is non-trivial and is based on an idea due to Kuttner and Maddox (cf. [138, proof of Theorem 2]). Now, let y = (Yk) V P be given, that is, Ek (ykI = oo. Without loss of generality we may suppose that y is a real sequence and that

yk = oo where yk := max{0, Yk}. k

Put k° := -1 and define inductively an index sequence (kp) such that kp+l

kp+, -P

- kp+p+1 and

yk>p+1

(2.3:1)

k=kp+1

-

and such that there are at least p + 1 values of k with kp < k < kp+1 p for which yk > 0. We divide those values into p + 1 sets, say Fp1i Fp2, ... , Fp,p+1, by putting the first into Fp1, the second into Fp2, ... , the (p+ 1)th into Fp,p}1, the (p+2)th into Fp1 and so on. For at least one i we must have on account of (2.3:1),

Y yk= kEFpi

11

yk>1.

kEFp;

We choose for each p E No one such ip and put U Fpi,, _: { yr I r E No } P

where v,. < v,.+1 (r E NO). Because, by choice of the sets Fpi,, , the distance between any two distinct elements of Fpi,, is greater than p + 1, the index

sequence (v,.) satisfies v,.+1 - v,. -+ oo (r -> oo) and consequently it defines uniquely a thin sequence x = (xk) with

E YkXk = E yv,. = E k

r

L,

yk = 00,

P kEFp;,,

that is y 0 TO, which proves t D TO.

0

As a corollary of 2.3.2(e) and 2.3.3(a) we get the desired characterizations of CO C WA, C C WA and M C WA in terms of the matrix A.

Corollary 2.3.4. Let A = (ank) be any infinite matrix. Then E Iankl < co, that is (ank)k E f for each n E NO k

if and only if one (thus each) of the inclusions m C WA, C C WA, CO C WA, MO C WA, and T C WA holds.

42

Matrix methods: basic classical theory

If A is conservative, then c C WA and, a fortiori, A maps each convergent sequence in a bounded sequence. That is,

C C WA and A(c):= {Ax jx E c} C m, or equivalently, c C mA := {x E WA I Ax E m} . We now characterize the last property.

Theorem 2.3.5 (c C MA). For any matrix A = (ank) the following statements are equivalent: (a) m C mA, that is m C WA and A(m) C m. (b) c C mA, that is cCLOA and A(c) C m. (c) CO C mA, that is CO C WA and A(co) C m. Supn Lk lankl < co. (d) IIAII The implication `(c) . (d)' is the only non-trivial implication, and we

establish it by using the so-called gliding hump method, even though it may be established more easily by applying the uniform boundedness principle (see, for example, Section 7.4). There are situations, however, where the gliding hump method works but the uniform boundedness principle cannot be applied. Thus we take this opportunity to demonstrate the gliding hump method in this relatively straightforward case.

Proof of 2.3.5. The implications (a) = (b) . (c) are obviously true. (d) = (a) : If IIAII < co, then we get m C WA from Corollary 2.3.4 and

E ankxk S llxll.IlAll < oo (n E N° and x = (xk) Em). k

Thus Ax E m as required. (c) = (d) : Let co C WA and A(co) C m. In particular, Ae' = (ank)n E

m (k E N°) and (ank)k E t for each n E N°. Thus, for each r E N° we may choose an M(r) > 0 with Iankl < M(r)

(n E N°).

(2.3:2)

k=0

We also have, by hypothesis, that

E Iankl < OG

(n E N°).

(2.3:3)

k

If we assume that II-4II = sup 57 Iankl = co, n

(2.3:4)

k

we can establish the existence of an x E co with Ax 0 m which contradicts the statement A(co) C m. We now inductively choose index sequences (np) and (kp). We put k_1 := 0 and pick an no E No with

Conservative matrix methods

E lank) > M(k_1) + 1

43

[cf. (2.3:4)]

k

and then a k° E N° with ko > k-1 and 00

E lankI < 1

[cf. (2.3:3)].

k=ko+1

If nP_1 and kp_1 have been chosen for some p > 1, then we can choose an np E No with np > np_1 and E IanPkl > (p+ I)M(kp_1) +p2 + 1

(2.3:5)

[cf. (2.3:4)].

k

After that we can choose kp E No with kp > kp_1 and 00

Y" IanPkl < 1

[cf. (2.3:3)].

(2.3:6)

k=k,+1

Applying (2.3:2) and (2.3:5) and (2.3:6) we have for any p E N that k,

kP-1

CO

Ian,kI - E IanPkl - E IanPkl

IanPkl = k=k,_1+1

k

(2.3:7)

k=0

k=k,+1

> (p + 1)M(kp_1) + p2 + 1 - 1 - M(kp_1)

= pM(kp_1) + p2.

The choice of the index sequences (np) and (kp) is illustrated in Figure 2.3.1, where the matrix A and the `gliding hump' are shown schematically. Now, the chosen index sequences (np) and (kp) enable us to define an x E co with Ax in. Define x = (xk) E co by xk

Then for p > 1,

if0
1

(2.3:8)

if kp_ 1 < k < kp (p = 1, 2, ...).

sgil an,k g kP

E an,kxk

L an,kxk -

>

k=kp-,+1

k

kP

>

1

L IanPkl -

p k=k,-i+1

cc

kP-1

lan,kxkl -

Iaf9kxkl k=kp+1

k=0

kP -1

00

IanPkl

IanPkl k=O

k=k,+1

[using (2.3:8) and IIxIIx < 1]

>

M(kp_1) + p - M(kp_1) - 1 = p - 1 [by (2.3:7), (2.3:2) and (2.3:6)].

Hence, Ax V m.

0

44

Matrix methods: basic classical theory ky

0 0

np(2.3:7)

ny (2.3:2)

(2.3:6)

<,1f{ky_,)

<

>{y+I)b1(ky)+(y+i), (2.3:7)

n,+ 1

(2.3:2)

(2.3:6)

E

<1

n',+ 2

hump 'gliding' to +x if p is increasing.

Fig. 2.3.1: Illustration of the `gliding hump' in the proof of 2.3.5

Next we characterize the matrices A for which co C CA and those for which CO C cOA; that is, the matrices which are conservative for null sequences and those which are regular for null sequences. After that we will characterize the conservative and the regular matrices.

Theorem 2.3.6 (conservative, regular for null sequences). Let A = (ank) be an infinite matrix. 1. The following statements are equivalent: (a) A is conservative for null sequences (that is, Co C CA ). (b) CO C WA and A(co) C c.

(c) A satisfies' (Zn) h1AII < oo (row norm condition) and (Sp) V k E N° : ak := limn ank exists (column condition). LIMIT FORMULA: If A is conservative for null sequences, then

(ak) E e and

limA x = 1: akxk

(x = (xk) E co).

k

II. The following statements are equivalent: (a) A is regular for null sequences (that is, Co C CA and limA Ico = 0)(b) co C wA and A(co) C co. 8 (Zn) and (Sp) stand for the German words Zeilennormenbedingung (= row norm condition) and Spaltenbedingung (= column condition), respectively.

Conservative matrix methods

45

(c) A satisfies (Zn) with ak = 0 for all k E N°, that is (Spo) Vk E N° : (ank)n E co. Proof. II. We show that part II of the theorem follows from part I. The equivalence of (a) and (b) is obvious. (b) . (c) : If (b) is true, then (b) of part I holds, that is the matrix A satisfies (Zn) and (Sp). Moreover, since Aek = (ank)n E co (k E N°), we have that (Spo) holds. (c) . (b) : If (c) is valid, then so is (c) of part I. Thus co C WA and A(co) C c, and, using the limit formula and (Spo), we obtain

akxk = 0 for each x = (xk) E co,

limA x = k

therefore A(co) C co.

I. The equivalence of (a) and (b) is obvious. (b) ' (c) : If (b) is satisfied, then IIAII < oo follows from 2.3.5, and (Sp) holds because ek E co and Aek = (ank )n E c for each k E No . (c) = (b) : Assume that (c) holds. From IIAII < oo we obtain co C WA by Theorem 2.3.4. Using (Zn) and (Sp) we prove first that K

for each K E N° and e > 0.

lakl <- IIAII + E k=0

This implies (ak) E 2

and 11(ak)IIi < IIAII

(2.3:9)

Hence Ek akxk converges for each x = (xk) E co. So, if K E NO and e > 0 then, using (Sp), we choose nK E N° with K

(n > nK).

Iak - ankl < E k=0

This gives, for n := nK, K

K

K

E Iakl < E Iak - and + E Iankl < e + IIAII k=0

k=0

k=0

Now, let x = (xk) E co be given. We will be finished when we show Ax E c and

limA x = E ak xk. k

For E > 0 we may choose ko E N° with 1xk1 <

2(2IIA1I + 1)

(k > ko)

(2.3:10)

46

Matrix methods: basic classical theory

and then, since (Sp) holds, choose N = N(ko) E NO with ko

(n > N).

<2

Dank - ak)xk k=0

Thus for n > N,

T, ankxk k

akxk k

ko

lank - ak)xk

+

E

-+ 2 <

lank - ak)xk

It

k=ko+i

k=0

00

e

2(211A11 + 1)

2+2=E

{

00 l a nk +

+

E lakl k=ko+1

k=ko+1

[cf. (2.3:9)].

0

This gives (2.3:10) since e was arbitrary.

As a corollary of Theorem 2.3.6 we get the well-known characterizations of both the conservative and the regular matrices due to Silverman (special methods, 1913), Toeplitz (regularity, 1911) and Kojima and Schur (conservative methods, 1917-1920).

Theorem 2.3.7 (of Toeplitz, Silverman, Kojima and Schur). Let A = (ank) be an infinite matrix. 1. The following statements are equivalent: (a) A is conservative (that is, CC CA ). (b) c C WA and A(c) C c.

(c) A satisfies (Zn), (Sp) ands (ZS) >k ank (n E N°) and a := limn Ek ank exist (row sum condition). LIMIT FORMULA: If A is conservative, then

limA x = X(A) lim x + T akxk

(x = (Xk) E c),

k

where

X(A) := limA e -

n

lim ank = a -

ank -

IimA ek = lim k

k

k

n

ak k

is called the characteristic of A. II. The following statements are equivalent: 9 (Zs) stands for the German word Zeilensummenbedingung (= row sum condition).

Conservative matrix methods

47

(a) A is regular (that is, c C CA and limA Ic = lirn). (b) A is regular for null sequences and e E CA with limA e = 1. (c) A satisfies the conditions (Zn), (Spo) and the condition (Zsl), that is (Zs) with a = 1. 11foreover, if A is regular, then X(A) = 1.

Proof. II. We show that part II of the theorem follows from part I. The implication '(a) ' (b)' is obviously true. If (b) holds, then (Zn) and (Spo) follow from the regularity of A for null sequences (cf. Theorem 2.3.6 II). Furthermore, (Zsl) follows from e E CA and limA e = 1. So (c) holds. In particular, we get X(A) = 1.

Now from part I of the theorem we get that (c) implies c C CA and therefore, using the limit formula, limA lim because X(A) = 1 and ak = 0 (k E NO). Thus (a) holds. 1. That (a) implies (b) is obvious.

(b) = (c) : Condition (b) implies (Zn) and (Sp) since, in particular, A satisfies (b) in Theorem 2.3.6I. Also, (b) implies (Zs) because e E WA and Ae = (> k ank)n E C. (c) (a) : Suppose (c). Then Co C CA (cf. 2.3.61). Let x = (xk) E c be

given. Then x = (x - ae) + ae with a:= lim x. We obtain (a) from Ax = A(x - ae) + aAe E c since co C CA, x - ae E c° and (Zs) holds. In addition, (Zs) and the limit formula in 2.3.6 imply limA x

=

as + E ak(xk - a) = a(a - yak) + k

k

x(A) lim x +

!:

akxk k

akXk.

k

We now have (a) and the limit formula for conservative matrices.

As an application of Theorem 2.3.7 we will prove that a matrix is conservative if it sums the sequence e and all thin sequences. We use this result in Section 2.4 to prove a well-known theorem due to Hahn which tells us more generally that a matrix sums every bounded sequence if it sums each sequence of zeros and ones.

Theorem 2.3.8. If A is a matrix which sums all thin sequences, then A is conservative for null sequences, that is T C CA implies co C CA and, in particular, 11AIl < oo. Moreover, A is conservative, if T U {e} C cA. Proof . The second part of the theorem follows immediately from the first, since A is conservative if e E CA and if A is conservative for null sequences. Now, let T C CA. Because ek is clearly the difference of two thin sequences,

48

Matrix methods: basic classical theory

we have ek E CA (k E N°), that is, A satisfies (Sp). Moreover, we know

(ank)k E Ta = e (n E N°) by Theorem 2.3.3(c). If we can show that IIAII < oo, then Theorem 2.3.6 implies that A is conservative for null sequences. To that end we assume sup E I ank I = 00 n

k

and construct a thin sequence x 0 CA, contradicting T C CA. Without loss of generality we may suppose that A is a real matrix and that

sup E ank = oo where a k := max{0, ank }. n

(2.3:11)

k

Now, using the main idea of the proof of Theorem 2.3.3(c), we adapt the proof of Theorem 2.3.5 to the current situation. Because Aek E c C m (k E N°) we may choose for each r E NN° an M(r) > 0 with r (n E N°).

E lank) < M(r)

(2.3:12)

k=°

We put k_1 := 0 and assume that np_1 and k._1 are already chosen for some p > 1. We choose np E No with np > np-1 and (cf. (2.3:11))

k

EanPk>(p+1)M(kp_1)+p2+1.

(2.3:13)

After that we choose kp E NO with kp > kp_1 + p + 1 and 00

E Ianpkl < 1.

(2.3:14)

k=kp -p

Applying (2.3:13), (2.3:14) and (2.3:12) we get for any p E N, kp-p-1

00

kp-I

E anPk > E a nk - E Ianpkl - E lanpk I k=kp_7+1

k

k=kp-p

(2.3:15)

k=0

> (p + 1)M(kp_1) + p2 + 1 - 1 - M(kp_1)

= pM(kp-1) +p2. Now the idea of the proof of 2.3.3(c) comes in. For a fixed p we divide the values of k with kp_1 < k < kp - p - 1, for which anpk > 0, into p sets, say Fp1, Fp2 , ... , Fpp, by putting the first into Fp1, the second into Fg2 , ... , the eh into Fpp, the (p + 1)th into Fp1 and so on. Note, if there

Conservative matrix methods 49

are fewer than p such values, some of the sets FP2, ... , FP , will be empty. Nevertheless, for at least one i = iP we must have

E anpk = E anpk > M(kP_1) +p. kEFpi

(2.3:16)

kEFF;

We choose for each p E Na such an iP and put

U Fps, =: Iv, ( r E N°} P

where v,. < v,+l (r E N°). Because, by the definition of the sets FPEp the distance between distinct elements in FpjP and the distance from F to FP+1,1 are greater than p, the sequence (v,.) satisfies v,+1 - v,. -+ 00 (r -+ oo). Consequently (v,) defines uniquely a thin sequence x = (xk); 1

hence x E WA. We now show that x

kp-1

kP

1:k

CA. For each p E N we have

anpkxk

00

anpkxk - 57 lanpkxkl -

k=kp-,+i

k=0 00

kp-1

>

anpk -

IanpkI -

k=0

kEFp:p

Ianpkxkl

k=kp+1 lanpkl

k=kp+1

[definition of x and 1Ix1f .

>

< 11

M(kp-1)+p-M(kP_1)-1 > p-1 [cf. (2.3:16), (2.3:12) and (2.3:14)].

Therefore Ax m D c. So X

CA.

0

By means of the characteristic of conservative matrices we get a classification of these matrices: those with X(A) = 0 and those with X(A) $ 0.

Definition 2.3.9 (coregular, conull matrix). A conservative matrix A, and the corresponding matrix method, is called coregular if X(A) # 0 and conull if X(A) = 0.

Remark 2.3.10. (a) Regular matrices are coregular, since X(A) = 1. (b) This classification of matrices is important and we will discuss it further, particularly in Chapter 3. Conull matrices, for example, sum all sequences that are `sufficiently slowly oscillating', but this is not true for coregular matrices (cf. Theorems 2.5.5 and 2.5.10). We close the section with some simple examples.

Examples 2.3.11. (a) The zero matrix is conull. (b) The identity matrix I and the matrices Z4 , C1 and Bi are regular, and thus also coregular.

50

Matrix methods: basic classical theory

(c) The matrix E-1 (cf.. 2.2.10) is regular for null sequences (cf. Theorem 2.3.611) and conull since it satisfies the conditions (Zn), (Spo) and (Zs) with a = 0. (d) If t = (tk) E f, then the matrix to

A :=

to to

ti tl

t2

is conull. In the case of t ,'£ 0 it is not regular for null sequences, because at least one column limit fails to be zero. A

Exercise 2.3.12. Prove mo = P independently of the proof of 2.3.3(c).

Exercise 2.3.13. Let A and B be matrices which map m into m. Show that the product AB exists, that AB(m) C m, and IIABII < IIAII IIBII Exercise 2.3.14. Let A and B be conservative matrices. Prove that the product matrix AB is also conservative and X(AB) = X(A)X(B). In particular, AB is conull if A or B is conull. Exercise 2.3.15. Let A = (ank) be any matrix and A := (ank - an,k+l ). Show the equivalence of the following statements: (a) cs C C,1.

(b) (ank )k E co (n E N°) and A is conservative. (c) A satisfies the conditions (i) (ank)k E co (n E N°), (ii) Sup,y Ek lank - an,k+l I < oo. (iii) (>k(a,,k - an,k+1))n E c, (iv) (ank - an,k+1)n E C (k E N°). Hint: Make use of Abel's partial summation formula and apply the ToeplitzSilverman theorem.

Exercise 2.3.16. Let A = (ank) be a given matrix. Prove analogously to 2.3.5 equivalence of the following statements: (i) the 4(e) C co (ii) Supn,k lankl < oo

and

(ank),, E co (k E N°).

Exercise 2.3.17. Simplify the proof of Theorem 2.3.8 for the case where X replaces T. . Bibliography: [196], [108], [254], [128]

Coercive and strongly conservative matrix methods 51

2.4

Coercive and strongly conservative matrix methods

The essential aim of this section is to prove two inclusion theorems. The first, Schur's theorem (1920), characterizes the `coercive matrices' (matrix

methods) and we will also apply it to obtain a characterization of the `strongly conservative matrices' (matrix methods) due to G. G. Lorentz (1948). The second, Hahn's theorem (1922), extends Schur's theorem by telling us that a matrix is coercive if it sums all sequences of zeros and ones (the converse is trivially true). To prove Schur's theorem we use the gliding hump method. There is, to date, no known `purely functional analytic' proof of Schur's theorem. Some proofs found in various sources may appear to be purely functional analytic, but close inspection reveals that these proofs all rely on results established by means of gliding hump arguments. Combining

analytical and functional analytic arguments we will give a more elegant proof of an extended version of Schur's theorem in Section 11.3.

Throughout this section A = (ank) denotes an infinite matrix with coefficients in K, and, in the case of its existence, ak is the limit of the kth column (k E N). That is, A satisfies (Sp).

Theorem 2.4.1 (of Schur). For a matrix A = (ank) the following statements are equivalent: (a) A is coercive, that is m C CA. (b) A satisfies (Sp),and Ek lankl converge uniformly for n E NO. (C) CO C CA and 1° h(A) := lim supra Ek lank - ak l = 0. LIMIT FORMULA: If .4 is coercive, then iimA X = 57 akxk

for each x = (xk) E m.

k

To understand Schur's theorem better, it will help if we first give a corollary, some examples and a remark. We will then state Hahn's theorem and prove Schur's and Hahn's theorems together.

Corollary 2.4.2. Every coercive matrix is conull. In particular, a matrix cannot be both regular and coercive. Proof. For a conservative matrix A we have X(A) = lima e - Fk limA ek (cf. Theorem 2.3.7I). Recalling that limA ek = ak and using the limit formula in 2.4.1 for a coercive matrix with x = e, we see that X(A) = 0. Since X(A) = 1 when A is regular we see that A cannot be both regular and coercive.

Examples 2.4.3. (a) The regular matrices I. Z., Cl and Bi are not coercive on account of 2.4.2. is defined as lim sup. b :_ io The limit superior of a real sequence inf.Eyo supk>. bk. Note, h(A) is defined for each matrix A being conservative for null sequences.

52

Matrix methods: basic classical theory

(b) The matrix E-1 (cf. 2.2.10) is conull as a direct check shows. However,

it is not coercive since h(E-1) = 2. Alternatively, we see that E-1 is not coercive because ((-1)k) E m \ cE-,. (c) If t = (tk) E e, then the matrix A defined in 2.3.11(d) is conull, and coercive since X(A) = 0 and h(A) = limsupn E' n+1 Itkl = 0. Remark 2.4.4. Let us remark that there is a formulation of Schur's theorem in functional analytic terms. In (t, 11 11,) the so-called weak convergence and the 11111 -convergence of sequences are equivalent. We will return to this version of Schur's theorem in Theorem 6.6.24.

Theorem 2.4.5 (of Hahn, cf. [1051). If a matrix A sums all sequences of zeros and ones, then it sums all bounded sequences. That is, X C CA implies m C CA.

Proof of 2.4.1 and 2.4.5. Both theorems will be proved if the circle of implications (b) (c) = (a) = (a*) = (b) is established where (a*) X C CA (or equivalently m° C CA) (b) (c): Let (b) be true. On account of the uniform convergence of the series Ek lank I for n E NO, there exists a K E No with -

00

Iankl < 1

(n E N°).

k=K+1

This implies-since A has bounded columns-that K

Iank) C E II(ank)nll. + 1 < 00 (n E 1O); k=0

k

thus IIAII < oo. Thus, A has (Zn). Since it also has (Sp) by hypothesis, it is conservative for null sequences (by Theorem 2.3.61) which also gives

(ak) E t, that is E lakl < oo.

(2.4:1)

k

Now, let e > 0. Using (2.4:1) and the uniform convergence of the series Ek lankI for n E No we may choose an N E No with 00

(lank) + lakl) <

(n E N°)

(2.4:2)

k=N+1

and then (note that A has (Sp)) choose an no = no (N) E No with N

Iank - akI < 6

(n > no).

k=0

Therefore, for any n > no we obtain N

lank - akI k

00

< E lank - akI + E (Iank) + lakl) k=0

k=N+1

(2.4:3)

Coercive and strongly conservative matrix methods 53

<

2+2 =e

[by (2.4:2) and (2.4:3)).

That proves

lank - akI = h(A).

lank - akI = lim sup

0 = littm

n

k

k

So we have (c). (a): If (c) holds, then m C WA and (ak) E £ (cf. 2.3.4 and 2.3.6). (c)

Furthermore, noting e6 = m we get for every x = (xk) E m that

1: ankxk -

akxk

E lank - akllxkl k

k

lank - ak I -++ 0

IIxII.

(n-+oo)

k

which implies x E CA and limA x = F_k akxk (x = (xk) E m). Thus (c) implies (a) together with the limit formula. (a) => (a*): This implication is trivial. (a*) . (b): In this part of the proof we apply the method of the gliding hump. It may be useful for the reader to illustrate the choice of the index sequences (np) and (kp) analogously to the figure in the proof of 2.3.5. Now, let X C CA (or equivalently ma C CA) be satisfied. Without loss of generality we assume ank E li$ (n, k E N°) since otherwise we may consider the matrices" (Rank) and (`sank). On account of Theorem 2.3.8 the matrix A is conservative for null sequences, which implies (cf. 2.3.6 I) IIAII < oo,

(ank)n E c (k E N°) and

(ak) E P.

(2.4:4)

So it remains to prove that Ek lankl converges uniformly for n E N°. Because (ak) E P it is enough to prove that the series >k lbnkI is uniformly convergent for n E No where the matrix B = (bnk) is defined by

bnk := (ank - ak)

for all n, k E No.

We note from (2.4:4) that B satisfies the conditions IIBII < oo and (Spo) Let us assume that L.k IbnkI does not converge uniformly for n E N. Thus co

3e>0 VKENo 3nEN°: EIbnkl> e. k_K

Now it is easy to see that for each K E N° there are infinitely many n E N° Thus, replacing e by 5, we may choose for for which >k K IbnkI

every K E N° an index sequence (vj(K))j such that 11 In the case of a E C we denote by Ra and Q a the real and imaginary parts of a, respectively.

54

Matrix methods: basic classical theory 00

Ib;(K)kl ? 5e

(j,K E N°).

(2.4:5)

k=K

Then, we use the properties IIBII < co, (Spo) for B and (2.4:5) to define inductively index sequences (np) and (kp). Let k_1 := 0, and let no, k° E NO be chosen such that Ibnpkl > 5e,

Ibno°I < e

and

Ibnokl < 6k=ko+1

k

If p > 1 and np_1 and kp_1 are already chosen, then, using (2.4:5) and (Spo), we may choose an np > np_1 with kp-1 Ibnpkl ? Sc

and E Ibnpkl < e

(2.4:6)

k=°

k

Since Ek Ibnkl converges on account of IIBII < oc, we may find a kp > kr_1 with 00

Ibnpkl < e.

(2.4:7)

k=kp+1

By (2.4:6) and (2.4:7) we obtain for each p E No, kp

kp-1 Ibnpkl

=

k=kp-1+1

k

>

00

Ibnpkl - E Ibnpkl - E Ibnpkl k=°

(2.4:8)

k=kp+1

5e-e-e = 3e.

(Note, the 'big' coefficients bnpk (with kp_1 < k < kp) form the `hump in the nptl' row of B' which `glides to oo' as p tends to oo.) Now, we define x = (xk) E m° with 11x1100 < 1 by x° := 0 and

xk := (-1)psgnbnpk

(kp_1 < k < kp

and p E N0),

and we prove Bx V c, that is x V CB. From that we get x V CA on account

of (ak) E e and the way B is defined. This contradicts our assumption that x C CA, since obviously x E m° = (Y). To see that Bx c, use (2.4:6), (2.4:7), (2.4:8) and tnp

1 to obtain for each p E No

E bnpkxk k

kp-,

kp

cc

bnpkxk + (-1)p L Ibnpkl + E bnpkxk k=°

k=kp-l+1

> -e+3e-e=e ifpiseven < e - 3e + e = -e if p is odd.

Hence (tnp)

c, so Bx 0 c.

k=kp+1

Coercive and strongly conservative matrix methods 55

We introduced in 1.2.17 the notion of `almost convergence' and the set f of all almost convergent sequences. In the second part of this section we now characterize strongly conservative matrices, that is matrices A with f C CA (cf. 1.3.1). Since there is a limit functional F-lim defined on f, we are also interested in a characterization of strongly conservative matrices which are consistent with almost convergence, that is limA If = F- iim .

Definition 2.4.6 (strong regularity). A summability method V = (V, Nv, V-lim) is called strongly regular if V is strongly conservative (that is, f C Nv) and V-lim I f = F-lim . A matrix A is called strongly regular if the corresponding matrix method is strongly regular. In the following remarks and examples, we omit the easy proofs.

Remarks 2.4.7. (a) A strongly conservative summability method is conservative and a strongly regular summability method is regular. (b) If an infinite matrix is coercive, then it is strongly conservative, but not strongly regular (cf. (a) and Theorem 2.4.2).

Examples 2.4.8. (a) The identity matrix A := I is not strongly conservative since CA = c and c C f. (b) The zero matrix is strongly conservative, but not strongly regular. The same is true in the case of the matrix A in 2.4.3(c). (c) The matrix CI is obviously strongly regular (see Exercise 1.2.23(c) and the definition of almost convergence in 1.2.17). Applying Exercise 1.3.10 we

find that the Abel method Al is strongly regular. (d) The matrix Z2 is not strongly conservative (however, it is regular) . since x := (1,1, 0, 0,1,1, 0, 0, ...) E f, but x D cZ

In Theorem 2.4.9 characterizing strongly conservative matrices, we consider only conservative matrices since strongly conservative matrices are necessarily conservative. We will come back to this theorem in 7.4.7 where we will give an extended version and-using functional analytic methods-a `soft' proof of it.

Theorem 2.4.9 (G. G. Lorentz (1948)). Let A = (ank) be a conservative matrix and ak be the limit of the kt' column of A. Then the following statements are equivalent: (a) .4 is strongly conservative.

(b) lim sup, Ek Jank - an,k+1 - ak + ak+1l = 0 LIMIT FORMULA: If A is strongly conservative, then

limit x = r(A) F-lim x + > akxk

(x = (xk) E f ).

k

First we need a lemma about sequences that are almost convergent to zero. We start with some notation and a remark.

56

Matrix methods: basic classical theory

Notation and Remark 2.4.10. If fo := {x E f I F-limx = 0}

(see

also Exercise 1.2.23) denotes the set of all sequences almost convergent to 0, then f = fo ® (e) because codim f (Kern (F- lim)) < 1 and F-lime = 1.

Lemma 2.4.11. bs C fo (see 1.2.3 for the definition of bs).

Proof. Let x = (xk) E b.9 be given. Then M := SUPK Irk oxkI < 00, and for all n, p E N° we get12 p+n

p+n

p-i

Exk - Exk

1

xk n+1 E k=p

k=0

2M

<

-* 0

(n -* oo)

k=o

which implies 1

n+1

p+n

E xk -+ 0

(n -* oo)

uniformly for p E N° .

k=p

0

Thus X E fo.

Proof of 2.4.9. (a) =:>. (b): Let A be strongly conservative and B = (bnk) be defined by bnk := (ank - an,k+l) (k,n E N°). Obviously, B is conservative since A is, and ak - ak+1 is the limit of the kt' column of B. Thus, by Schur's theorem (cf. 2.4.1), to prove (b) it is enough to show that (b*) B is coercive, that is m C CB. Let y = (yk) E m. Then (cf. 2.4.11) if xk := yk - Yk-1 (k E No, y-1 := 0), we have x = (xk) E bs C fo, and for each n E N° we obtain (note 2.1.1 and x E WA because x E fo C f C CA) ank(yk - Yk-1) = E(ank - an,k+1)yk =

ankxk = k

k

k

bnkyk k

which implies y E CB since x E CA. Therefore (b*) holds.

(b)

. (a): Let A satisfy (b). Then D = (dnk) with dnk := ank - ak

(n, k E N°) has the following properties: (i) D is conservative and satisfies (Spo) [cf. 2.3.6 and 2.3.71, (ii) X(D) = X(A) (cf. 2.3.71, (cf. (b)]. (iii) lim sup,, Ek (dnk - dn,k+1 I = 0 W e are going to prove f C CD and limD I f = X(D) F-lim. Then, using (ii) and the fact that (ak) E e we get f C CA

and

limA x = X(A) F-lim x + E akxk (x = (xk) E f ). k

lz As usual, we put F- to ...:= 0.

Coercive and strongly conservative matrix methods 57

Thus (a) and the limit formula hold. To prove f C CD and limD If = X(D) F-lim it is enough to show fo C COD. (Because for each y E f with

a:= F-limy we get y = (y - ae) + ae and y - ae E fo

[cf. 2.4.101.

Thus, since e E CD, X(D) = limD e, and D satisfies (Spo), we have

y E CD and limD y = limD(y - ae) + a limD e = X(D) F-limy.) Now, let x = (xk) E fo and c > 0 be given. In particular, x E WD since D is conservative. Because x E fo we may choose an N E No with 1

p+N

(p E N°).

E xk < 4(1 +IIDII) N+1 k=p

(2.4:9)

Then, using (iii) and (Spo), we choose no = no(N) E NO such that

E Idnk - dn,k+ll

6

<

and

2N(1 + IIxII00)

k

(2.4:10)

N-1 Id,,

<

(2.4:11)

4(1 + IIxII 00 }

k=0

for each n > no. Therefore, for every n > no, we obtain 00

dnkxk-N+1

k

E do E xk µ=k-N

k=N

k

=:R

00

1

µ+N

k

E xk E dnµ N+1 k=N µ=k-N

dnµ µ

=:7

µ+N

1

+N+1

dnµ k=µ

µ

xk

=:a

because

s < 1 + IIDII IIDII < 4

[by (2.4:9)]

and k

?'

=

00

k=N

µ=k-N

k

µ=k-N

xk k=µ

Matrix methods: basic classical theory

58

[dnµ = 0 for It. < 01

N-I

1

k

1

Exkµ=k-N E d,µ

N+1

N+1

k=0

N-1

1

N-1

1

1: dnµ1: xk N+1 µ-0 k=µ NllxllOO

4

N+1 l+Ilxlloo

<

k N-1 1 xk E dnµ

k=0

µ=0

N-1

E IdnINlixlI00 N+1 µ=0 e 4

[cf. (2.4:11)]

and

N-1 f3

1

57. dnµ-(N+1)dnk

Ednkxk_N+1 E

=

N-1

IlxllV+ I

IIXIIo E Idnkl + k-0

4+N+1 µ=1 k=N

4

+

k

Idnµ - dnkl

00

IV

4

00

k=N µ=k-N IIXII..

E

µ=k-N

k=N

k=0

-

k

CO

Il Nxll+1

Eµ µ=1

+ NI121Ia

Idn,k

[by (2.4:11)]

k-1

I dn., - dn,, +1I V

d,,,, - dn,v+1 I

[since E µ =

N(.2+i) ]

V

4+4<2

[cf. (2.4:10)].

Thus, we have proved fc C COD which imply (a). The limit formula holds as we already rioted.

As an immediate consequence of 2.4.9 we get a characterization of strongly regular matrices.

Theorem 2.4.12 (strongly regular matrices). If A = (ank) is any matrix, then the following statements are equivalent: (a) A is strongly regular. (b) A is regular and strongly conservative. (c) A is regular and satisfies lim supra E k lank - an,k+1 I = 0. We now apply 2.4.9 and 2.4.12 to some examples.

Examples 2.4.13. (a) The Cesaro method C1 is strongly regular (as we already stated in 2.4.8(c)).

Coercive and strongly conservative matrix methods 59

(b) E-1 is conull but not strongly conservative. (c) The discrete Borel method Bi (cf. 1.2.13(d)) is strongly regular.

Proof. (a) The Cesaro matrix C1 = (cnk) is regular and it is strongly regular by 2.4.12 since lim sup,, Ek I Cnk - Cn,k+1 I = lim sup,, n+11 = 0-

(b) The matrix E-1 = (ak) is obviously conull. However, on account of lim supn Ek Iank - an,k+l I = lim sup,, 4 = 4 # 0, 2.4.9 shows that E-1 is not strongly regular.

(c) The discrete Borel method Bi = (b,,k) is regular as is well known (cf. Exercise 1.3.11). To show that B2 is strongly regular, we verify (b) of 2.4.12 for the matrix Bl . Now, EIbnk

- bn,k+I

k

-nnk le

-n

(k + 1)!

k!

nk

e-,,

nk+1

- e

1-

TI -

n

k+1

k

/n-2

k

e-n

k! e

n

n (,__ {k+1 -1 +E k! k °O

(n-2 nk k=O

e-n

n-1

k!

{

k

00

k

Ek, -Eki+Ev k=n k=0

k=1

=

oo nk n k+1-1}+k=n

n-2

k

k

nn-1

e-n

71k

! -- 1 +

(n - 1).

k=n

k.

< e-" nn n.I

+ e-n k=n

nk k.

Stirling's formula,

n! =

nne-n

27rn(1 +en)

with a suitable (F,2) E co,

gives

limn sup

e-,, I

E 00

I

This completes the proof.

We close this section with a result that will play a role in the following chapters. As we have already stated in 1.2.23, the thin sequences T are contained in fo and therefore each strongly conservative matrix sums all

60

Matrix methods: basic classical theory

thin sequences. As a further application of Schur's theorem and Hahn's theorem we give a necessary condition for matrices that are conservative for null sequences, to sum all thin sequences.

Theorem 2.4.14 (T C CA). If A = (ank) is a matrix which is conservative for null sequences and has column limits ak, then ( sup lank - akl)

E co

(2.4:12)

n

k

is necessary for T C CA.

Proof. We assume that (2.4:12) fails, that is there exists an co > 0 and an index sequence with supk lank - akI > 2eo (v E N°). Moreover, since (ank)n converges to ak and (ak) E co, we can choose a subsequence which we again denote by and a further index sequence of (without loss of generality) with k,, - + oo (v - oo) such that Jan,.k I eo. Now, applying Schur's theorem (cf. 2.4.1(b)), we see that the matrix A = is not a Schur matrix. Applying Hahn's theorem 2.4.5 to A, we get a sequence z = of zeros and ones with z 0 cA. Thus the sequence x = (xk) E X defined by

xk=Iz,, 0

otherwise

is a thin sequence with x 0 CA. That is, T ¢ CA.

O

Exercise 2.4.15. (a) Let the map h in 2.4.1 be defined for matrices A = (ank) and B = (bnk), let X(A) exist and A E 1K be given. Prove that h(AA) = JXIh(A),

h(A + B) < h(A) + h(B)

and

h(A) > X(JAI)

with

!AI := (lankl)

are valid.

k) = diag (tn) be the diagonal matrix with

(b) For (tn) E w let A =

ank.=

Jtn ifn=k 0

otherwise

(n,kEN°).

Determine the set of all (tn) with h(A) = 0.

(c) Determine the set of all t = (tn) E co such that h(A) = 0 for the matrix A = (ank) defined by ank .= j to 0

if k < n otherwise

(n, k E N°).

Abundance within domains; factor sequences

61

Exercise 2.4.16. Prove that a matrix A is a Schur matrix if and only if V C CA and Ek Supn lank) < 00. (Consequently, a regular matrix satisfies Ek supn lankI = 00 on account of 2.4.2.)

Exercise 2.4.17. Let A = (ank) be a matrix with column limits ak (k E !k0) and let A be conservative for null sequences. Characterize the inclusion ` fo C CA' and give a limit formula for limA x (x E fo).

Exercise 2.4.18. Prove that in Theorem 2.4.9 the assumption `A is conservative' cannot be replaced in general by 'A is conservative for null sequences'.

Exercise 2.4.19. Let A = (ank) be any matrix and A := (ank - an,k+1) Show the equivalence of the following statements: (a) bs C CA.

(b) (ank)k E Co (n E N°) and A is a Schur matrix. (c) A satisfies the conditions (i)

CS C CA (see Exercise 2.3.15),

(ii) lira supn Ek lank - an,k+1 - limn(ank - an,k+1)I = 0. Bibliography: [196]; [254], [211], [148], [76], [267]

2.5

Abundance within domains13; factor sequences

In the main theorem of this section, we consider a matrix A and sequences x = (xk) in CA that satisfy a mild additional condition that is automatic if .4 is conservative. We deduce sufficient conditions for a sequence y = (Yk) to ensure that yx = (ykxk) E CA. This theorem, which is due to K. Zeller (cf. [260]), enables us to conclude from the summability of just one sequence the summability of a huge number of other sequences. This is what we mean by `abundance' or Reichhaltigkeit. As a further consequence of the main

theorem we get that every conull matrix sums both bounded divergent and unbounded sequences and that a conservative matrix which sums a bounded divergent sequence also sums an unbounded sequence. First we introduce the notion of factor sequences.

Definition 2.5.1 (factor sequence). Let A be a matrix, let x = (xk) E CA and y = (yk) E w. Then y is called a factor sequence for x and A (or for x in CA), if yx = (ykxk) E CA. The following statements are immediate consequences of this definition. 13 We will use the phrase `Abundance within domains' as a translation of Reichhaltigkeit von Wirkfeldern introduced by K. Zeller. While other authors have used the English `Bigness' for the German Reichhaltigkeit, the author feels that the English `Abundance' better captures the sense of Reichhaltigkeit'.

Matrix methods: basic classical theory

62

Remarks 2.5.2. (a) If A is conservative and x E co, then every y E m is a factor sequence for x and A. (b) If e E CA and y is a factor sequence for e and A, then y E CA. (c) If A = (ank) is a matrix, x E CA and G = (gnk) := (ankxk), then y is a factor sequence for x and A if and only if y E cG. The following theorem is the main result of this section.

Theorem and Definition 2.5.3 (factor sequences, Zeller [262]). Let A = (ank) be a matrix which satisfies (Sp). Let ak denote the limit of the kth column of A and let x = (xk) E CA be given such that Ek akxk converges. If

lima x = E akxk

(2.5:1)

k

and

sup F, ankxk v,n

< 00,

(2.5:2)

k=0

then there exists an index sequence (r?) such that each sequence r,+1

y = (Yk) which satisfies

F_ 1Yk - yk-11 ---f 0

(? -> 00)

(D)

k=rj+1

is a factor sequence for x and A. Moreover,

E ak ykxk converges

and

k

limA yx =

akykxk. k

A sequence which satisfies the condition (il) for some index sequence r =

(r?) is called slowly oscillating (with respect to r). To get a feel for the meaning of condition (D) the reader should now solve Exercise 2.5.15.

In Theorem 2.5.11 we will show that (2.5:1) and (2.5:2) are necessary for the validity of that theorem. On account of Remark 2.5.2(c) we may reduce Theorem 2.5.3 to the following theorem whose proof is simpler and which gives information on the summability of slowly oscillating sequences. To simplify the notation we extend the definition of the functional X as follows.

Definition 2.5.4. More generally than in 2.3.71 we define

x(G):=linmF_9nk-Flimn9nk=9- 1: 9k k

k

k

for any matrix G = (gnk) that satisfies the conditions (Zs) and (Sp) and for which >k 9k converges. Here g and gk denote the limit of the row

Abundance within domains; factor sequences

63

sums and the limit of the kth column of G, respectively. When we write X(G) = a, a E 1K, we assume (Zs) and (Sp) are satisfied and that the series Ek 9k is convergent. A

Theorem 2.5.5 (Zeller (262]). Let G = (9nk) be a matrix for which X(G) = 0 and CO

E 9nk =: M < oo.

sup

(Zr)

r,n k=r

Then there exists an index sequence (rj) such that if y E w satisfies (Q) in 2.5.3, then y E cc.

(2.5:3)

Moreover, for such y = (yk),

E 9kyk converges and lime y = > 9kyk.

(2.5:4)

k

k

Corresponding to the remark following 2.5.3, the conditions X(G) = 0 and (Zr) are necessary for the validity of the implication (2.5:3), cf. 2.5.10.

Proof of 2.5.3. We show that 2.5.3 is a corollary of 2.5.5. Let A = (ank) be a matrix with (Sp), ak be the limit of the kth column of A, and let x = (xk) E CA satisfying (2.5:1) and (2.5:2) be given. First we verify that the matrix G = (gnk) := (ankxk) satisfies the assumptions of Theorem 2.5.5. Since

ankxk = lim

limA x = inn n

[note that x E c. ]

9nk

at

k

k

and akxk = limn ankxk = limn gnk =: 9k, the matrix G satisfies (Zs) and (Sp). Furthermore, X(G) is well-defined since the series Ek 9k = Ek akxk converges by hypothesis. So we have

k

akxk = 0

9k = limA x -

gnk -

x(G) = lin 1

k

[cf. (2.5:1)].

k

The matrix G satisfies (Zr) because (2.5:2) implies V

CO

1:

sup 9nk r. n k=r

= sup

r,n k=r

< 2 sup E ankxk
Now applying 2.5.5 we get the existence of an index sequence (rj) such that each y = (Yk) E w which satisfies (i:)) is in cc. Thus y is a factor sequence for x and A (cf. 2.5.2(c)).

64

Matrix methods: basic classical theory

Furthermore,

Egkyk = Eakxkyk k

k

and (2.5:4) implies the convergence of Ek akxkyk and yields

limA yx =1 m E ankxkyk = lima y = k

akxkyk-

9kYk = k

k

0

Altogether, the theorem is proved.

Proof of 2.5.5. First, to facilitate the proof, we introduce an auxiliary matrix H = (hnr) defined by hnr

(n, r E N- ).

gnk

k=r

It is well-defined since (Zs) is satisfied by G (cf. 2.5.4). Use of the matrix H obviates the need for triple summation. That H has the following properties is easily seen using the properties X(G) = 0 and (Zr) of G :

sup IhnrI = M < oo,

(2.5:5)

hnr -+ 0 (r -+ oo) for each n E No,

(2.5:6)

n,r 00

hnr -> hr

gk (n -+ oo)

for each r E No,

(2.5:7)

k=r

hr -+ 0

(r --> oo)

sup Ihrl < M < oo r

[by (2-5:7)],

[by (2.5:5) and (2.5:7)].

(2.5:8) (2.5:9)

Using the gliding hump method for H we choose first an index sequence (r3) and an auxiliary index sequence (n?) and show that (AD) is a summa-

bility criterion for H, that is that (2.5:3) holds with H in place of G. Next we verify, by means of Abelian partial summation, that (0) is also a summability criterion for G, that is (2.5:3) holds. The validity of (2.5:4) is established in this step of the proof. I. Choice of an index sequence (r?) and an auxiliary index sequence (nj). We choose an arbitrary sequence (aj) E 2 which satisfies

a° = 2M + 1,

a? > 0 (j E N°) and ((j + 1)aj) E co.

(2.5:10)

Put n_1 = r_1 = 0. Then for each j E N°, assuming nj_1 and ri_i have been chosen, we choose an nJ E No with n? > nj_1 and (cf. (2.5:7)) rj_t 57 (hnr r=0

- hd < ai

(n > n?).

(2.5:11)

Abundance within domains; factor sequences 65

n._

Fig. 2.5.1: Illustration of the choice of (rj) and (n.,)

After that we choose rj E No with rj > ri_1 and (cf. (2.5:6) and (2.5:8))

hl
and (hnI
(nrj).

(2.5:12)

In Figure 2.5.1 we illustrate the choice of nj and rj; that is, we sketch a gliding hump for the matrix H. 11. (51) is a summability criterion for H. Let y = (yk) satisfy (Q) in 2.5.3, that is ri+1 Iyk - Yk-1I + 0

(j

-+oo),

(2.5:13)

k=rj+I

and let u

_ (uk) be defined by Uk

Yk - yk-1

(k E N° and y-1 := 0).

We prove

hrur converges and lime U =

u E cH,

r

hrur. r

Note that u E m by (2.5:13). For all j > 1 and each n E No with nj_1 < n < nj we obtain

E hnrur -E hrur r

r

66

Matrix methods: basic classical theory

rj_2

E(hnr

- hr)Ur

+

(hnr - hrur

r=rj_2+1

r=0

00

E (hnr - hrur

r=rj+1 rj

C

rj_2
max

f

Ilull,,.aj-1 +

r=rj_2+1

00

r,+1

+E 57 (Ihnrl + lhrl)Iurl

[cf. (2.5:11)]

i=j r=r,+1 r7

00

Ilullooaj-1 + 2M E lurl + E 2ai r=r,_2+1

i=j

ri+11

)

Iurl

r=ri+1

[cf. (2.5:5), (2.5:9) and (2.5:12)]

--* 0 (j -i oo) r +1

[since (a,) E t C co and (2.5:13) holds we have

E lurl

E co C m].

r=ri+l

i

Thus, the convergence of the series in the first line of the above calculation is justified by the convergence to zero of the series and terms at the end of the estimation. Using a similar calculation we may show the convergence

of Fr hnrur (0 _< n < no) which gives us u E WH. Altogether, we have proved u E CH and limH U = Er hrur. III. (Q) is a summability criterion for G. Let y = (yk) satisfy (0), and let u = (Uk) be defined by Uk := Ilk - yk-1 (k E No and y-1 := 0). Then from part II we get hrUr converges and limes u = hrur. r r Using (2.5:14) we verify now that

(2.5:14)

u E CH ,

y E CC, E 9kyk converges and lime y = k

gkyk, k

for which it is sufficient to prove

llnrur = r

9nkyk (n E 1N1°)

and

hrur = E9kyk

k

k

Here we prove the first identity; the proof for the second is analogous. Since Ek gnk converges because of (Zs) we obtain, by applying a variant of Abel's partial summation formula (cf. Exercise 2.1.14), v

v

E 9nkyk k=0

=

E(Yr r=0

00

CO

yr-1)1: gnk - yv k=r

gnk

k=v+1

Abundance within domains; factor sequences

67

hnrur - yvhn,v+l r=0

for each n E No. The desired identity follows since >r hnrur converges and yvhn,v+1 -* 0 (v -+ oo) holds as we verify using (2.5:12), ((j + 1)aj) E co and lyvl

=

JE,(yk

- yk-1)1

k_o

ri

Y,

IYoI + T,

fyo1+(j+1) <

(j E N and v > rj )

fyk - yk-1 f

i=O k=ri_,+1 Max i

r;

!yk-Yk-1 k=ri_i+l

(j + 1)K (K > 0 suitably chosen using (Q)).

Before stating corollaries of Theorems 2.5.3 and 2.5.5, we introduce an auxiliary sequence (f3). We will use it to define, for any index sequence (rj ), a bounded divergent sequence which satisfies (Q) in 2.5.3. We make use of this auxiliary sequence at several places in the book and always denote it by. (fj).

Notation and Remarks 2.5.6. We denote by (f j) the sequence of the partial sums of the series

1-1+2+2-2-2+...+-+...+n-n-...-n+..., 1

1

1

1

1

1

1

1

where n appears (in series) n-times with positive sign and then n-times with negative sign. The sequence (f j) has the following properties: (i) 0
As corollaries of 2.5.3 and 2.5.5, respectively, we get statements on the summability of various kinds of sequences connected with the famous names Agnew, Mazur, Orlicz, Zeller and Wilansky.

68

Matrix methods: basic classical theory

Corollary 2.5.7. A conull matrix G must sum both bounded divergent sequences and unbounded sequences. That is, X(G) = 0 implies c c m n CG C CG.

Proof. Since X(G) = 0 and IIGI1 < oo (note, by definition conull matrices are conservative) G satisfies the assumptions of 2.5.5. We choose an index sequence (rj) according to 2.5.5, put, without loss of generality, ro := 0 and determine sequences y, z E w with z 0 m and y E m \ c which satisfy (10) in 2.5.3. Then, by 2.5.5, we get z E cG \ (mflcG) and y E (mflcG) \ c. Let z = (zk) be defined by

zk :=

i=O

a

+1

(rj < k < rj+l and j

° ),

and let y be defined in accordance with 2.5.6(v). Then z = (zk) is obviously unbounded and it satisfies (i2) in 2.5.3 because ri+1

1 Izk-zk-11=3+2 -10

(j-+oo).

k=ri+1

Further, the sequence y = (yk) is bounded since IIyIloo = II(fj)I1. = 1 (cf. 2.5.6(i)), and y c on account of 2.5.6(iii) and (iv). Thus y satisfies (Q) because r,+1

Iyk - yk-11 =

Ifj+1 - fjI

(j E N°)

k=ri+1

and 2.5.6(ii) holds.

Corollary 2.5.8. If a conservative matrix A sums a bounded divergent sequence, then it also sums an unbounded sequence. That is, CA C m implies CA = C.

Proof. If A is a conull matrix, then m fl CA C CA is always satisfied as we showed in 2.5.7. Now, let A be coregular, that is X(A) 0 0, and let x = (xk) E (m fl CA) \ c. Then, without loss of generality, we may assume limA x = Ek akxk since otherwise i := x - x{Ale with limA x - Ek akxk has the desired property. (Note, Ek akxk converges because (ak) E P. See 2.3.6 I.) If we choose (rj) in accordance with 2.5.3, and z as in the proof of 2.5.7, then we get zx m because x has a cluster point14 different from zero. Furthermore, we obtain zx E CA by 2.5.3 since z satisfies (l). Thus we have proved m fl CA C CA. As a corollary pf 2.5.8 we get the first statement in 1.2.18(b) which we have not yet proved. For convenience, we recall 1.2.18(b) here. 14 a E K is called a cluster point of a sequence x, if there exists a subsequence of x which converges to a.

Abundance within domains; factor sequences

69

Corollary 2.5.9. Almost convergence (as a summation method) is not a matrix method; more precisely, f # CA for every matrix A. Proof. Apply 2.5.8 in connection with c C f C m (cf. 1.2.18(a)). 0

It is in fact true-but we will not prove it yet-that there is no matrix A for which f = m fl CA. With the aim of deducing further well-known theorems from 2.5.3, like the bounded consistency theorem due to Mazur and Orlicz, we will show that Theorems 2.5.3 and 2.5.5 are the best possible in the following sense. Theorem 2.5.3 fails if we omit either (2.5:1) or (2.5:2) from the hypotheses and Theorem 2.5.5 fails if either X(G) = 0 or (Zr) is omitted. We deal first with Theorem 2.5.5.

Theorem 2.5.10 ('converse' of 2.5.5). Let G = (gnk) be a matrix for which X(G) is defined (cf. 2.5.4). If

(Zr) is not satisfied by G or if X(G) ¢ 0,

then (0) in 2.5.3 is not a summability criterion for G. That is, for each index sequence (r?) there exists a y = (yk) E w such that ri+t

E Iyk - 1Yk-1I -40 (j -+oo) and i O CG-

(2.5:15)

k=ri +1

We can even require y E m.

The theorem corresponding to Theorem 2.5.3 is as follows.

Theorem 2.5.11 ('converse' of 2.5.3). Let A = (ank) be a matrix with (Sp) and let x = (xk) E CA such that >k akxk converges. If limA x #

akxk k

or if sup

u.n k=o

ankxk

00,

then for each index sequence (rj) there exists a y = (Yk) E m with ri+i

E I yk - yk-1 I --+ 0 (j 4 oo)

and yx

CA.

k=r,+1

This theorem may be deduced from Theorem 2.5.10 just as Theorem 2.5.3 was deduced from Theorem 2.5.5. The details are left to the reader. We remark that 2.5.10 can be proved using the uniform boundedness principle. The disadvantage of such a proof is that we get only the existence of a y which satisfies (2.5:15). We do not get an explicit y. Since, for the case X(G) 96 0, we will need an explicit y we use the gliding hump method to prove 2.5.10. In the following, the case `G does not satisfy (Zr)' plays no role. Therefore the corresponding part of the proof is left to the reader (cf. Exercise 2.5.17).

Matrix methods: basic classical theory

70

Remark 2.5.12. In the case X(G) # 0 in the proof of 2.5.10 we will show* that for every index sequence (rj) with ro = 0, there exists a subsequence

(kj) with ko = 0 such that the sequence y chosen with respect to (kj ) according to 2.5.6(v) satisfies the condition (2.5:15) in 2.5.10.

Proof of 2.5.10. Let f3

X(G) $ 0 which, according to our convention, includes the existence of gk := limn gnk (k E NO), Ek 9nk (n E N°) and

Ek A. Let (rj) be an index sequence with ro = 0. Similarly as in the proof of 2.5.5 we choose a subsequence (kj) of (rj) and an auxiliary index

sequence (nj). Here we do not need the auxiliary matrix H since the y we choose will be piecewise constant and the corresponding estimations are simpler to handle. Take an arbitrary sequence

(aj) E t with aj > 0.

(2.5:16)

Put ko = no = 0 and define inductively a subsequence (kj) of (rj) and an index sequence (nj) as follows. For j E No choose nj+i with nj+i > n3 so that kJ

LI9nk - 9kl < aj+I and

6 + E 9k - E 9nk < aj+1 (n > nj+1)

k=o

k

k

(2.5:17)

Now choose kjtl to be a member of the sequence (rj) such that kj+1 > kj, v

E 9nk

< aj+1 and

(n < nj+i, v > p > kj+1).

9k < aj+1 k=µ

k=µ

(2.5:18)

We now define (cf. 2.5.6(v)) y = (Yk) E m by

(kj < k < kj+1 and j E N°).

yk := f j

In view of k;+,

57, Iyk - yk-1I = Ifj+1 - fJ I -} 0 1J -4 00)

[cf. 2.5.6(ii)]

k=kj +1

and since (kj) is a subsequence of (r.,), the statement rj+1 Iyk - yk-1I

0 U -i 00)

k=r,+1

is obviously true. To show that y

CG we prove that E k 9npkyk) j g c

We may assume y E wG since otherwise there is nothing to prove. We show kJ-1 I

E 9kyk k=0

E c

)JENO

(2.5:19)

Abundance within domains; factor sequences

71

using Cauchy's criterion (for sequences) and the fact that (aj) E 1. For all r, p E N° with r > p we have k,-1

kP-1

r-1

9kYk

9kyk -

V=P

k=0

k=0

1:

E 9kYk k=k [definition of yj

9k

k=ka. r-2

[1I(fj)II00 < 1 and (2.5:18)1.

av v=p

To show (>2k9n,kyk)j g C, we consider for each j E N the identity k,+i-1

k,-1

00

E 9n,kyk + E 9n,kyk + E

E 9n,kyk

9n,kyk

k=k,+i

k=k,

k=0

k

Aj+Bj+C3

and prove (Cj) E co, (Aj) E c, but (Bj) y 0 cc;. First, (Cj) E co holds because

c. This implies Gy 0 c, that is

CO

00

lcjl < E fv ,.=j+1

9n,k

< E a

k=k

[cf. 2.5.6(v), (i) and (2.5:18)1

v-j+1

and (aj) E t. We get (Aj) E c using (2.5:19) in conjunction with the following calculation:

k,-1

k,-1

Y, I(9n,k-9k)yk

Aj - E 9kyk

k=0

k=0

k,-1

E I9n, k - 9k

[note Ilyll. < 1]

k=0

aj To see that (Bj)

[cf. (2.5:17)].

c we note first that k,+i -1

(j--300)

E 9n,k k= k,

which is an immediate consequence of k,+i -1

0 - E 9n,k k= k,

Q+E9k - E9n,k k

k

(2.5:20)

72

Matrix methods: basic classical theory

ki -1

+

00

<

9k +

I9nik - 9k1 +

k=0

k=ki

aj + aj + aj+1 + aj

0o

9nik

k=k,

[cf. (2.5:17) and (2.5:18)].

Altogether we obtain (cf. 2.5.6(iii)) ki+1-1

9nikyk=0 if j=(q+1)(q+2)-1 and qEN°,

B,7= k=ki

and (cf. 2.5.6(iv) and (2.5:20)) ki+i -1

Bj

=

9nik

(j = (q+ 1)2 - 1 and g E NO)

k=ki

-} /3 0 0

as

q -> oo (that means j -* oo );

thus (Bj) V c. This proves 2.5.10 in the case X(G) $ 0. The proof of the 0 case where (Zr) fails is left to the reader (cf. Exercise 2.5.17).

By using 2.5.5 and 2.5.10 we now deduce a well-known theorem in summability. In the next section we will deal with further applications of 2.5.3 with 2.5.11 and 2.5.5 with 2.5.10, respectively. Applying 2.5.5 and 2.5.10 K. Zeller proved in 1952 that the assumptions

that A is conull and CA C cB imply that B is conull. If, in addition, we make use of 2.5.12 and 2.5.6(v) we get the following stronger result which was proved by J. Copping [70] (1958) and A. Wilansky [251] (1964) using classical methods, and by G. Bennett and N. J. Kalton [26] (1972) using functional analytic methods.

Theorem 2.5.13 (Copping [70], Wilansky [251]). If A is conull and B is b-stronger than A, then B is conull.

Proof. Since X(A) = 0 and 11AiI < oo the matrix A satisfies the assumption of 2.5.5. Thus we may choose an index sequence (rj) according

to 2.5.5. We assume m fl CA C cB and X(B) 0 0. Then we apply the proof of 2.5.10 to B to get a subsequence (kj) of (rj) and the sequence y = (yk) defined by Yk = f j (kj < k < kj+1 i j E NO). Now Y E m and y CB. But (rj) was obtained by applying 2.5.5. Hence y E CA. This contradicts m fl CA C CB. Thus X(B) = 0, that is B is conull. (Note, for the application of 2.5.10 we needed the existence of X(B) which came from c C m f1 cA C CB.)

El

As a consequence of Theorem 2.5.3 and using arguments similar to those in the proof of Corollary 2.5.7 (cf. also 2.5.8) we may show that each conservative matrix A which sums a bounded divergent sequence x sums a lot of further bounded divergent and also a lot of unbounded sequences,

Abundance within domains; factor sequences

73

so that CA is much bigger than c ® (.x) and m tl CA. In this sense we may state that the domain CA is big if A sums a bounded divergent sequence. In closing this section we show that this is no longer true if A is regular and x is unbounded.

Theorem and Definition 2.5.14. For all unbounded sequences d = (dk) and a E K there exists a regular triangle A such that CA = c ® (d) and limA d = a. A matrix method A with domain CA = c ®(d) with a suitable d E w \ c, more precisely with d E w \ m because of Corollary 2.5.8, is called a one-sequence method".

This theorem goes back in this form to K. Zeller (cf. [265, Satz 1]) and-without the condition 'triangle'-to V. Darevsky [72] and, originally, in the case of certain unbounded sequences d, to S. Mazur [165]. Proof. Following K. Zeller we prove the theorem in four steps (I)-(IV).

(I) We assume a = 0, dk 54 0 (k E N°) and -d'-' =: Ok -} 0 where d_1 := 1. Then we put

A := diag (-dk_1) E-1 diag (

1

O1

=

02

1

k

0 where E denotes the summation matrix. Obviously, A is a triangle which is regular by the Toeplitz-Silverman theorem 2.3.711. Moreover, Ad =

(-1,0 ....), that is d E coA. Thus c ® (d) C CA and limA d = 0 = a. To prove CA C c ® (d) we consider the inverse matrix of A. By the definition of A we have 1

i 1

1

A-1 = diag (dk) E diag (-dk111) = -diag (dk)

0 d a d 1 1 0 d,

da

...

1

1

- diag (dk)

do

1

1

do

i

do 1

do

d,

_L d,

1 d,

... .

.

.

U =:B

=:C

The matrices B and C satisfy (Zn) by Cauchy's ratio test and the assumption on Ok. Thus C is conservative since the rows are (pairwise) is Note, in German a 'one-sequence method' is called Einfolgeverfahren.

74

Matrix methods: basic classical theory

equal and the columns are constant. Moreover, B is regular since it obviously satisfies (Spa) and because it satisfies (Zs1) as one may easily check by the assumption on Ok. Therefore A-1 y = By - limC y (dk) E c + (d) for each y E c; hence CA = A` (c) C c E6 (d). (II) We now assume a = 0, dk # 0 (k E NO) and lim supk I Ok I < 1. Then we put A := diag (hk) diag (-dk_1) E-1 diag {dk )

where (hk) is chosen such that all row sums of A equal one. With the same conclusions as in (I) the reader may prove that A has the desired properties (cf. Exercise 2.5.18). (III) Let a = 0, do = 1 and d1 = 2. Then we choose a subsequence (dik)

of (dk) with d10 := do and d11 := d1 such that IdVI < 2IdlkI <- Idlk+lI

(lk < V < 1k+1

,

k E N°).

(2.5:21)

Since d* :_ (dik) satisfies the assumptions on (dk) in (II) we may choose for

(dik) a regular triangle A* = (ank) with CA- = ce (d*) and limA. d* = 0. Now, we define A = (ank) as follows. We put a1n`k := ank (n, k E N°) and fill the gaps (in the definition) 1k < V < 1k+1 (k E N°) by coefficients ank satisfying the conditions -aylk aVLk-ldlk-1 +aVLkdik +aVVdv = 0, aVtk-1 av2 = 0 (i lk-1, lk> v)

1,

Therefore, we have for Ik < V < 1k+1 (k E lkl°) the solution a.t k-1

= -aVt k =

-dV d1k-1 - dtk

and, applying (2.5:21), the inequality

d

2d1k

d1k-1 - dlk

d1k-1 - d1k

IaVlk-lI <

<1.

Now,-it is an easy exercise to show that A is regular. Moreover, d E COA by the definition of A and because d* E COA-. Further, CA C c EB (d) since CA- = c ED (d*). Hence. CA = c EB (d) and limA d = 0 = a.

(IV) Let a E K and d E w \ m be arbitrarily given. Without loss of generality we may assume do - a = 1 and d1 - a = 2. (Otherwise we vary suitably do and d1.) By (II) there exists for (dk - a) a regular matrix A with CA = c Ei3 (d) and limA (d - ae) = 0, that is limA d = a since A is regular.

Comparison and consistency theorems

75

Exercise 2.5.15. Let r = (rj) be an arbitrary index sequence and let (cf. (12) in 2.5.3)

rj+'

Qr

y = (yk) E ; , J

( E fYk - Yk-l I -+ 0 f

(j --* 00)

k=rj+1

Establish the following facts about Qr. (a) by < Q, <w with by C Qr C W.

(b) If rj+1 - rj -+ oo (j -+ oo), then CO 0 Qr. (c) If (rj+1 - rj) E m, then CO C Qr. (d) m fl Qr \ c O O and Qr \ m O O. Exercise 2.5.16. Let G = (gnk) be a matrix which satisfies X(G) = 0 and

sup

t

r,n k=r

9nk _: M < oo.

Prove (without using 2.5.5 and by proceeding similarly as in the proof of 2.5.10) that there exists an index sequence (rj) such that for y = (Yk) defined according to 2.5.6(v) the statements y E eG and limo y = >k gkyk hold.

Exercise 2.5.17. Give a proof of the case `G does not satisfy (Zr)' in Theorem 2.5.10.

Exercise 2.5.18. Let d = (dk) with dk # 0 (k E N°) and lim supk IekI < 1 be given. Prove that the matrix A in part (II) of the proof of 2.5.18 satisfies CA = c ® (d) and limy d = 0. Bibliography: [262]; [251), [70], [26]

2.6

Comparison and consistency theorems

In the present section we examine the comparability of matrix methods (cf. 1.3.3) and in the case of comparability the question of whether the methods

are consistent. In the first part we consider only matrices A and B such that B = CA where C is a suitable transition matrix. Then, under suitable assumptions, the property `B is stronger than A' and in such a case the property `B is consistent with A' are equivalent to C being conservative and C being regular, respectively. In the second part of the section we deduce the well-known bounded consistency theorem from Theorems 2.5.3 and 2.5.11.

Theorem 2.6.1 (comparison by using a transition matrix). Let A, B and C be (infinite) matrices with B = CA such that (CA)x and C(Ax)

exist and (CA)x = C(.4x) holds for each x E cA. Then the following

76

Matrix methods: basic classical theory

statements hold: (a) If C is conservative, then B is stronger than A (that is, CA C CB)-

(b) If C is regular, then B is stronger than and consistent with A (that is, CA C CB and limB lc,t = limA ).

Proof . If C is conservative then under the hypothesis of the theorem we get CA C wB and Bx = C(Ax) E c for each x E CA. Thus CA C cB. If in addition C is regular, then IimB x = limo Ax = hmA x (x E CA). The associativity assumptions in 2.6.1 are satisfied if C is row-finite

(cf. 2.2.5(i). If B is row-finite and A is a triangle, then B = CA obviously implies C = BA-1 so that C is row-finite (cf 2.5.5(i)). Under these assumptions we strengthen the conclusions in 2.6.1 as follows:

Theorem 2.6.2 (comparison, consistency). Let A be a triangle, B be row-finite and C := BA-'. Then the following statements hold: (a) B is stronger than A if and only if C is conservative. (b) B is stronger than and consistent with A if and only if C is regular. Proof. Because B = CA the implications `=' in (a) and (b) follow immediately by 2.6.1(a) and 2.6.1(b), respectively. If C is not conservative, then there exists a y E c with y cC. Then

for x := A'y we obtain x E CA and Bx = (CA)x = C(Ax) = Cy

c,

that is B is not stronger than A. This proves (a). If C is conservative but not regular, then there exists a y E c with limcy # limy. Therefore, if we put x := A-1 y we get x E CA C cB by (a). But IimB x = limC y -A limy = limA x. This proves (b).

In Exercise 2.6.13 we give a criterion for the equivalence of triangles. Now, we turn to the question of consistency. Criteria for consistency are contained in 2.6.1 and 2.6.2 under certain assumptions. From the inclusion theorems in Sections 2.3 and 2.4 we can deduce some consistency statements that we now give. Remarks 2.6.3. (a) Let A and B be matrices which are both conservative for null sequences; then the following statements are equivalent: (i) A and B are consistent on Co. (ii) A and B are consistent on W. (iii) ak = bk (k E N°) where ak and bk are the column limits of the matrices A and B, respectively. If A and B are both coercive, then A and B are consistent on m if and only if either (ii) or (iii) holds. (b) For conservative matrices A and B with column limits ak and bk and with limits a and b for the row sums, respectively, the following statements are equivalent:

Comparison and consistency theorems

77

(i) A and B are consistent on c. (ii) A and B are consistent on cp ® (e).

(iii) ak = bk (k E N°) and a = b. (c) A strongly conservative matrix A is strongly regular (that is, consistent with almost convergence) if and only if it is consistent with (almost) A convergence on W ® (e).

Schur's theorem tells us that there is no regular matrix A for which CA D m, so there is certainly none for which CA = w. Questions of consistency among regular matrices thus arise naturally. The rest of this section explores this issue. We begin with a theorem due to R. P. Agnew.

Theorem 2.6.4 (Agnew, [1, Theorems 1 and 10]). Let x,y E w be given. If either

(a) xEm\candyEmor (b) xEw\mandyEw, then there is a regular matrix A with Ax = y.

By taking y = ae for a E K we obviously obtain the following corollary which shows that there are no constraints on what the limit of a divergent sequence may be under the set of all regular matrices.

Theorem 2.6.5. Let x E w \ c and a E K be given. Then there exists a regular matrix A with x E CA and limA x = a.

Proof of 2.6.4. (a) Let x E m \ c and y = (yn) E m be given. A bounded divergent sequence x = (xk) must have at least two distinct finite cluster points, say a and b. We choose index sequences (vk) and (µk) with µo < vo < µl < v1 < µ2 < ... such that xiyk -4 a, x,,,, -3 b and xuk $ xvk (k E N°). Then the matrix A = (ank) defined by if k = IA,, ank

3!n - xun

x,,,, - xw 0

if k=vn

(k,nEN°)

otherwise

carries x into y. Now, A is regular by Theorem 2.3.7 II since it clearly satisfies (Spo) and (Zs1). To see that it also satisfies (Zn), we make use of the boundedness of y and the fact that (xvk) and (x,k) converge to different limits.

(b) Let x E w \ m and y = (yn) E w be given and let (8n) be an arbitrary sequence of positive numbers with the limit zero. Since x = (xk) is unbounded, we can choose an index sequence (vn) with vn > n (n E NO) such that I xyn - YnI + IIn - xnI <1 + 9n (n E N° ). (2.6:1)

Ixv -xnI

Then the matrix A = (ank) defined by

78

Matrix methods: basic classical theory

xv - yn

xv - xn if k = n ank

yn

xn

if k = vn

xv - xn

(k, n E N°)

otherwise

0

carries x into y. It is easily seen that A satisfies (Spo) and (Zsl). It follows from (2.6:1) and the choice of (On) that it satisfies (Zn). Thus, A is regular by Theorem 2.3.7IL Agnew's theorem 2.6.5 tells us, for example, that we need assumptions additional to regularity for consistency theorems. When considering ques-

tions of consistency between matrices A and B, there are at least three ways of proceeding. First, we could consider only special methods (cf. 3.4.4 and Exercise 2.6.14), second we could make various assumptions about the matrices A and B (cf. 2.6.1, 2.6.2 and Exercise 2.6.14), and third we could assume comparability between parts of CA and CB (cf. 2.6.10 and 2.6.12). Using functional analytic methods in Chapter 8 we will characterize all

regular matrices which are consistent with all stronger regular matrices. These are the so-called perfect matrices. In the present section we use 2.5.3 and 2.5.11 to show, amongst other results, that every regular matrix is b-consistent with each b-stronger regular matrix. This result is the well-known bounded consistency theorem due to Mazur and Orlicz and, independently, to Brudno. First we introduce some auxiliary notation.

Notation 2.6.6. Let A be an infinite matrix with (P C CA and let ak be the limit of the kth column of A. Then we put IA := S x = (xk) E CA

I > akxk converges k

11

(inset of A),

J`I

AA (X) := lima x - 1: akxk (x = (xk) E IA), k

AA := Kern AA := {x E IA I AA(x) = 0} .

Remarks 2.6.7. Let A be a matrix which is conservative for null sequences. Then the following statements hold: (a) m n cA C IA and co C m n AA.

(b) m n CA = (m n AA) ®(u) with u = 0 or u E m n'A \ AA. (c) If A is coregular, then m n CA = (m n AA) e (e). (d) If A is regular for null sequences, then m n COA = m n AA and more generally, CO.4 = AA

Proof. The proofs are very simple, so we only sketch them. (a) By 2.3.6 we have (ak) E f; thus Ek akxk converges for each (xk) E m (cf. 2.3.3). Furthermore, again by 2.3.6, we get AA(x) = 0 (x E co) which

Comparison and consistency theorems

79

forces co c m fl AA. (b) Note, m fl cA = m fl IA and codimmnca m fl AA < 1. (c) If A is coregular, then AA(e) = x(A) # 0, that is e E (m n CA) \ AA . 0 (d) This is a trivial consequence of ak = 0 (k E N), see 2.3.6 II.

We prove the bounded consistency theorem using two theorems. The first is said to be of Mazur-Orlicz type and tells us that AA and AB are comparable relative to m if A and B are matrices which are b-comparable and conservative for null sequences. The proof of this is a further application of 2.5.3 and 2.5.11. The second is a simple corollary of the first and gives, for matrices which are conservative and b-comparable, a limit formula valid on the bounded domain of the weaker matrix.

Theorem 2.6.8 (of Mazur-Orlicz type). If A and B are matrices which are conservative for null sequences, then

mflAACcB

mfAACAA.

In particular, limB X = Lk bkxk (x = (xk) E m fl the limit of the kth column of B.

AA)

where bk denotes

Proof. Let mflAA C cB. We assume mfl AA AB-L. Since we obviously have m fl AA C IB (cf. 2.6.7(a)) we may choose an x = (xk) E m fl AA with AB(X) # 0. Because x E AA it satisfies (2.5:1) and in view of < I!AII IIxIIc < oo

(n,v E f °)

k=0

x also satisfies (2.5:2) in 2.5.3. Thus, for A and x we may choose an index sequence (ri) in accordance with 2.5.3. Since B and x satisfy the assumptions of 2.5.11 we may choose a y = (Yk) E in with rf+i

E JYk - Yk+1 l -+ 0 and yx

cB.

k=rj+1

On the other hand, 2.5.3 shows that yx E m fl AA which contradicts the assumption m fl !AA C CB.

0

Remark 2.6.9. Theorem 2.5.13 follows immediately from Theorem 2.6.8 since x(A) = 0 means that e E m fl AAJ-. Therefore, since B is b-stronger

than A, we may apply 2.6.8 to get e E A'. Thus x(B) = 0 and B is conull.

L

A further trivial consequence of 2.6.8 is the limit formula mentioned above.

80

Matrix methods: basic classical theory

Theorem 2.6.10 (limit formula on m fl CA ). Let A and B be matrices which are conservative for null sequences and let ak and bk be the limits of the kt' column of A and B, respectively. If B is b-stronger than A and if u E IA with m fl CA = (m fl AA) ® (u) is chosen according to 2.6.7(b), then

lima x = a (umAx -

akxk) + E bkxk 1: k k

where

t0

(x = (Xk) E m fl CA) (2.6:2)

ifu=0

ABU AA(U)

ifuEIA\AA.

Proof. In the case of u = 0 we have m fl CA = m 1 AA L, and (2.6:2) reduces to lima x = >k bkxk which is just the second statement in 2.6.8. If u E IA \A A then we write x E m fl CA as x = (x - axu) + axu, where ax E K is chosen so that x-a2u E mflAA holds. Obviously, ax := AA(u) So we have AB(x)

=

AB(x - au) + axAB(u)

= axAB (u) =

[because x - axu E m fl AA C AB ]

AB(26)AA(x) AA (u)

0

which proves the limit formula (2.6:2).

Now, as a corollary of the limit formula in 2.6.10 we get a consistency theorem in the case of b-comparable matrices which are conservative for null sequences.

Corollary 2.6.11 (consistency). Let A and B be matrices which are conservative for null sequences, and let B be b-stronger than A. Let u E IA with m fl CA = (m fl AA) ED (u) be given according to 2.6.7(b). Then the consistency of A and B on W ED (u) implies their b-consistency.

Proof. Note that consistency on cp is equivalent to ak = bk (k E N°). If u = 0, then a = 0 in the limit formula (2.6:2); thus bkXk = 1: akxk = HMA x

lima x = k

(x = (xk) E m fl CA).

k

In the other case we have a = 1 since HMA U = IimB u; thus

lima x = HMA x - 1: akxk + k

This proves the corollary.

bkxk = HMA x k

(x = (Xk) E m fl CA).

Comparison and consistency theorems

81

The (following) well-known bounded consistency theorem is a special case of Corollary 2.6.11. In the case of regular matrices it was formulated and proved by Mazur and Orlicz in [167] and [168], respectively. It was also proved for regular matrices independently by Brudno [52]. Other proofs were given, for example, by G. M. Petersen [194] using factor sequences, as well as by Orlicz [191], Bennett and Kalton [26] and Snyder and Wilansky [224] using functional analytic methods. However, we should mention that, ultimately, all the proofs are based on gliding hump arguments.

Theorem 2.6.12 (bounded consistency theorem). Let A and B be conservative matrices where- A is coregular and B is b-stronger than A. Then the consistency of A and B on c implies the b-consistency of A and B. Special case: If A and B are regular matrices and B is b-stronger than A, then A and B are b-consistent.

Proof. If A is coregular we have m fl cA = (m fl AA) ® (e) by Remark

0

2.6.7(c). So we may apply 2.6.11 with it := e.

A further consistency theorem due to Volkov [245] is contained in Exercise 2.6.16. In Chapter 9 we will establish some very general consistency theorems of `Mazur-Orlicz type' which are due to Boos, Leiger and Fleming (cf. [42] and [40]) and which contain, as special cases, those of this section.

Exercise 2.6.13. Let A and B be conservative triangles such that A-' B-L is conservative. Show that A and B are equivalent. Exercise 2.6.14. Let A and B be given regular row-finite matrices with AB = BA. Prove that A and B are consistent. Exercise 2.6.15. Prove that the matrix A defined in the proof of 2.6.4(a) satisfies (Zn).

Exercise 2.6.16. Let A be an arbitrary matrix, let JAI

j x = (xk) E w I sup L

n

Ja.kxkI < 00 k

be the set of all absolutely A-bounded sequences, and let JAI fl CA be

the absolutely A-bounded domain of A. Prove that if A and B are regular matrices with JAI fl CA C IBI fl cB, then A and B are consistent on IAl fl CA. Bibliography: [196], [254], [267]; [1], [167], [168], [52], [194], [191], [26], [224], [245]

82

2.7

Matrix methods: basic classical theory

Triangles of type M

Theorem 2.6.2 tells us that a row-finite matrix B is stronger than and consistent with a triangle A if and only if the transition matrix C := BA-1 is regular. However, even if we know that B is stronger than A, it is not easy in general to determine the matrix C and check its regularity. Therefore, motivated by the bounded consistency theorem, we are interested in classes of regular matrices A with the property that each stronger regular matrix is consistent with A. In the present section we will restrict our attention to regular triangles. The methods of proof will still be classical in this section, but using functional analytic methods in Section 8.6 we will get more general results.

Convention 2.7.1. Throughout this section we denote the inverse matrix of the triangle A = (ank) by A-1 = (ank). 0 The next theorem is a corollary of the Toeplitz-Silverman theorem and will be useful subsequently.

Theorem 2.7.2. Let A = (ank) be any triangle and Gn

Ek_o Iankl

(n E N°). Then the following statements hold:

(i)Vx=(xk)EmA: (i )Em. nEcoforallkEN°.

E

The statements (i) and (ii) fail if we replace

by

for (A,,)

m.

Proof. (i) The matrix B = ( ) has norm IIBII = 1, so m C MBThus, if x = (xn) EmA, that is y := Ax E m, then (-x) = By E m. If (A,) 0 m, then D = (An

) has IIBII = SUpn Ianl = oo. Hence there

exist z 0 m and y E m such that Dy = z. Now x = A-'y EmA and

DAx=(5G)=z0m.

(ii} Now ()n E co for each k E N° if and only if the matrix B = () satisfies (Spo). Since IIBII = 1 < oo, we see that (G )n E co for each k E NO if and only if B is regular for null sequences (cf. 2.3.6). But if

y = Ax, then By = (c -). Thus, because A is a triangle, so that A-1 exists, (G) E co for all x = (xn) E co., if and only if B is regular for null sequences. The additional statement can be verified similarly to the corresponding part in the proof of (i). We now turn our attention to the main notion of this section.

Definition and Examples 2.7.3. Let A be a matrix with bounded columns. Then A is defined to be of type M if to = 0 implies t = 0 for every t E t. The identity matrix I is obviously of type M. That the Cesaro matrix Cl is of type M is seen as follows. If t E e with tCl = 0, then, since Cl-1 is column-finite", (tC,)C1-' and t(C,C1-1) exist, and 113 A = (ank) is called column-flnite:t=* V k E NO 3nk E NO b n > nk : ank = 0.

Triangles of type M 83

t = t(C1C1-1) = (tC1)C1-1 = 0 by 2.2.6. Clearly, this argument shows that any triangle A for which A-1 is column-finite is of type M (for example, any band matrix). d Indeed we can go further.

Theorem 2.7.4. A regular triangle A = (ank) is of type M if A-1 has bounded columns.

Proof. Let t E e with to = 0 be given. Since A-1 has bounded columns 0 we get (cf. 2.2.4) t = t(AA-1) _ (tA)A-1 = 0. Theorem 2.7.5. A regular triangle A = (ank) is of type M if there exists a (zi) E m with zi # zi (i ¢ j) and 1I(zi)II < 1 such that n

V i E N°

S (xki )k E m V n E N°

:

z i n= E ankxki k=0

Proof. Let (zi) and (xki)k (i E N1°) be chosen in accordance with the assumptions in the theorem. Let t = (tn) E e with to = 0 be given. Then we get n

E

tnzin

n

= E to n

0.0

ankxki = k=0

thank = 0-

xki k

n=k

is holomorphic for lei < I and II(zi)II,' < 1, we have shown that f vanishes at the distinct points zi (i E NO). These

Since PO = En

points have a cluster point ° with 1l;°j < 1. Thus, by the identity theorem for holomorphic functions, f (i;) = 0 for all 1e1 < 1. Hence t = 0 since to = fl 0 = 0 for each n E N°. °}n!

More examples of matrices of type M will be given in Chapters 3 and 8. Theorems 2.7.4 and 2.7.5 will be applied in Chapter 3 to some special matrices like Cesaro matrices of non-negative order, regular Riesz matrices and regular Euler matrices. The next theorem is the basis for the consistency statement which we are aiming for.

Theorem 2.7.6. Let A be a regular triangle of type M and let B be a triangular matrix satisfying (Spo). Then COA C MB implies COA C c°B.

Proof. Let c°.4 C MB and C = (cnk) := BA-1. Then for each y E co we have x := A-ly E COA and therefore we get Cy = C(Ax) = (BA'1)Ax = B(A-1(Ax)) = B((A-'A)x) = Bx E in, which implies IICII < oo by 2.3.5.

Assume that there exists an x E c°A \ c°B. Put y = (yn) := Ax and z = (zn) := Bx. Since z co, there exist a 6 > 0 and a subsequence (zn ) with f zn I > b (v E N). Since the columns of C are bounded (by u1CII ), we may inductively choose subsequences

of

and values tk E IK

84

Matrix methods: basic classical theory

(k E N°) such that (nk.) is a subsequence of (nk_1,,) for each k E N, and converges to tk as v tends to co. Now put rv := n,,,, such that Therefore (v E N°) so that is a subsequence of converges to tk as v tends to oo. By construction (cf. 2.3.6) the matrix is conservative for null sequences and t :_ (tk) E f with 11t1l1 < llCll holds for its column limits. Next, we prove that to = 0. (This yields t = 0 since A is of type M. Thus C is regular for null sequences (cf. 2.3.6). But this contradicts our assumption Cy = Bx = z 0 co since y E co.) Let n E No be fixed. For each v E NO and k < N < r we have r

0o

N

57 (tn -

tnank = n=k

n=k

n=k

r

00

+ >2 (tn n=N+1

I=

1: tnank

0 and IVv --> 0 as v -i oo since B satisfies

(Spo), t E f and (ank)n is bounded. Moreover, 00

Sup land >2 (Itnl + lCr.nl)

n>N+1

< 2 sup

n> N+1

n=N+1

lank) IICII --+ 0 (N -3 cc).

Hence, for a given e > 0 we can choose N E N° and vo E No with so large that III + IIIINvI + IIVv4 < 2 for all v > vo. Now k vo such fixing N and noting that IIINvh < 2. That implies 00

m

E thank < e , and therefore 1: thank = 0, n=k

k=n

The theorem is now proved.

Corollary 2.7.7. Let A be a regular triangle of type Ib7 and let B be a regular triangular matrix: If CA C CB, then A and B are consistent. Proof. Since A is regular, we have CA = COA ® (e). Thus A and B are consistent because 1 = limA e = limB e and since A and B are consistent on COA by 2.7.6.

In 2.7.7 we saw that a matrix of type M is consistent with each stronger regular triangular matrix. Now, we are going to prove that the converse is also true. This is due to Banach (12].

Triangles of type M 85

Theorem 2.7.8. Let A be a regular triangle. Then A is of type M if and only if A is consistent with each stronger regular triangular matrix B. Proof. We assume that A is not of type M and aim to give a regular triangular matrix B which is stronger than A and is not consistent with it. By assumption, there exists a t = (tn) E f \ {0} such that to = 0. We put a :_ E 000to F,k=o ank and define B = (bnk) by n

(1 - a)ank + E tµak if k < n

bnk

(n, k E N°).

µ=k

otherwise

0

The (triangular) matrix B is regular by 2.3.711 since n <-

Ibnkl

11

- al

n µ=0

k=0

k=0

µ

Iankl + E ItPI E IaµkI k=0

I1- alIIAII+IIAIIIt,I < oo, n

n

n

µ

Ebnk = (1-a)Eank + Et,Eaµk k=0

k=0

k=0

11=0

(1-a)+a = 1 (n -oo), n

=

bnk

(1 - a)ank + E tµaµk µ=k 00

- 0 + E tµaµk = 0

(n -+ oo, k E NO).

µ=k

Moreover, CA C CB since x = (xk) E CA implies

E bnkxk k=0

n

n

n

=

µ

(1 - a) E ankxk + E tµ L..: aµkxk µ=0

k=0

- (1-a)limAx+Et,.

k=0 µ

aµkxk

0 'o

µ=0

if n-+ oo.

k=0

We now consider y :_ {n+i sgntn) and x := A-1y. Obviously, Y E co; thus x E c°A. On the other hand,

,i 1i, 0 a

lima x = (1 - a) IimA X + E tµyµ = µ=0

that is A and B are not consistent.

n

0

We continue to explore the consequences of Theorem 2.7.6 relating to regular triangles and the property `type M'.

86

Matrix methods: basic classical theory

Theorem 2.7.9. Let A be* a regular triangle of type M and Gn :_ Ek=o lank I (n E N). Then x = (xk) E C°A implies ( ) E co, and the result is the best possible in the sense of Theorem 2.7.2.

Proof. Let B := diag (Gn 1) be the diagonal matrix with diagonal (Gn 1) Then we obviously have COA C mA C mB by Theorem 2.7.2(i) and therefore COA C COB by Theorem 2.7.6.

The sequence (Gn) and the result in the preceding theorem give the reasons the for the following definition.

Definition and Example 2.7.10 (summability order). Let A = (ank) be any triangle and Gn := Lk=o lank) (n E N°). Then the sequence (Gn) is called the summability order of the method A. In the sense of Theorem 2.7.9 the summability order gives us a measure for the maximal growth of A-summable sequences. For example, in the case of

the Cesaro method C, we have Gn = 2n + 1 (n E N°). Therefore each x = (xk) E coc, satisfies (k 1+1+ ) E Co.

ln

Theorem 2.7.11. If A and B are regular triangles of type M, then AB is also of type M.

Proof. Let t = (tn) E t with t(AB) = 0 be given. Then s := t A E I since h1AII < co, and we get sB = (tA)B = t(AB) = 0 by 2.2.4. Therefore

to = s = 0 because B is of type M. Thus t = 0 since A is of type M. Consequently, AB is of type M.

Corollary 2.7.12. Let A and B be regular triangles which are equivalent and consistent. Then A is of type M if and only if B is.

Proof. Let A be of type M. Then B = CA, where C = BA-1, and C-1 is regular (cf. 2.6.2): in particular, C-1 has bounded columns and consequently C is of type M by 2.7.4. Therefore, B = CA is also of type M by Theorem 2.7.9. The proof of the converse statement follows by symmetry.

The following result, which is a consequence of 2.7.9, is contained in Corollary 2.5.8.

Theorem 2.7.13. A regular triangle A which sums no unbounded sequences is equiconvergent, that is CA C m implies CA = C.

Proof. Let CA C in. Thus, since A is a triangle, c C MA-1. Therefore I IA-1II = sup,, Gn < oo by 2.3.5, which implies that A is of type M (cf. 2.7.4). Thus, by 2.7.9 we get (fr-) E Co for every x E COA. Hence COA = CO so that CA = COA & (e) = c.

As we have already mentioned, we will return to matrices of type M and consistency problems in Section 8.6. There we will continue our study of matrices of type M in a more general context.

The mean value property

87

Exercise 2.7.14. Let a E JR be given. Prove that aI + (1 - a)Cj is of type M if and only if a > 0.

Exercise 2.7.15. Let A and B be regular triangles. Verify that A is of type M if AB is. Bibliography: (201], (12], [167], [166], [168], [115]

2.8 The mean value property In this section we discuss the mean value property of matrices which will prove to be sufficient for a regular triangle to be of type M. Since it is not easy to check the mean value property, we will give-amongst other things-tractable necessary and sufficient conditions for a triangle to satisfy the mean value property. In this we will essentially follow the lines of the corresponding section in Peyerimhoff's Lectures on Summability [201].

Definition and Examples 2.8.1 (mean value property). A matrix A = (ank) is said to satisfy the mean value property with a constant K > 0 (or MK(A) for short) if r

00

(O
sup E ankxk 1: ankxk < K 0
k=0

(2.8:1)

In general condition (2.8:1) is called a mean value condition. The identity matrix I obviously satisfies M1 (I). For the Cesaro matrix C1 we get M, (Cl) as follows. For each x = (xk) E w we have Er

+ l k=0

xk

1

r

1

1

+ 1 k=0

xk

rr xk

r+1 O

(r
Subsequently we will prove that other (classes of) matrices satisfy the mean value property.

For convenience we restrict our interest to triangles in this section.

Proposition 2.8.2. Let A = (ank) be a triangle and A-1 = (ank) be its inverse. Then A has MK(A) if and only if

E 1 ankakv

< K (0 < r < n E N°).

(2.8:2)

v=0 k=v

Proof. ForO
r

k

r

r

E ankxk = E ank E akvyv = E yv E ankakv k=0

k=0

v=0

v=0

k=v

(2.8:3)

88

Matrix methods: basic classical theory

Hence, (2.8:2) is sufficient for MK(A). To prove that (2.8:2) is also necessary, we assume that A enjoys MK (A) and consider for 0 < r < n E NO the sequence y = (yv) defined by

if0
sgn E ankakv

yv

(v E N°).

k=v

otherwise

0

Then, using (2.8:3) and MK(A), for (xk) := A-Ly and 0 < r < n E No we have r

r

r

E E ankakv

r

r

ankakv =

E yv v=0

v=0 k=v

k=v

< K sup

ankxk

0
k=0

I yv I < K.

That is, (2.8:2) holds.

Theorem 2.8.3. A regular triangle is of type M if it enjoys the mean value property.

Proof. Let t E P with to = 0 be given, and let A-1 = (ank) be the inverse of A. Then we get for all 0 < is < r E N° the identity r 0

=

00

E akµ E thank k=Ji

n=k

r

r

r

akJ+

k=µ

n=k

k=µ

n

r

CO

n=r+1

k=Jc 0o

t$ +

n=r+1 r

ankakv + E to E ankak/s

to n=Js

00

thank + E akµ E thank

k=µ

r

!,

to E ankakp.

n=r+1

k=Js

Now, if A satisfies MK(A) for some K, then, using (2.8:2), we have for

all p < r E N° that CO

tµI < K E Itn -3 0 (r -> oo). n=r+l

Thus to = 0, that is t = 0. Now, we are going to give two different necessary conditions for a matrix to satisfy the mean value property.

Theorem 2.8.4. Let A = (ank) be a regular triangle which satisfies the mean value property. Then (xk) E COA implies (annxn) E co.

The mean value property

89

Proof. Let A satisfy MK(A) for a certain K > 0. Then for each x (xk) E COA and y = (yn) := Ax we have n-1

yn -

lannxnl =

Ilyll. + Kn<sup

ankxk

v < n- 1

k=0

I

< (1 + K) liyll. Goo (n E N°), that is (annxn) E M. Consequently, A and the matrix B := diag (ann) satisfy the condition COA C mB of 2.7.6 and thus we get COA C cos, which is the statement of the theorem slightly reformulated.

Theorem 2.8.5. If A = (ank) is a regular triangle which has the mean value property, then (arr)

f.

Proof. Suppose that A satisfies MK(A) for a certain constant K. The regularity of A implies (cf. Exercise 2.4.16)

Esuplankl = 00. k

(2.8:4)

n

Furthermore, applying MK (A) to x = e' (r E N°) we get

land < Klarrl for r < n; thus sup land < Kl aryl (r E NO), n

and therefore >k lard = oo by (2.8:4), that is (arr) 0 1. Theorem 2.8.6. Let A and B be triangles which are equivalent and consistent. Then A enjoys the mean value property (for a certain constant K) if and only if B does (for a suitable constant L).

Proof. By Theorem 2.6.2 we get that, if A = (ank) and B = (bnk) are equivalent and consistent, the transition matrix C = (cnk) := BA-1 and its inverse C-1 = (cnk) = AB-1 are regular. In particular, 11C11 < oo and 11C-111 < 00.

For each x E w we use temporarily the notation yA = (yn) := Ax and yB = (yB) := Bx. We have Ax = A(B-1yB) = C-1yB; thus

A=

lynl

cnvyB

<_ IICr-111 sup 1y8 I. 0
v=0

Assuming that A satisfies MK(A) and noting B = CA we obtain for

r < n, that r

n

E bnkxk

E xk E cnvavk

k=0

k=0

r

v=k v

n

r

E cnv E avkxk + E cnv E avkxk v=0

k=0

v=r+i

k=0

Matrix methods: basic classical theory

90

<

sup lyV I IICJI + K sup iyV

0
<

O
IICII(1 + K) sup IY

I

IICII

< L sup IyBI

0
0
where L := IIC-1II IICII(1 + K). The proof of the converse statement can 0 be done by interchanging the roles of A and B.

Next we seek a sufficient condition for triangles to satisfy the mean value property. First we need a lemma.

Lemma 2.8.7. Let A = (ank) be a triangle with ank > 0 (k < n E NO) and an+1,k > an+1,k+1 for all k < n E N°. ank

an,k+1

Then akv < 0 for all v < k E N° where A-' = (ank) is the inverse of A. Proof. We are going to prove the lemma by induction. First we note that a,0 = - a ¢ aoo < 0. Then, for a given n E N, we assume that akv < 0 for all k < n and v < k. For all v < n we obtain17 n+1

0=

an+l.kakv k=v

+

n

an+l,n+lan+l,v

(an+l,k

k= v

ank

- an+l,v) ankak,, anv

an+l,v

dnv

anv

> an+j,n+lan+l,vi

0

thus an+l,,, < 0 for all v < n + 1 since an+l,n+1 > 0.

Theorem 2.8.8. If A = (ank) is a triangle with inverse A-' = (ank) such that ann > 0 and anv < 0 (v < n E N°), then the following statements hold:

(a) ank>0 for all k
(b) A satisfies Ml (A) if Fk_O ank > 0 for all n E No .

(c) V x = (xk) E w V r < n E N0 :

I

E ank < k=0

(k=0ank)

sup

0
llyuj An

where y=(yv):=Ax and An:=Ek=O ank. Moreover, if A = (ank) is a triangle with ank > 0 (0 < k < n E NO), then (c) implies anv < 0 (v < n E NO). 17 5,i, denotes the Kronecker symbol, that is 5,iv = 1 and 5,v = 0 if n = v and n

v, respectively.

The mean value property

91

Proof. (a) We fix n E N. (For'n = 0 there is nothing to prove.) The statement is obviously true in the case of k = n. We proceed by induction and assume that an,, > 0 for k + 1 < v < n and get n

ankakk > ankakk + E anvavk = 0; thus ank > 0 v=k+1

(b) Let _k=0 ank > 0 be satisfied for all n E NO. To prove that A satisfies M1(A), we apply Proposition 2.8.2. For v < r < n we have r n E ankakv = k=v

ankakv > 0 k=r+1

since ankakv < 0 if v < k. Thus, we have r

to

k

ank E akv

E ankakv k=v

k=0

v=0

n

k

< E ank k=0

(2.8:5) n

akv =

Snv = 1

(r < n E ld°).

v=0

v=0

J

(c) To prove (c) we consider the matrix B = (bnk) = I -) which obviously has row sums equal to one and inverse B-1 =

(\bnk)

= (ankAk).

Corresponding to the equality (2.8:5) we get r

to

k=v

k

bnk

bnkbkv

k=0

bkv

(r < n E NO) -

v=0

Now, applying the proof of the sufficient part of 2.8.2 to the matrix B, the desired inequality follows from the definition of B. The remainder of the proof is left to the reader (cf. Exercise 2.8.12). As an application of Lemma 2.8.7 and Theorems 2.8.8 and 2.8.4 we get the well-known theorem of Mercer (cf. [169] and also [106]).

Theorem 2.8.9 (Mercer's theorem). For each a E ]0,1] the matrix A, := al + (1 - a)C1 is equiconvergent (and regular).

Proof. Let a E ]0, 1] be given. Using the Toeplitz-Silverman theorem 2.3.711, we see immediately that Aa = (ank) is a regular triangle, so that c C CA. . If we show that A,, also satisfies the mean value property, then, by applying Theorem 2.8.4, we see that (annxn) E co for all (xk) E c°A. On the other hand,

054 ann=a+(1-a)n+l -4 a540 as n-+oo,

92

Matrix methods: basic classical theory

which implies (xn) E co. To prove that Aa satisfies M1(Aa) it is sufficient to verify that Aa satisfies the assumption of Lemma 2.8.7, and therefore those of Theorem 2.8.8. But, noting that 1=a n+1

Onk =

1

na

n+1 0

ifk
(k, n E N°),

otherwise

this is obvious.

0

We close the section with a remark which gives us a further motivation for using functional analytic methods.

Remark 2.8.10. Let us consider in the case of a regular triangle A the following conditions:

(a) A satisfies the mean value property. (b) Y (xk) E CA : Sllpn,rENO IF_k=O ankxkI < 00. It is easy to verify that (a) implies (b), whereas the proof of the converse implication, which is also true, requires complicated gliding hump arguments. However, using functional analytic methods we are able to may prove it at the end of Section 8.2 (cf. 8.2.21) by an easy argument. In this connection we remark that A is said to have sectional bound-

edness, if it satisfies (b). We will extend this notion in the functional analytic part of the book (cf. Chapters 7 and 8) to sequence spaces which carry suitable topologies. IL

Exercise 2.8.11. Let A and B be regular triangles such that B is stronger than and consistent with A. Show that in general MK(B) does not imply ML(A) for any L > 0. Exercise 2.8.12. Complete the proof of Theorem 2.8.8. Bibliography: [2011

2.9 Potent matrix methods Hahn's theorem (cf. Theorem 2.4.5) tells us that a matrix which sums all sequences of zeros and ones sums all bounded sequences. In connection with the comparison of bounded domains the question arises whether for a given matrix A another matrix B has to be b-stronger than A, if B sums all sequences of zeros and ones which are summable by A. (Note, if A is coercive, then the answer is obviously `yes' by Hahn's theorem.) In 1948 Agnew (cf. [21} proved that in the particular case of A = C1 all matrices B that sum all sequences of zeros and ones summable by A are conservative. Then, in 1981 Kuttner and Maddox (cf. [138]) showed that all such matrices B must be b-stronger than C1. In this section, we deal further with this question and start with two basic definitions due to Kuttner and Parameswaran (cf. [1401).

Potent matrix methods 93

Definition 2.9.1. A matrix method A is called potent if for each matrix B the inclusion X fl CA C CB implies m fl CA C CB, and a matrix A is potent if the corresponding matrix method is. Let KP denote the set of all potent conservative matrix methods (matrices) and KG denote the set of all matrix methods (matrices) A such that each matrix B satisfying X fl cA C CB is conservative. Clearly, KP C KG.

Examples and Remarks 2.9.2. (a) Each coercive matrix is potent by the Hahn theorem (cf. 2.4.5) and is consequently in KP. (b) The identity matrix and therefore any other matrix A with mflCA = c is not potent so not a meptber of KP. [Note, (x fl CA) = p ® (e) and that a matrix B satisfies CB D W ® (e) if and only if it satisfies (Sp) and (Zs). For example, the inverse of CL is not conservative and enjoys (Sp) and (Zs). 1

(c) Obviously, A E KP if and only if A E KG and for each matrix B the inclusion X fl CA C CB forces m fl CA C CB.

(d) If T U {e} C CA, then A E KG by Theorem 2.3.8. In the remainder of this section we make use of the notion of a dense subset" in a normed space. This is a topological notion, but our methods are still classical. We will not apply here any topological or functional analytic methods.

Theorem 2.9.3. If A E KG and (X fl CA) is dense in (m fl CA, II Iloo), then A is potent. Proof. Let B = (bnk) be a matrix with X fl CA C CB. Since A E KG the matrix B is necessarily conservative; in particular, we have IIBII < oo. Now, let x E m n CA be given. Clearly, the theorem is proved if we can show that Bx is a Cauchy sequence. Let e > 0. Because (X f1 CA) is dense in mflCA, we can choose a y E (XfCA) with Ilx-ylloo < 2(1+21811)' Then y E CB and, hence, there exists an no E No such that E(bnk

E bnkyk -

- brk)Yk

k

k

brkyk

< 2 for all n, r > no.

k

Therefore, for all n, r > no, we have

E bnkxk - E brkxk k

k

< I- brk)(xk - yk)I + 11>nk - brk)yk k

<

k

2011 llx - yllco + 2 < 2 + 2 = e.

Thus Bx is a Cauchy sequence, and mfCA C CB has been verified.

0

18 In a semi-normed space (X, 11 11) a subset A of X is called dense in X (relative to 1111 ), if for each x E X and e > 0 there exists a y E A with Ilx - yll < e.

94

Matrix methods: basic classical theory

Remark 2.9.4. In the proof of 2.9.3 we made use only of the density of (X n CA) in m n cA and of JJBJJ < oo, but not of the fact that m n cA is a (bounded) domain. Therefore we have in fact proved: if E is a subspace of m such that (X n E) is dense in (E, JJ JJ,,,,) and B is a matrix with JJBJJ < oo and X n E C CB, then E C cB. In Section 11.5 this enables us to generalize the notion of potency of matrices to sequence spaces. This will lead to the matrix Hahn property. IL

As an application of Theorem 2.9.3 we will obtain a result which goes back to Kuttner and Parameswaran [140]. However, the original proof looks

complicated and is based on the same idea due to Kuttner and Maddox [138] that we have already used in the proofs of 2.3.3(c) and 2.3.8. We will follow the lines of a proof given by G. Bennett (cf. [24]) which is also based on the idea of Kuttner and Maddox, but allows for a more general approach in Section 11.5.

Theorem 2.9.5. Every strongly conservative matrix is potent. To prove this theorem, we will first prove two propositions which are of interest in themselves.

Proposition 2.9.6. m = mo + bs. Moreover, for all x E m and e > 0, there exist a y E mo and z E bs with x = y + z and JJzIJ,,,, < E. In particular, mo is dense in (m, JJ

Proof. For convenience we assume K = IR, since otherwise we may deal separately with the real and imaginary parts of the sequences being considered. Let x E m and s > 0 be given. We are going to construct a y E mo such that z := x - y E bs and IJzIJ0 < s. (2.9:1)

By adding ae to x (and then to y) for a suitable a E R we may assume

that Xk > 0 (k E N°). (We refer to this as a translation.) Furthermore, multiplying x by a suitable scalar allows us to assume 0 <_ Xk < 1

(k E N°). (We refer to this as a scaling.) Now, take yo := 0 and define zo, yl, z1, y-2.... inductively by k

Zk := Xk - Yk

and Yk+1 :=

1

7=0

0

k

if O y3 < E xi l=°

(k E N°).

otherwise

Then z E bs, in fact 114b5 < 1, and JJzIJ,,, < 1. In particular, we have shown that m = mo + bs. (We should note that the `inverse' scaling and translation for y and z does not influence z E bs and y E mo.) To prove statement (2.9:1) in general, we choose an n E N with n > and split x (satisfying 0 < xk < 1) into n disjointly supported" parts, 19 Two sequences x and y are called disjointly supported if min{fxk1,IykI} = 0 for each k E Ne.

Potent matrix methods

95

,x(n), and apply the above argument to each of the parts. For each i = 1, 2, ... , n let x(i) be defined by

say x(1)

x(a} = J xk k

0

if and < xk <

(k E N°).

n

otherwise

Applying suitable scalings and translations (the trivial details are left to the reader), the above argument provides sequences y(a) E mo and z(a) E bs having the same supports as x(a), and such that x(i) = y(a) + z(a)

and

< 1 < E. n

IIz(x)II

The setting y := ELI y(Zi and z F_k=1 z(2) gives us the desired y E mo 0 and z E bs satisfying (2.9:1). Proposition 2.9.7. Let E be any sequence space. ). (a) If bs C E C m, then mo fl E is dense in (E, 11 (b) If bs U {e} C E C m, then (X fl E) is dense in (E, (c) If f C E C m and B is any matrix with (X fl E) C CB, then E C CB. I

Proof. (a) If bs c E C m, x E E and E > 0, then we may choose y E mo and z E bs with fix - yii,,,, = UUzIIoo < e in accordance with 2.9.6. Since y = x - z E E + bs = E we obtain y E mo fl E as required. (b) Now, the above-mentioned idea of Kuttner and Maddox [138, Lemma 3] comes in. Looking at the proof of 2.9.6 we see that, after translation and

scaling, the sequence y takes only values 0, 1, ... , N for some N E No . (Since e E (X fl E), we may perform translations and remain inside E.) Therefore, it suffices to prove that any sequence x E E fl mo, taking only values 0,1, ... , N actually, belongs to (X fl E). Let Ej = {k E Nl° lxk = j } (j = 1, ... , N) and let L be the least common multiple of {1, ... , N}. Now, we define sets Fl,..., FL as follows: the nth member of Ej (in the usual ordering of N°) goes into the set Fk if and only if k =- n mod (j } . (Note, members go into more than one Fk, and the process terminates if Ej is finite.) By the construction it follows that20

r

E XE,nFk (n) differs front

L EXE,(n) n=1

n=1

, by no more than one. Summing on j and observing x = FN 1 3XEJ, we see that r 1 r

E XF,, (n) differs from n=1

L

E xn n=1

20 If M is a non-empty subset of No, then XM : X -a 1K denotes the characteristic function of M on NO, that is XM(n) - I if n E A and xM(n) = 0 otherwise. By the definition of a sequence, yst E w.

96

Matrix methods: basic classical theory

by no more than N. In other words, XF,. - ix E bs, so that XF,, E (XnE), and hence x E (x fl E) because N

N

L

L

N

x = EXE; = E EXE, nFk = E j=1 k=1

j=1

L

EX_,, nFk =

k=1 j=1

EXFk.

k=1

(c) Because of T C f (cf. 1.2.23(d)) we have T C X n E C cB; thus IIBII < oo by 2.3.8. Since bs U {e} C f C E we get from (b) that (X n E) 0 is dense in (E, II II,,). Thus, E C cB by 2.9.4.

Proof of 2.9.5. Let A be strongly conservative, that is f C CA. Then we 0 may apply Proposition 2.9.7(c) to the case E := m n CA. Knowing from Theorem 2.9.5 that strongly conservative matrices are potent, the question arises whether for a matrix to be potent, it is necessary for it to be strongly conservative. In Sections 3.4 and 3.2 we will verify that the answer is `yes' in the case of conservative (real) Hausdorff matrices and `no' in the case of conservative Riesz matrices. For the characterization of potent conservative Hausdorff matrices we make use of a particular class of matrices, namely the class of all triangles

A = (ank) with liminfn lannI > 0. As we will now prove, such a matrix cannot be in KG. Thus these matrices are not potent. This result is due to Kuttner and Parameswaran (cf. [139]).

Theorem 2.9.8. For any countable set M of sequences there exists a matrix B = (bnk) with limn E. Ibnkl = oo and M C COB. Note, such a matrix is necessarily non-conservative.

Proof. Let M = { x= = (xk) k i = 0, 1, 2, ...1 be a given (at most) countable set. For each n E No we choose a suitable non-negative integer kn with I

kn > n + 1 and take bnk :=O for k > kn. Thus M C cB =w for any choice of the remaining coefficients of B. We define inductively bnk for k < kn so that not every bnk is zero and such that bnk xk = 0 for each i < n and n E N° . k

This is certainly possible since kn > n + 1. Furthermore, by multiplying by a suitable constant, we can arrange that

L I bnk I= n + 1

for every n E No.

k

For any given i, we obviously have Bx` E cp C co; thus M C coB, and the 0 matrix B has the desired properties. *

Corollary 2.9.9. Let A be any matrix and let X n CA be at most countable. Then A V KG and hence A 0 KP.

Notes on Chapter 2

97

Proof. According to Theorem 2.9.8 we may choose a non-conservative matrix B with x n CA C CB. Thus A 0 KG KP. Theorem 2.9.10. Let A = (ank) be a triangle with liminfn Iannl > 0. Then the set X n CA is at most countable. In particular, A V KG D KP by Corollary 2.9.9.

Proof. By the assumptions on A, there exist N E N and 5 > 0 such that

Iannl > 5 for all n > N.

(2.9:2)

The set of all A-summable sequences is contained in the set of sequences x for which there exists an ro (which may depend on the particular sequence

x) such that no > N and to - tn+1 I < 2

for all n > no,

(2.9:3)

where to = Ek ankxk (n E No). So it is enough to show that there is an at most countable set of sequences of zeros and ones having the property (2.9:3). For this it is sufficient to show that, for any no, there exists at most a finite set of sequences of zeros and ones satisfying (2.9:3). Supposing

throughout that x = (xk) E x, there is only a finite number of ways of choosing xn for n < no. Corresponding to each such choice, there is at most one way of choosing xn for n > no such that (2.9:3) is satisfied. For, if it is possible to satisfy (2.9:3) at all, the values of xn for which (2.9:3) holds are determined inductively in a unique way. For suppose that x has

been determined for v < n (where n > no > N). This fixes tn. The two possible values of to+1 (corresponding to xn+1 = 0 and xn+1 = 1, respectively) differ by an+1,n+1 Thus by (2.9:2), there is at most one of them which satisfies (2.9:3). (Of course, it may happen that neither does.) This proves the theorem. We close the section with the reminder that-as we already mentioned in Remark 2.9.4-in Section 11.5 we will deal with so-called Hahn properties which include potency of matrices as a special case.

Exercise 2.9.11. Show that bs is dense in (fo, II

11c o).

Bibliography: [2], [138], [139], [140], [47], [214], [215], [28]

2.10 Notes on Chapter 2 In Chapter 2 we characterized-in terms of the matrix coefficients-in a series of theorems matrices A with the property that E C CA where E is an arbitrarily fixed sequence space. More generally we may consider a further sequence space F and FA := {x E w I Ax E F}

(domain of A with respect to F)

and give necessary and sufficient conditions for A to satisfy E C FA, that is, for A to fulfil E C WA and to map E into F. Since in the special case

98

Matrix methods: basic classical theory

E = c = F the Toeplitz-Silverman theorem 2.3.71 contains the desired characterization, we speak in the general case about Toeplitz-Silvermantype theorems. In order to get an overview of Toeplitz-Silverman-type theorems in the case of standard sequence spaces we refer the reader to the overview paper [231] of Stieglitz and H. Tietz and the papers [156] and [157] of Y. Luh. Further notes on Toeplitz-Silverman-type theorems are contained in Section 2.10.

3

Special

summability methods After having become acquainted in Chapter 2 with several inclusion, comparison and consistency results it is time to put some life into the theory by considering (classes of) examples of summability methods. In this way we shall have the opportunity to apply theorems, proved in Chapter 2, to special matrix methods. In particular, we investigate the members of the classes concerning the properties defined in Chapter 2 like conservative (for null sequences), regular (for null sequences), coercive, type M, mean value property, potent, and so on. In Section 3.1 we consider briefly the `Holder methods' Ha (a E NO) and then in depth the `Cesaro methods' CQ which play a central role in connection with applications of summability to different fields of mathematics. By means of the Toeplitz-Silverman theorem we characterize those Cesaro methods which are regular (these are all conservative Cesaro methods). Moreover, for instance, we prove that C,3 is stronger than and consistent

with each Ca where -1 < a < 8 ('monotonicity' of the Ca-methods) and then we deduce from that-using a well-known theorem of Knoppthe equivalence of Ca and Ha for each a E No . Generalizing the Cesaro methods we consider in Section 3.2 the class of `weighted means' which are also called `Riesz methods' and are matrix methods ('Riesz matrices'). On account of the simple structure of the defining matrices, the handling of Riesz methods is in general simple in com-

parison with other classes of matrix methods. So, for instance, it is easy to verify standard properties and it is possible- but highly non-trivial-to characterize those Riesz matrices which are potent. Section 3.3 is dedicated to the class of `Norlund methods (matrices)' which also contains all Cesaro methods. Besides the standard considerations we characterize the comparison and the consistency of Norlund matrices. In Section 3.4 we are concerned with the class of `Hausdorff methods (matrices)' which play an essential role in certain applications of summability, for example in connection with `moment problems'. As special Haus-

100

Special summability methods

dorff matrices we present the Ceshro and Holder methods. Moreover, the definition of Hausdorff methods allows us to extend the definition of Holder

methods HI to general indices a E C and to introduce the well-known class of `Euler methods (matrices)'. In contrast to the characterization of the comparison of Hausdorff methods it is extremely non-trivial to characterize those Hausdorff matrices which are regular; the difficulties lie in the norm condition and lead to `moment sequences'. Another non-trivial and mathematically interesting part of this section is the characterization of those Hausdorff matrices which are potent. In Section 3.5 we deal with the class of `Sonnenschein methods (mar trices)', in particular with the class of `Karamata methods (matrices)' by way of introduction. The class of Karamata methods contains the classes of Euler, Taylor and Laurent methods, respectively. We are concerned mainly with regularity considerations in the case of certain Karamata matrices. In Section 3.6 we are interested in summability methods which are defined by power series. They are in general non-matrix methods. The best known examples of power series methods are the Abel method and the Borel method which are stronger than and consistent with the conservative Cesaro and Euler methods, respectively.

3.1 Cesaro and Holder methods We introduce in this section the Holder methods (matrices) H° (a E N°). They result from the Ceshro methods C1 in an obvious way, namely by iteration. The advantage of the matrices H° is that many of their properties can be derived from corresponding properties of C1. A decisive disadvantage is the direct handling of those matrices since there does not exist a simple formula for their matrix coefficients. This circumstance is one reason, among others, that we consider also the Ceshro methods (matrices) Ca (a E R and -a N) which turn out to be equivalent and consistent with H° for a E No . They have the advantage that calculation with the coefficients of the matrix Cam, and thus with the transform CCx (x E w), is usually simpler than in the case of Ha. However, unlike the Holder matrices, it is more difficult to handle products of Cesaro matrices because, for example, the product of two Cesaro matrices is not in general a Cesaro matrix. In such cases it is advantageous to switch to the equivalent Holder matrices as we will demonstrate in the proofs of 4.1.13 and 4.1.14.

Definition 3.1.1 (0. Holder, 1882). Let C1 be the Cesaro method (of first order) from 1.2.13(c) and let a E NO. Then

H" :_ (Cl)' (that is, H° := I and H° := C1H"'1 (a > 1)) is called the Holder matrix (of order a) and the associated matrix method is named the Holder method (of order a) or briefly the H"matrix and -method, respectively.

Ces3ro and Holder methods

101

Following Hausdorff (1921), we will introduce Holder methods (matrices) for arbitrary order a as special Hausdorff methods in Section 3.4.

Remarks 3.1.2. (a) By definition H' = C1. (b) H" (a E N) is well-defined and a triangle as a product of triangles. (c) H" = H"-1C, for each a E N (cf. 3.1.1 and 2.2.6). (d) H"+$ = Ha HO for all a, 0 E No (cf. 3.1.1, 2.2.6 and (c)).

Theorem 3.1.3 (monotonicity of H° ). For each a E NO the method H"+1 is (strictly) stronger than and consistent with H", that is CH- 9 CH.-+1

and

limes- x = limn, +i x (x E CH-)-

In particular H" (a E N°) is regular.

Proof. Because H"}1 = CLH", using the regularity of C1, we obtain CH' C CH-+1 with consistency as an immediate corollary of 2.6.1(b). Furthermore, cHa 54 cHa+, follows easily from c g cc,.

Theorem 3.1.4 (type M). For any a E N° the matrix H" is of type M. Proof. If a E No, then (H")-1 = (Ci 1)" is a column-finite triangle since C1 1 is. In particular, it has bounded columns. Thus H" is of type M by 2.7.4.

For the definition of the Ceshro methods we make use of binomial coefficients. To that end we recall their definition and some of their properties.

Definition and Remarks 3.1.5 (binomial coefficients). For 0 E R and v E No we define the binomial coefficients by

(

:= l1

1 3()3-1)...(/3-v+1) ifv # 0' v! ifv

1

0.

The following identities are satisfied: (a) (Q) = v, (a 1 (/ ,) (,d, v E No with 0 < v < (3). (J3, L, E No with v > ,8). (b) 0 (,6 E R and v E N). (c) (,,"1) + ('5) = (0+1) v 11

n

(d) (n+, 3) =E (v+0-1)

(n E N° and 8 E R).

v=0

n

(v-k+6-1) _ (n-k+'y+d-1 (e) v=k F (n-v+7-1) n-k n-v v-k 1

(n,kEN° with k
Proof. The statements (a), (b) and (c) follow immediately from the definition of the binomial coefficient.

Special summability methods

102

(d) For any 8 E R we may prove the statement by induction where the induction step from N to N + 1 can be realized in the following way: N+1

1

E

+lN+1 1

V=O

V=O

+ (N +,1 )

CNN 13)

CN + 1+,6 \

[induction hypothesis]

[cf. (c}].

N+1

(e) For Jtl < 1 we consider the absolutely convergent power series

n+y-1tH n

n

1

(1 - t)7

and

(n+o_1) to n

n

-

J

-

1

(1 - t)6

Then the Cauchy product of these series is absolutely convergent and we obtain (cf. 2.1.9)

-

1

(1- t)7+6

En E"-o

jnv+^yv+tF-11 n- v

1

to

J

V

Since on the other hand 1

-

(1 - t)+6

(n+7±5-')tn n

(Its < 1)

we get by comparison of coefficients that

Cn+y+6-1) n

- n (n-v+y-11 (v+6-1 v=o

n- v

v

Replacing n by n - k in this equality we get

n-k+y+6-1 n-k

n-k

E

(n-k-v+y-1'\ (v+6-1

n-k-v v E (n-v+y-1 v-k+6-1

lJ

V=O

v=k

which is statement (e).

n-v

v-k

0

The following calculations serve to prepare the definition of the Cesaro

matrix Ca for a E R with -a 0 N.

Cesaro and Holder methods

103

Preparatory calculations 3.1.6. Let n E No and x = (xk) E w be given. Then we define iteratively n

Sn (x) := xn

n

_ v=0 E

E X"

Sn(x)

a

v=0

and

S'O

/

(x))

n

S,-1(x) (aEN with a>1).

Sn(x) V=o

If E denotes the summation matrix from 2.2.10 and if we put E° then (Son (x))n = E°x and

(x)) =: Eax

(S,"_1

(Sn (x))n = E

(a E N and x E w)

obviously hold. Now, for a given a E N we wish to determine the coefficients

of the matrix Ea = (s(nk)). That is, we determine n

snk'

with

1: Snk) xk = S.n (x)

(n E F ° ).

k=0

To that end we do some formal calculations. We have

(1 - z)

(Sn (x) - sn_1 (x)) zn =

Sn (x) zn = n

n

Sna-1 (x) zn n

(with S°1(x) := 0); thus =

Sn (x) zn

(1 - z)

so (x) zn

n

n

xnzn

(1 - z) _a n

(v+a-11 z [\

V

V

xnz n

(n-k+a-1 xkz n n-k

n k=0

[binamial series]

lJ

[Cauchy product[.

Therefore, by comparing coefficients we obtain

Sn(x) =

(3.1:1) kjj=o

Thus we set (a) Snk

:

(n-k+a- 1) n-k J

(n,k E N°, k < n)

104

Special summability methods

(note, snk) := 0 if k > n). To justify the above formal calculation we may

start with (3.1:1) as a definition for Sn (a E N) and verify by induction on a that the original equation n

Sn(x) _

S.'-'(X)

xn)

(a E N; Sn(x)

v=0

is true. Using 3.1.5(d) we have n

(a)

E snk

= n (n-k+a-1) = L,n (k+a-1) n - k

k=0

k

k=0

k=0

and the matrix (cnk)) defined by 1

(a)

ink

(a)

(n, k E 1`

1;7+a) snk

)

n

satisfies the row sum condition (Zs1) which is necessary for regularity. Obviously we may extend the definition of cnk) to any a E R with -a ¢ N. So, the following definition makes sense.

Definition 3.1.7 (Cesaro method). Let a E R with -a ¢ N be given. Then Ca

{c(nk)) defined by (n-IC +a-L ) cnk)

0

(a?) n

if k < n

(k, n E N°)

ifk>n

is called the Cesaro matrix (of order a), and the associated matrix method is called the Cesaro method (of order a) or, in short, the Ca-matrix and -method, respectively. In the following example we write down explicitly the Cesaro matrices Co, CL and C2.

Examples 3.1.8 (C°, C1 and C2). (a) Co = I since (nn-k L) = 0 if k < n, and (nnkkt) = 1 if k = n. Frthermore, by definition we have 1

CL =

1

L

2

2

(n-k)

L

L

L

3

3

3

Using 3.1.7 and 3.1.5(a)) we get

because

(k < n). 1;7+nL)

+ 1

Cesi}ro and Holder methods

(nn-k t

(2)

2(n-k+1)

(n+2)

Cnk

l

(n + 1)(n + 2)

2

105

if k
ifk>n,

0

that is 1

12

1

3 3

_ C2

0

3

4

2 6 3

6 2

I

10

1°

1°

i°

1

.

0 l

We are now going to check how far the conditions (Zsl), (Spo) and (Zn), which are necessary and sufficient for regularity, are satisfied in the case of the Cesaro matrices.

Facts 3.1.9. Let n E N° and a E R with -a ¢ N be given. (a) Ek c(nk) = 1; in particular, each Ca enjoys the property (Zsl ). (b) Ca satisfies (Spo). (c) If a > 0, then Ca is regular. (d) If a < 0, then (Zn) is not satisfied; in particular, if a < 0, then Ca is not conservative for null sequences (thus it is not conservative or regular). Proof. (a) This follows immediately from 3.1.5(d) and Definition 3.1.7. (b) For any given k E No and n > k we have the identity (tt-k+a -1 )

n-k+a-1 n-k+a-2...a

n- k

(n-k !

(n+a)

(n+a) (n+a-1)...(a+1)

t

11

n

nl

n-1

n

n+a-In+a-2

n-k+1 a n+a-k n+a

from which (Spo) follows since (n+a) E co and because each of the other k factors converges to one if n tends to infinity. (c) If a > 0, then cnk >_ 0 (n, k E N); thus the row norm condition (Zn) is implied by (Zs1). The regularity of Ca, a > 0, now follows from (a), (b) and the Toeplitz-Silverman theorem (cf. 2.3.711).

(d) First we show that Ca does not satisfy (Zn) if -1 < a < 0. If we note (cf. 3.1.18)

{n n

a} n-a

{a1+

r

1}

(n -+ oo, a > -1),

(3.1:2)

where I' denotes the gamma function, sup,, n-Q = oo and c(a) = (nna)-1, then we obtain

106

Special summability methods

IIC0II = sup n

IC;,k) I > Sup Term) I n

k

= 00,

that is C0, does not satisfy (Zn).

Next we prove that C0, does not satisfy (Zn) if a < 0, -a V N. For oo by induction on

that it is obviously sufficient to verify supra

v E No for a fixed -1 < 6 < 0. This statement is true for v = 0 as we have just shown. If it is true for a v E No, that is if supra le(-V+b)I = 00, then it is also true for v + 1 since Cnn

n

b-v

CnfV+b}

b-v / n-v+S

Cn

1

-v+& 0

and limn na v+b = 0. Thus the induction is complete.

In 3.1.3 we have verified the monotonicity of the Holder methods H° (a E N°). We now establish the monotonicity of the Cesaro methods

C0, (a > -1).

Theorem 3.1.10 (monotonicity of Ce ). For all a, ,f3 E R satisfying -1 < a < 3 the method CQ is stronger than and consistent with C0,, that

is cc. C cca and lime, x = limes x (x E cc.). Remarks 3.1.11. (a) If we put a = 0 in 3.1.10, then the statement yields the regularity of C,6 for $ > 0. (b) In 3.4 we will extend the definition of the H0,-methods from a E No to all a E P.. Then we will again derive the monotonicity of the C0,-methods from that of the H0,-methods (a E P) and from the equivalence of Ca and

0

H".

Proof of 3.1.10. Let a,/3 E P with -1 < a < (3 be given and let x = (xk) E w, y = (yv) := Cax and z = (Zr) := C8x. We are going to determine the transition matrix B with Cq = BCa which is a triangle since Ca and C are. By the definition of Ca we have

v-k+a-1

1

yv = k=0

k

(v E N°).

From this we have

(n-n-a-I (vva)

yv

rn_u_a-1 rv-v+k-l)xk kJo[` v_ol`

Cesaro and Holder methods

107

EXk E jn-n-v-1) {v-v+k-1 v=k

k=0

t

Ank

-a and 5:= a in 3.1.5(e) we get

Obviously, Ann = 1. Putting y Ank

= (n-k-a+a-1) =0 n-k

(k
Hence, we have

n-v-a-1

n

Xn =v=0 E{

+a )C

n-v

(3.1:3)

v ) yv.

Next we determine zr for a given r E NO in terms of yv (0 < v < r) by means of (3.1:3): Zr

r

1

Xn

r-n

Q (r l

n= 0

jval

r. (r-T+/3-1 )

(r

vyv

V=0

n=O 1

E((r_n+8_1)(n_u_a_1\ (v+a)

r

v=0

1

n- v

r n

n=v

r. jv+a) r-v+,3-a-11 J yv t v r-v v=0

(r+N) V=0

J

v

yV

[by 3.1.5(e)].

If the matrix B = (br(v) is defined(( by -1) brv

lr-v /3-

(r+131

r

0

J

(va) ifv
ifv>r

then C,3 = BC,,, as we have just verified. Moreover, we obtain cc. C ccs including consistency if B is regular (cf. 2.6.1). Now, we are going to prove the regularity of B with the aid of the Toeplitz-Silverman theorem (cf. 2.3.711).

Obviously, brv > 0 (r, v E N°) because of 6 - a > 0 and a,)3 > -1. Thus, for all r E No we get brv V=0

r (r+,3)

E

v-0

(r-vJ3a-1va) ) Tv

\v

108

Special summability methods

-a+(a+1)-11 )

1)

(

r+(3

Cr-0+,8 r - 0

(3.1.5(e) for n := r, k:=0, y:=/3-a and 5:=a+11 =

1.

Hence, (Zn) and (Zsl) are verified for B. It remains to show that B satisfies (Spo), that is (brv), E co (v E N°). We start with v = 0. Putting y :_ 8 - a - 1 we obtain

=

bro

_

(rrry)

(r + y)(r + y - 1) ... (y + 1)

(r+s) - (r+(3)(r+(3-1) r+f3) r3--

(/3+1)

-1) ... [1-/3+1}.

C1

Thus, because 1 + t < et (t E R), 0 < bro

e-P e-- ... er

exp -(1 - y) E k + -+0 (r-*oo) [#--y=a. 1>0 and Ev +l+i =oo], therefore (bro)r E co.

In the case of v > 1 and r > v we have irrvv7) lvva) (r-v+1+'Y) (V-1+a)

brv

br,v_ 1

r-v+l

v-1

r-v+1 v+a

r-v+1+y

v+a (r

v

v

oo

)

which implies (including the existence of the limits)

0 = lim bro = ... = limr br,v_1 = limr brv. r 0

Our theorem is proved.

Remark 3.1.12. Let a > -1. By (3.1:3) the inverse C,,,-' = (

Vii) of

the Ca-matrix is determined by(

a) = j (n_n-v-1) (vva) ifv
0

(n,v E N°).

otherwise

Moreover, C,,,-' enjoys (Spo). To prove the last statement we should note that the transition matrix B in the proof of 3.1.10 is just Ca 1, if 3 := 0, and that we did not make use of a < 0 when we showed that B satisfies (Spo). A

Cesaro and Holder methods

109

Theorem 3.1.13 (type M). For any a > 0 the matrix Ca is of type M. Proof. If a > 0, then Ca is a regular triangle by 3.1.9(c). Furthermore, its inverse satisfies (Spo); thus it has bounded columns and consequently Ca is of type M by 2.7.4. o

Theorem 3.1.14 (mean value property). The matrix C, satisfies M1(Cl), whereas for each a > 1 the matrix C. does not enjoy the mean value property. Moreover, we will show in 3.3.6 that for any a E ]0, 1] the Cesaro matrix Cc, satisfies M,(C0).

Proof. We have already proved in 2.8.1 the statement concerning C1. Let a > 1 be given. By Theorem 2.8.5 the matrix C« does not enjoy the mean value property, if its diagonal is an absolutely summable sequence, that is ((n+o)-1) E However, this follows immediately from the asymptotic n relation' (n+o) n0 since (n-a) E t if a > 1. 0

On account of the definition of Ha and Cc, (a E N°) it is of interest to compare CQ and Ca-IC,. The following result is due to Knopp (1923). It was also proved by Andersen [5] (1928). We will use this theorem to conclude the equivalence of the Ha- and Cc,-methods (a E N°), which may be surprising because the corresponding matrices are very dissimilar.

Theorem 3.1.15 (Knopp). For each a E N the methods Cc, and Ca-1 CF are equivalent and consistent.

Proof. Since the statement is obviously true if a = 1, we assume a > 1. For any y = E w and all /3, n E N° let Sn (y) be defined as in 3.1.6. Using Abel's summation formula (cf. 2.1.1) we obtain n

n

E(v+J3)y v=0

=

n

v

-EEYk + (n+/3+1) Eyk v=0 k=0 n

k=0

- E S,', (y) + (n +,8 + 1)S,', (y) =0

-Sn (y) + (n + (3 + 1) S', (y).

(3.1:4)

Furthermore, we put

(t(n') (x)) := Cax (x E w and 03 E N°).

-

Since S11(x) = (n + 1)tn')(x) the identity (3.1:4), applied in the case of /3 = 1 and y = Cl x, implies 1 Let D C IR, let to E RU{-oo,oo} be acluster point of D, and let f,g : D -i R be functions. Then f and g are called asymptotic equal at to, if there exist a function c : D -+ R with limt-+to c(t) = I and f(t) = c(t)g(t) for t E D\ {to}. In such a case we use the notation f (t) - g(t) (t to), or f (t) - g(t) for short.

110

Special summability methods

n

(v + 1)tyl}(x) = -Sn(Clx) + (n + 2)Sn(Clx).

SS(x) = v=a

From this we get by summation and by (3.1:4), applied for 0 = 2, n

Sn(x)

2

S,

n

n

(x)

_

SS (C1x) + E(v + 2) S' (Clx) v=o

V=o

V=o

_

-SS(C1x)-Sn(C1x)+(n+3)Sn2 (Clx)

_

-2Srz(C1x)+(n+3)Sn(Clx).

[application of (3.1:4) for /3 = 2)

Continuing inductively we obtain

Sn (x) = (-a + 1) Sn (Clx) + (n + a) Sn-l (Clx).

(3.1:5)

Division of (3.1:5) by (nna) gives us

Cax = (-a + 1) C. (C1x} + aCa_1 (C1x)

(3.1:6)

(note that Sn = (nna) teal ). Now, if

x E cc,_,c, ,

that is Cjx E cc,_, C cc,

[cf. 3.1.10),

then (cf. (3.1:6)) x E cc, and = = =

lime, x

(-a + 1) limo, C1x + a limc,_, C1x (-a + 1) limc,_, C1x + a limc,_, C1x

[cf. 3.1.10]

limc,, Clx = limc,_,c, X.

Therefore cc,_,c, C cc, is satisfied with consistency. Now, we are going to verify the inclusion `D'. Using (3.1:5) we get by the definition of Sn (x) the equalities

S'(x)

= =

(-a+1)Sn(Clx) +(n+a)(Sn(Clx)-'Sn-1(Clx)) (n+1)Sn(Clx)-(n+a)S.,'_1 (CI x);

thus, dividing by (nn a), the identity

t'(x) = (n + 1) tn(C1x) - ntn_1 (C1x), therefore

n

t°(x) = (n + 1) to (C1 x), =o

that is Ci(Cax) = Ca(C1x) = (CaC1)x,

and thus cc, C cc, c, using the regularity of C1. From this we see from 0 (3.1:6) the inclusion cc, C cc._,c, which is what we had to prove.

Ceshro and Holder methods 111

As an immediate consequence of 3.1.15 we obtain, as was our purpose, the equivalence statement:

Theorem 3.1.16 (of Knopp and Schnee).2 For each a E NO the methods Ha and Ca are equivalent and consistent. Proof. We prove the statement of the theorem by induction. It is obviously

true in the case of a = 0 and a = 1 since H° = I = Co and H1 = C1, respectively. If we assume that the statement is true for an a E No, then by 3.1.15 it is true for a + 1, if we can verify the equivalence of Ha+1 and Ca C1. From the induction assumption we have the identity

{xEw I C1xEcH-} = {xEw I ClxEcc,}. Since Ha+1 = H0C1 the desired identity CH-+1 = cC,C, follows. Moreover, for any x E PH.+1 we have limH-+1 x = limH. CI X = lime, CI X = lime, c, x

0

and thus the consistency of Ha+1 and C0 C1.

Naturally, the monotonicity of the Ca-methods raises the question, under what conditions, with -1 < a < Q, does the Ca-summability of a sequence imply its Ca-summability? We will deal in 4.1 with this and related general questions, that is with Tauberian theorems for the Ca-methods. For

example, we will prove the important result that all Ca-methods (a > 0) are (pairwise) b-equivalent. As a consequence of that result we will see that Ca (a > 0) is strongly regular since C1 is. It will then follow that Ha (a > 0) is strongly regular because Ca and Ha are equivalent. Now, strongly regular matrices are potent by 2.9.5. so Ca as well as Ha are potent. Strong regularity and potency of Ca and Ha will also be a corollary of the characterization of the strong regularity of Hausdorff matrices in Theorem 3.4.21.

Exercise 3.1.17. Determine the first column of Ca (a E 1R, -a ¢ N) as well as the first four rows of the Cesaro matrices C3 and C-1.

Exercise 3.1.18. (a) Prove n+a n

n

-a

-->

1

l'(a + 1)

(n - oo),

for any a > -1 (that is, (3.1:2) in the proof 3.1.9(d)). (b) For a E NO and each x E w the equivalence x E coC,

n-a Sa(x)

-4 0

(n -+ oo)

holds. Prove it by using (a). 2The statement of the theorem holds even for each a E k with a > -1 as we will prove in 3.4 when we have generalized H° from a E No to a E R.

112

Special summability methods

Exercise 3.1.19. Let A = (ank) be a conservative matrix with column limits ak (k E N°). Use the transition matrix to prove the equivalence of the following statements:

(a) A is stronger than C1. (b) sup. Ek (k + 1) lank - an,k+i I < 00. Hint: First show that both (a) and (b) imply the boundedness of (kank)k for every n E N° . Then make use of Abel's summation formula and the fact that cc, = c°c, ® (C1-le). Bibliography: [108], [196], [201]; [267], [5], [173]

3.2

Weighted means, Riesz methods

An obvious generalization of the Cesaro method C1 is given by the weighted

means which are also matrix methods. Since they were first examined in detail by Riesz, they are also called Riesz methods. We should mention, however, that they were used earlier by Kronecker, Cesaro and Hardy. An advantage of the weighted means is the simple form of the generating matrix and of its inverse. This makes the method very convenient to handle.

In this section we show that each weighted mean is conservative and

characterize their regularity by a simple application of the ToeplitzSilverman theorem. We also characterize those conservative matrices which are stronger than a given Riesz method. In the third part of the section we give a characterization of those Riesz methods which are potent. This is of interest since there are many potent Riesz methods which are not strongly conservative and, as we have shown in 2.9.5, strongly regular matrix methods are potent. The proofs in this part are quite deep and methodologically interesting.

General assumption 3.2.1. Throughout the section let (pk) be a real sequence with p0 > 0 and pk > 0 (k E N), and Pn := F_k_0 pk (n E NO).

Definition 3.2.2 (Riesz methods). Let p = (pk) and (Pn) be given in accordance with 3.2.1. Then the matrix Rp defined by P-k

Tnk

P

10"

ifk
(R,p) := (R,pn) := (rnk)

(k,nENO)

otherwise

is called a Riesz matrix (associated with the sequence p). The associated matrix method is called a Riesz method (associated with p). We often refer to a Riesz method as a weighted mean method. We now give some simple examples.

Examples and Remark 3.2.3. (a) The identity matrix I is not a Riesz matrix since the first column contains the value `1' as well as V. (However, the corresponding method is equivalent and consistent with some Riesz

Weighted means, Riesz methods 113

methods (cf. 3.2.23).)

(b) The matrix Re has rno = 1 (n E N°) and rnk = 0 for k > 0 (n E NO). Thus cReo = w. Similarly, if p E W so that pk = 0 for k > ko for some ko, then RP has rnk = 0 for k > ko and again cR, = w. (c) If p := e, then RP = Re = Ci obviously. (d) If pn > 0 for each n E No, then the inverse Ry 1 = (rnk) of RP is given by P.

ifk=n

0

otherwise

- P2=. ifk=n-1 Pa

rnk

(n,kEN°).

[For a proof show that Rp(rnk) = I.]

IL

Theorem 3.2.4 (mean value property). Let Rp be a Riesz matrix. Then RP enjoys the mean value property with K = 1.

Proof. We adapt the corresponding proof for the case of CI (cf. 2.8.1). For each x = (xk) E w we have

1: pkxk r

1

P P.

P,.

P.

k=0

E Pr 1

'

k=0

pkxk

<

1

'

Pr k=0

pkxk

(r
thus Ml (RP) is satisfied.

Theorem 3.2.5 (type M). If pn > 0 for each n E No, then Rp is of type M.

Proof. The columns of the matrix Rp are obviously bounded. Now, let

t = (tn) E 8 with tRp = 0 be given. For each k E N° we get 0 = E n=kP k

->

n =k}1 P=

Pk

and consequently to = 0 (n E N°), which

proves the theorem.

We remark that the statement in Theorem 3.2.5 is, in the case of regular Riesz matrices, an immediate corollary of 3.2.4.

Theorem 3.2.6 (summability order). If pn > 0 for each n E NO, then the summability order (Gn) of Rp is given by Gn = P -+P` (n E N°). Thus x = (xk) E cons yields (2w) E co. Proof. See Definition 2.7.10, Theorem 2.7.9 and Theorem 3.2.5.

Applying the Toeplitz-Silverman theorem 2.3.7 to Riesz matrices we get that each Riesz method is conservative, and a simple characterization of those Riesz methods which are regular.

Theorem 3.2.7 (conservativity, regularity). Let Rp be a Riesz matrix. (a) R. is conservative. (b) Rp is regular if and only if Pn -' oo if n -> oo.

114

Special summability methods

(c) Rp is either regular or coercive. If Rp is coercive, then it is conull. LIMIT FORMULA: If we put p := limn p , then, for each k E N°, PPk is

the limit of the kth column of Rp and

limRP x = X(Rp) limx + E ppkxk

(x = (xk) E c).

k

Proof. (a) and (b): Rp is conservative on account of the Toeplitz-Silverman theorem 2.3.7. Since all row sums of the matrix Rp equal one and all

pn are non-negative, the conditions (Zsl) and (Zn) are trivially satisfied. Furthermore, (Pn) is a non-negative monotonically increasing sequence,

and therefore it converges to a finite limit or diverges to cc and (P ) converges to a finite value p > 0, where p = 0 if and only if Pn -* oo (n -+ oc). Thus, Rp satisfies (Sp) (with ppk the limit of the kth column), and (Spo) if and only if Pn -+ oo (n -# oo).

0

(c) The proof is left to the reader (cf. Exercise 3.2.22).

Theorem 3.2.8 (comparison). Let Rp be a regular Riesz method with pk > 0 (k E N°) and A = (ank) be a conservative matrix method. Then A is stronger than Rp if and only if the following conditions hold: )k E co. (i) V n E No : (

(ii) Sup,, >2k Pk I p - pk+11 i < cc. Remark 3.2.9. If, in addition, A is row finite, then condition (i) is obviously satisfied.

Proof of 3.2.8. First we verify that (i) is a necessary condition for CA J To that end we assume CA D cRP and choose an arbitrary n E N. In particular, the series >2k ankxk converges for each x E CRP. If we put y := Rpx, that is x = Rp 1y (cf. 3.2.3), then cRP.

Pkyk - Pk-iyk-1

ankxk =

ank

k

pk

k

We now consider the particular sequences (yk) E co given by Yk = (-1)kEk where (Ek) E co and Ek > 0 (k E NO) implies and note that the convergence of Ek ank(-1) k Pkck+Pk-1Ek-1 pk

r

1\ank

Pkek + Pk-16k_11 Pk

E co;

li k

thus rank PkEkl 1\

A

E Co.

(3.2:1)

l1 k

Since (e.) is an arbitrarily given positive null sequence, (QapkP*)k is a fac-

tor sequence for co, that is

(

Pk)

E m. Thus (apn,")k E co because

Pk -+ oo. Thus we have shown that () is necessary for CA D cRP .

Weighted means, Riesz methods

115

In the next step of the proof we determine the transition matrix V = (vnk) with A = VRP. Since RP is a triangle, we obviously obtain V = ARF-1, that is

- an,k+r

vnk = Pk r ank

for all n, k E No .

)

Pk+l

Pk

(3.2:2)

Now, using Abel's partial summation formula (cf. 2.1.1), we will prove that for (VRP)x we can calculate associatively for each x E C°Rp, that is

Ax = (VRP)x = V(RPx)

(x E CORE)

holds, provided that CA D cRp or (i) and (ii) are satisfied. To that end let x = RP-'y with y E co be given. Then for all n, K E No we have K

K

k=O

k=O

Yk-1 E ankxk = E ank i'k Ik-rk-t Pk K k=O

ank -

an.k+l

k

Pk+t

v=O

Pk

+ K

k-0

a"kPk

an,K+1

PK+t

(PKYK - PK-1YK-1)

) Pkyk + P=X (PKYK - PK_lyK-1).

P

We put y_1 = 0 = P-1. Now, if we can show that in the cases under consideration the last term converges to zero when K -3 oo, then the series Ek ankxk converges if and only if Ek (Pkk - Pk+i') Pkyk does. However, in the case of CA D cRp this is an easy consequence of (3.2:1) and

in the case that (i) and (ii) hold, we obtain it obviously from ( ank Pk) Pk

E co

for each n E No

k

which comes from the inequalities anKPK

<

(ank

PK k=K

PK

<

00

PK E

Pk

an,k+l Pk+1

a_ an,k+l1 Pk Pk+l

I

ank

kK A

_

an,k+1

Pk.

Pk+1

Therefore, if CA D cRp or (i) and (ii) hold, we have

Ax = Vy = V(RPx)

for each x E CORP

including the convergence of the corresponding series.

(3.2:3)

116

Special summability methods

To prove the theorem we now assume that CR, C CA. Then (i) is satisfied, as we have already verified, and x E CA holds for each x E COR, . Consequently, since Rp is a triangle, this means by (3.2:3) that the transition matrix V is conservative for null sequences; in particular V satisfies (Zn) (cf. 2.3.6) which is equivalent to (ii). Conversely, if (i) and (ii) are satisfied, then V satisfies (Zn) because of (ii), and since lim unk = lim Pk n

n

ank C pk

- an,k+l Pk+1 I

= Pk (ak Pk

ak+1

(k E N°)

pk+1

the matrix V satisfies (Sp) since A does. Consequently, V is conservative for null sequences by 2.3.6. Thus, if we consider an arbitrary sequence x E CR,, then there exist a y E co and a A E K with x = Rp 1y+ARy 1e. Thus we get Ax E c, that is x E CA, from Ax

= A(Rp 1y) + AA(Rp 1 e) = V y + AA(RP 1e) = Vy + AAe [cf. 3.2.3(d)]

[cf. (3.2:3)]

since V and A are conservative for null sequences and conservative, respectively.

Corollary 3.2.10. If pk > 0 (k E N°), then the Riesz method Rp is equiconvergent if and only if (g ) E m.

Proof. Apply 3.2.8 to the identity matrix A := I and notice 3.2.9. As a further corollary we get a characterization of those conservative matrices which are stronger than C1 (see also Exercise 3.1.19).

Corollary 3.2.11. If A = (ank) is a conservative matrix, then the following statements are equivalent:

(a) A is stronger than C1. (b) supra E,k(k + 1) lank - an.k+1I < 00.

Proof. This follows from 3.2.8 since C1 = Rp where pn := 1 (n E N°) and because 3.2.8(i) holds since A satisfies (Zs).

Examples 3.2.12. (a) If p = (pn) := e, then Pn = Ek=° pk = (n + 1) and Rp = C1. Thus Rp is obviously strongly regular.

(b) If pn := n+1 (n E N°), then P,, - log n and the generated Riesz method Rp is called the logarithmic weighted mean. It is regular by 3.2.7(b) and even strongly regular (cf. 3.2.23(b)).

(c) If a > -1 and pn«I :=

(n E N°), then we obviously have Rp(t) = C1. Furthermore, pn«i _ rta+1i and therefore Pn«i = n prz«) r(+2) n«+1 (cf. Exercise 3.1.18(a) and 3.1.5). Thus for each a > -1 the Riesz method Rp() is regular by 3.2.7(b) and even strongly regular (cf.

Weighted means. Riesz methods

117

3.2.23(b)). ene (d) If 0 < 0 < 1 and p;,O) = en-', then PO) g as one may prove (cf. Exercise 3.2.21). Thus for each 0 < ;8 < 1 the Riesz method R,($) is regular by 3.2.7(b) and even strongly regular (cf. 3.2.23(b)). (e) For p2, = 2' and p2,+1 = ,+i} (r E N°) we have3 [-I-'1 [1 P,, = E 2k + E k=0

k=0

[--'] =

2171+1

- 1 + E

(k + 1)2

2[z1+1,

k=0

(k + 1)2

Thus the corresponding Riesz method Rp is regular by 3.2.7(b).

IL

In the remainder of the section we deal with the characterization of the potency of Riesz methods Rp. Since non-regular Riesz methods are coercive, and thus trivially potent, we assume regularity. Moreover, we assume that all weights pn are positive. So, we make the assumptions

pn>0(nENO) and Pn->oo(n-+ oo).

(3.2:4)

In the following we aim to establish the following characterization of potency which is due to Kuttner and Parameswaran (cf. [141, Theorem 1]).

Theorem 3.2.13 (potent Riesz methods). If p = (pn) satisfies the condition (3.2:4), then the following conditions are equivalent:

(a) Rp is potent.

(b) (p) E co. (c) Rp E KG. We will show that `(a) * (c) a (b) * (a)' where `(a) * (c)' is trivially true. (Note that `(b) * (c)' is not needed for logical reasons, but is really part of the proof of `(b) * (a)'.) To that end we state and prove five lemmas where we essentially bring over the first three from [141]. Furthermore, we model the proof of `(c) * (b)' on the corresponding part of that of Theorem 3 in [141]. In [47] a new proof of `(b) * (a)' is given since the corresponding part of the proof in [141] contains a false conclusion.

Lemma 3.2.14. Let p = (pn) satisfy (3.2:4) and (P) E co. Further, let (k,.) be the index sequence with ko = 0 and

(rEN°).

k,+1 =min{vEN I

Then a non-negative sequence (xk) is Rp-summable to zero if and only if

E

lim _P r k. vEI,.

0,

(3.2:5)

where I, := N° n [k,, k,.+1 [. 3 For each a E R we denote by (a] the largest integer less than or equal to a.

118

Special summability methods

Proof. First we note that if p and (kr) are as stated in the hypothesis,

then, since P () E co,

Pk.+t - 2Pkr.

(3.2:6)

Suppose now that x E COR9 is a non-negative sequence. Then k,.+i -1

Pvxv < vEI,.

E

Pvxv = 0 (Pkr+,1) = O(Pk,.)

v=o

since Pk,+t_1 < 2Pk (by the definition of kr+1 ), that is (3.2:5) holds. Conversely, suppose that (3.2:5) holds for a non-negative sequence x and let e > 0 be given. Then there exist t, k` E N with k* > kt such that

E pvxv < ePk. (r > t)

(3.2:7)

LIE I,

and

kt-1

Sl :_ E pvxv < ePk

(k > k`).

(3.2:8)

V=O

Then for any k > k* > kt there exists an r such that k E Ir and k

(3.2:9)

E Pvxv = S1 + S2 v=0

where S1 is defined in (3.2:8) and k,.+i

k

Sz

r

E Pvxv

E Pvxv = 1: 1: Pvxv

v=kt

v=kt

P=t vEI,

Then we see from the relations (3.2:7) and the definition of kr that

S2 < EePk,

=2CPk,.<4ePk.

(3.2:10)

p=t

0 Lemma 3.2.15. Let n E N and let {zl, Z 2 ,- , zn} be a set of nonnegative real numbers. Suppose that Zn :_ Y=1 zv, that Bn > 0 with By (3.2:8)-(3.2:10) we see that x E coR,,, and the lemma is proved. .

zv < Bn (v E Nn) and let t E N be arbitrarily given. Then we can divide the set Nn into t (pairwise disjoint) subsets N1, N.2,. .. , Nt (some of them may be empty), such that zv < vEN.

1Zn+Bn (s=1, 2,...,t).

(3.2:11)

Weighted means, Riesz methods

119

Proof. We put no := 0 and for s = 1, 2, ... , t we define

n8:=max({0}UrENN

!Zn}).

zv < v=1

(3.2:12)

fff

We remark that n8 is increasing and nt = n. It follows that

E z,. >

t Zn - Bn

(s = 1, 2, ... , t).

(3.2:13)

V=1

Now let N8 := {v E N I n8_1 < v < n8} (s E Nt). Then n, n,-i

>2 zv=>2zv- E zv << tZn+Bn v=1

vEN,

v=1

by (3.2:12) and (3.2:13). This proves the lemma.

Remark 3.2.16. We note for later reference that To prove this we remark that n,

0 ,EN, z > !Zn - B.

n,_1

{Zn-Bn < Tzv =

zv

vEN,

V=1

v=1

vEN,

where we made use of (3.2:13) and (3.2:12) for '< ' and '<' respectively. Hence

zv > } Zn - Bn -

S

t

1 Zn

= t Zn - Bn.

vEN,

A

That completes the proof.

Lemma 3.2.17. Let p = (pn) satisfy (3.2:4) and (P) E co. Let X be any set of non-negative integers and t be any given positive integer. Then we can divide X into t pairwise disjoint subsets X1, X2,.. -, Xt such that for each j E Nt we have

E pk - l

Pn

A

0 (n goo),

(3.2:14)

kEX,k
that is XX, - t XX E coRP, where XM : No -- f {0, 1} denotes the characteristic function of M C NO.

Proof. We use the notation, definitions and conclusions from Lemmas 3.2.14 and 3.2.15. First we note that Lemma 3.2.14 shows that the relation (3.2:14) is equivalent to the assertion that

Pk1

,

kEXjf11,

Pk - t

pk '-1 0 (r -+ oo).

(3.2:15)

kE.Cf1!.

Hence, to prove the lemma, it is enough to show that (3.2:15) holds. Con-

sider now an arbitrarily fixed value for r. For k E N let zk := Pk

if

120

Special summability methods

k E X El Ir and zk := 0 otherwise. Then for any positive integer n > kr+1 we obtain n

E zk =

Pk =: Zn and

zk < sup Pk =: B. kEXnI,.

kEXnI.

k=1

Now by Lemma 3.2.15, we can divide the set of all integers in X which are

less than or equal to n into t subsets X1, X2, ... , Xt such that for each

jENt we get 1 Z. t

- Bn -<_ E zk < 1t Zn + Bn kEXj

[by (3.2:11) and 3.2.16].

That is, kEX;nI,

Pk - t E Pk < kkEXnI* kEXnI.

The hypothesis that (P) E co now yields (3.2:15), and the lemma is proved.

O

Lemma 3.2.18. Let p = (pn) satisfy (3.2:4) and x = (xk) be a sequence with 0 < Xk < 1. Then for each p E N there exists a sequence y which takes values only in {0, P, ... , 1, 1} and for which x - y E CORD and IIx - yIk. < holdProof. For the given x we fix the sequence (vk) E { 1, ... , vk := min { A E 11, ... , p} I xk <

p}N0

by

(k E NO ).

PI

Aiming at a definition of y = (yk) we construct inductively strictly increasing (finite or infinite) sequences (nr) and (pr) in No such that no := 0 and nr-1 < /r-1 < nr < l.tr

(r = 1, 2, ...).

We start with

yo := P and we obviously get (yo - xO) po > 0. Then we define µo to be the least integer such that MO

E(Yk -xk)Pk < 0 where Yk

vk-1 for0
If no such integer exists, then the process of defining the sequences (nk) and (µk) terminates, and we put

Weighted means, Riesz methods

yk =

121

yk-1 for all k>0. P

If for a certain r E N the integer n, has been defined as also 9k for k < n,., then we define p, as the least integer greater than n, such that

vk-1

(yk - xk)pk < 0 where Yk

for n,. < k < E.c,.

P

k=°

If no such integer exists, then the process of defining the sequences and (Pk) terminates at the integer n, and we put

9k :=

(nk)

yk-1 for all k>n,. P

If p, has been defined (so also 9k for k < p,), we choose n,+1 as the least integer greater than p, such that n,+,

(Yk - xk)pk > 0 where 9k := vk for ji, < k < n,+i.

P

k-°

Again, if no such integer n,+t exists, then the process of defining the sequences (nk) and (Pk) terminates, and we put 9k := vk for all k > P,. P

In any of the two cases (namely, that the process of defining the sequences

(nk) and (pk) terminates or it does not) we get a sequence is = (yk) in p { 0, p , ... , -1, 11 such that P 11y- x11. < P and

((Yk - xk)pk) k=0

Since Pn is proved.

E M. n

oo (n --a oc) we obviously have y - x E COR, and the lemma

Lemma 3.2.19. Let p = (pn) satisfy (3.2:4), (P) E co, and let x E rno fl CRP be such that x takes only values in {0, 1, ... , p}, p E N. Then

xE(XnCR,). Proof . Let p > 2 (the case p = 1 is trivial) and let x = (Xk) E cRP be such that xk E {0, 1,... , p} for each k E N1. Then we put

E? := {kEN I xk=j} (j=1,...,P) Note that at least one Ej is non-empty and that x = EJ=i j XR,. Let L be the least common multiple of 1, 2, ... , p and let p? := 1 (j E Ne, ).

122

Special summability methods

Then, applying Lemma 3.2.17 for each j E Np, we get pairwise disjoint sets of integers Eji with Pj

Ej=UEji and

1

XEj, -

i=1

XE, E CORP.

pi

Now, we define for each k E NL the set

Fk := U {Eji I j E Np, i E Np,

: (i-1)j
Then for each k E NL and each j E Np there exists a unique i(j, k) E NP, such that Fk fl Ej = Ej i(j,k) Accordingly, we see that

j

1

- XE, - XFkf1E,

Pi

XE, - XE,,;(j.k) E CORP

and therefore we have P

L X - XFk =

E (pj1 XEj - XE,.i(, k)E CORP.

Thus XFk E CR,, for each k E NL since x E CR,,. Noting L

E XFk k=1

L

p

LL`

E E XFkf1E, -

k=1 i=1 p

E

L

p

L I: X Ej,.(,,k)

k=1 i=1 P

XEj.¢(j.k) _ E jXE, = X j=1

j=I k=1

we obtain x = k=l XFk E (X fl cRP} and the proof is complete.

0

Proof of 3.2.13. (c)

(b) : We suppose that (b) does not hold for the matrix Rp (where p satisfies the hypotheses (3.2:4)). Then there exists a 6 > 0 and an index sequence (nr) such that

pn. > dP.n for all r E N.

(3.2:16)

Using the notation y = (y,) = Rrx (x = (xk) E w) we get

Pr,.yn,. - Pn,.-lyn,.-i x n, _ -

(3.2:17)

pn,.

Now, if x E cRP then yn and ynr-I converge to the same limit and hence we see from (3.2:16) and (3.2:17)-applying the Toeplitz-Silverman

is the transform by a regular theorem 2.3.71I-that the sequence matrix of the convergent sequence (or) where Qtr = and 92r+1 = yn. (r E N°), and hence limr xn,. exists. Therefore, in particular, if x E

Weighted means, Riesz methods

123

X n cRp, then there exists an integer ro such that xn. has constant value for all r > ro. Hence the matrix H = (h,k) defined by

r+1

ifknr

(r, k E N°)

-(r + 1) ifk = n,.+1

hrk

otherwise

0

sums every Rp summable sequence of zeros and ones. But H is not conservative, since IIHII = sup, E ° IhnkI = sup, 2(r + 1) = oo. Thus if (b) does not hold, then RP 0 KG, that is (c) implies (b). (c) : We now proceed" to show that (b) implies (c) and suppose that (b) (P) E co holds. Let H = (hnk) be any matrix with X n CR, C cH. Since Rp is regular, we have e, ek E (X n cRp) C CH. Consequently, in order to prove that H is conservative, by the Toeplitz-Silverman theorem 2.3.71 it is sufficient to show that IIHII = sup E Ihnkl < oo.

(3.2:18)

IhnkI < 00 for each n E N°.

(3.2:19)

n

k

First we verify that 00

k=0

Suppose that (3.2:19) is false. Then there exists some integer n for which Ek IhnkI = oo. We consider such an integer and keep it fixed. We may assume without loss of generality that each hnk is real. Then, since Ek hnk exists, it follows that hnk = oo

and

k

hnk = 00-

(3.2:20)

k

Since the condition (b) holds by assumption, the hypotheses of Lemma 3.2.14 are satisfied. Let the sequence (k,) be defined as in Lemma 3.2.14. Then by (3.2:20) we can define inductively an index sequence (rt) with r1 = 1 and such that for each t > 1 we have k°t+t _1 hnk

t2

and Pk < Pk,.

(k,. < k < k,.+1 , r > rt).

(3.2:21)

k=k,,

Consider a fixed integer k _> k1. There exist unique integers r and t such that k E I, = No n [k,., k,+1 [ and r E .It = NO n [rt, rt+1 [. Now, for any positive integers t and r E .Jt, it follows from the second inequality in (3.2:21) and Lemma 3.2.15 applied to the numbers pk, where k E Ir = NO n [k k,+1 [, that we can divide the set I, = NO n [kr, k,+1 [

124

Special summability methods

into t disjoint subsets, say N,.(1), N,.(2), ... , Nr(t), such that for4 j E Nt we have

57 kEN.(j)

1

1

Pk+ Pk. `

pk

1

kEl.

(Pk.+1 - Pk,. +pk. +pk.+1 +Pk,.) <

E

3

Pk.+1

(3.2:22)

Since for each r > 1 the set I, is a disjoint union of the subsets N,.(j), 1 < j < t, it follows from the first inequality in (3.2:21) that the contribution of at least one of these subsets to the sum in (3.2:21) must be greater than

or equal to t. Let j, be the least integer such that hnk > t. kEN,.(j,.)

We now define the sequence x = (xk) E x as follows. Let Xk = 1 if hnk > 0

and k E Nr(jr) for some r > 1 and let Xk = 0 otherwise. (We recall our assumption that n is a certain fixed integer for which >k Ihnkl = cc.) Then it follows from (3.2:22) and Lemma 3.2.14 that x E c°Rp. However, we also have for each fixed r > 1 the inequality hnkxk k

hnk

- t,

kEN,.(j.)

Since t -p co as r -* oo, we see that Ek hnkxk is not finite. Thus the Htransform of the sequence x is not defined, and hence x is not H-summable

contrary to x EX fl cRp C cH. This contradiction shows that (3.2:19) is true. Since relation (3.2:19) has been established, (3.2:18) can be proved by combining the ideas of the arguments presented above with the standard 'gliding hump' construction and is left to the reader (cf. Exercise 3.2.27). Thus, in Theorem 3.2.13, the implication `(b) (c)' holds, and so, along with (c)', we have so far `(a) the trivial `(a) (c) q (b)'.

(b) (a) : Let (b), that is (F) E co, be true. To prove RF E KP, it is sufficient, by Theorem 2.9.3, to verify that Rp E KG and that (X fl cRp ) is dense in (m fl cRp, II Since we have just proved that (b) implies (c), we have Rp E KG. The density statement follows from Lemmas 3.2.18 and 3.2.19 as we will now prove. Let x E m fl CRP be given. Without loss of generality we may assume that x is a real sequence (since otherwise we

consider separately the real and imaginary parts of x). Furthermore we assume x # 0 and, by translation (by ae) and scaling (by al -) of x (and then by scaling (by ((xI(,,.) and translation (by -ae) of y which we will choose), we may also assume 0 < xk < 1 (k E N°). Now, for any

given e > 0 we choose a p E N with n < e and according to Lemma 4 For n E N we use the notation N := {1,2..... n}.

Weighted means, Riesz methods 125

3.2.18 a sequence y which takes only values in {0, p, ... , ",1} such that x - y E coRp, that is y E cRp, and that lix - yll. < n < e. From Lemma 3.2.19 we obtain y E (XflCRp) (since py takes only values in {0,1, ..., p} ). 0 Thus, we have proved that (X fl cRp) is dense in (m fl cRp, iI 11"").

In 2.9.5 we stated that each strongly conservative matrix is necessarily potent. We close the section with an example of a regular Riesz matrix which is potent, but not strongly regular.

Example 3.2.20. Let pn = 1 and pn = 2 if n is odd and even, respectively. Then the Riesz matrix Rp is regular (cf. 3.2.7(b)), but not strongly regular (cf. Exercise 3.2.23(b)), and it satisfies T C CORp (because T C coc,) as well as bs ¢ cRp (since bs is dense in (fo, II cf. 2.9.11).

Exercise 3.2.21. Let 0 < ;Q < 1 and

pnF)

=

end . Show PnR)

- n, es ens

Exercise 3.2.22. Prove that Rp is either regular or coercive (see 3.2.7). Exercise 3.2.23. (a) Determine a regular Riesz mean Rp which is equivalent to (and therefore consistent with) convergence. (b) Prove that a regular Riesz mean Rp is strongly regular if and only

if (P) E co and (P Ek=° IN -Pk+11) E co- Furthermore, if (pn) is monotone, then Rp is strongly regular if and only if (P) E co. (c) Apply (b) to the Riesz method Rp() in Example 3.2.12(c) to prove that Rp( is strongly regular for each a > -1. (d) Give a detailed proof of the statements in Example 3.2.20. Exercise 3.2.24. Let (Pk) and (qk) be sequences which satisfy 3.2.1 and pk # 0 for each k E N. Prove the following statements: (a) Rq is stronger than and consistent with Rp if (2L) is monotonically decreasing.

(b) Rq is stronger than and consistent with C1 if and only if n

sup n

C Qn k=1

klgk-1-Q+

qn(n+1) < oa. Qn

Exercise 3.2.25. Let p = (Pk) satisfy 3.2.1. Prove the following statements:

(a) If pk # 0 for all k E N°, then cRp = c holds if and only if (p) E m. (b) If Pk := qk (k E No and q > 1), then cRp = c. (c) If (pk) is monotonically decreasing, then CR, D cc, with consistency. (d) If (pk) is monotonically increasing, then CR, C cc, with consistency. (e) If (pt) is monotonically increasing, then cRp = cc, with consistency,

if and only if (P ) E m.

126

Special summability methods

Exercise 3.2.26. Let a > 0. Verify that the Cesaro matrix Ca is a Riesz matrix Rp, where p satisfies 3.2.1, if and only if a = 1.

Exercise 3.2.27. Let p = (pn) satisfy (3.2:4) and (p-) E co. Further let H = (hnk) be a matrix with X fl cRP C cH. Then H has Sp and (hnk)k E Z (n E N°) as we have already proved in part `(b) =: (c)' of the proof of Theorem 3.2.13. Complete this proof by showing CtHIC < oc. (Note

the hints in the proof of 3.2.13.) Bibliography: [201], [196], [141], [47]

3.3

Norlund methods

Generalizing the Cesaro methods we consider in this section Norlund methods which are due to G. Th. Woronoj (1901), W. B. Ford (1909) and N. E. :Norlund (1920). We restrict our interest essentially to the characterization of those Norlund methods which are conservative and to those which are regular, to the proof of the pairwise consistency and to the comparison of regular positive Norlund methods. Norlund methods may be motivated as follows. The Cl-method maps every sequence into the sequence of its arithmetic means. Each CQ-method

(a E IR with -a (

N) transforms the sequence x = (xk) into the mean

1) E (n - k + a - 1) xk nn T+

n-k

k -o

k0

(n E NO).

For every n E N° we have (and this justifies the use of the word mean)

r

E

( k+k-11 [cf. 3.1.5(d)].

k=O

As a generalization of this we now introduce Norlund methods.

Definition 3.3.1 (Norlund methods). Let p = E w with Pn op. # 0 (n E N°) be given. Then the matrix N, := (N, p) (N, p.)

(pnk) is defined by Pnk

( °"-k t P. 0

if k < n

(k, n E N°)

otherwise

and the associated matrix method is called a Norlund matrix and Norlund method (with respect to the sequence p), respectively. As the following examples show, we have already encountered some Norlund methods.

Norlund methods

127

Examples 3.3.2. (a)'If po := 1 and pn := 0 (n > 1), then N. = I. (b) If p:=e, then N p = Ne = C1 = Re.

(n+,,-1)then N = C (d) Let a E 1K \ {0}, po := a, p1 := 1 - a, pn := 0 (n > 1). Then the

(c) If a E R with -a N and matrix

a ZQ ._

1-a

of

a

1-a

a

and the associated matrix method is called a Zweier matrix or Zweier method (of order a). Obviously, Za is equivalent to and consistent with the Norlund matrix

Np

_

a

a

1-a

a

Since the first column contains the value `0' the matrix Za and the above :Norlund matrix Np are not R.iesz matrices. L First we apply the Toeplitz-Silverman theorem 2.3.7 to Norlund methods to characterize those which are conservative and those which are regular.

Theorem 3.3.3 (conservativeness, regularity). For p =

with

Pn := F_'=o p $ 0 (n E N°) the following statements hold: (a) Np is conservative if and only if Pn

E c and

su

1

np TPn

n

Ipvf < Oo.

=o

(b) If N. is conservative, then Np is regular if and only if (y) E co. LIMIT FORMULA: If Nr is conservative and p := limn P. , then 11- pl < 1,

X(NN) = 1-Ep(1-p)k = k

1

ifp=0

0

if p 0 0,

and

limN, x = X(Np) lim x +

p(1 - p)kxk k

(x = (Xk) E C) .

128

Special summability methods

Proof. For each Norlund matrix N9 = (pnk) we obviously have: (i) N9 satisfies (Zn) if and only if supn it E'v_0 (p-( < oo. (ii} Ekpnk = 1 (n E N°); in particular, N9 satisfies (Zsl).

(iii) If Np = (pnk) is conservative, then (Pno)n = (p) E c. (iv) If Np is regular, then (Pno)n = (p) E co. Now, we prove

(v) If ( 3.) E c, then Np satisfies (Sp) and `d k E No

: limnpnk = limn' = p(1 - p)k.

To that end let (P) E c and p := limn P. . Then we get

P.-1 _ Pn - pn = Pn

i;?n

P.

Pn

p (n-00).

--3

Thus for every k E No : pn-k

pnk = Pn

pn-k

Pn-k

= Pn-k Pn-k+1

Pn-1

P.

-

p(1 - p}

k

(n -> oo).

The statement in (v) implies immediately:

(vi) If (P) E co, then Np satisfies (Spo). Thus we get (a) from (i), (ii), (iii) and (v). Then (b) follows from (i), (ii), (iv) and (vi). The limit formula comes from (v) and the limit formula in 2.3.7. The first equality for X(Np) follows from (ii) and (v), whereas the second one holds because p = 0 or 11 - pi < 1 which is a consequence of (limnpnk)kENo E P (cf. 2.3.6).

0

As an immediate consequence of 3.3.3 we have

Corollary 3.3.4. If po > 0 and pn >- 0 (n E N), then the Norlund method Np is conservative if and only if (P) E c, and regular if and only

if (P) Eco. Applying the general results of Section 2.8 we now give a sufficient condition for regular Norlund methods to satisfy the mean value property.

Theorem 3.3.5 (M1(NN), type M). Let N. with pn > 0 (n E N) be a regular Norlund method. If (pPk) is monotone increasing, then Np enjoys Ml (Np) and is therefore of type M. Remark and Examples 3.3.6. (a) The reader should note that, in view of the regularity of Np, the condition (k-) is monotone increasing implies that (pn) is decreasing. (b) For any a E10,1] the Cesaro matrix Ca satisfies Mj(CQ). This is an immediate consequence of Theorem 3.3.5 since Cc, = Np where pn = (n+«-i). That (pn) satisfies the assumptions of Theorem 3.3.5 may be easily n shown (cf. Exercise 3.3.17).

Ll

Norlund methods

129

Proof of 3.3.5. It is sufficient to verify that Np = (ank) satisfies the assumptions in Lemma 2.8.7 because then M1(Np) follows from 2.8.8 and consequently the property `type M' from 2.8.3. Obviously, ank > 0 (k < n E N°). Furthermore, for each fixed n E NO an+l,k

an,k

_ pn+1-k Pn Pn+1 Pn-k

Pn pn+l-k

(k < n)

Pn+1 pn-k

is monotonically increasing if (pti) is increasing. This proves the theorem.

0 To avoid unnecessary complications in surveying the results and proofs we restrict our discussion in the remainder of this section to regular Norlund matrices Np satisfying the assumptions in 3.3.4. The notion of a convolution of two sequences will prove useful.

Definitions 3.3.7. (a) If p = (pn), q = (q,) E w, then p * q :_ (y,,) defined by n

n

yn :_

pn-,,q.

(=

(n E N°)

pvgn-v)

is called the convolution of p and q. (b) Np is said to be positive, if p° > 0 and pn > 0 (n E N). The pairwise consistency of regular positive Norlund methods results easily from the following theorem.

Theorem 3.3.8 (convolution). If Np and N. are regular and positive, then cNP C cNP,q and cNq C cNp,q hold with consistency.

Corollary 3.3.9 (consistency). Regular positive Norlund methods are pairwise consistent.

Proof of 3.3.8. To prove Theorem 3.3.8 we apply 2.6.1(b); that is, we determine a regular transition matrix A with Np,q = AN.. To that end let x = (xk) E w, (vn) := Np,gx and (yn) := Nqx. Then for each n E No the equalities n

k

E xn-k E pvgk-v

vn

=

v=0

k=0

n

n

n

E pv E xn-kqk-v v=0 k=v

n

k

E E pvgk-v

k=0 v=0

n

E pvgk-v

v=0k=v

n-v

n

E pv F xkgn-v-k v=0

k=0 n

n-v

E pv E qk v=0

k=0

E PvQn-vyn-v

V=0

n

E pvQn

V=0

130

Special summability methods

>k=o qk

hold where Qn

(n E N°). We now define A =

Pn-PQP

if v < n

n

E PItQn-o

anv

by

(n, v E N°)

11=°

ifv>n

0

and verify the regularity of A by applying 2.3.711. Since

anv =

Ian,I = 1

(n E N°),

the matrix A satisfies (Zn) and (Zsl). Also, n

n

EPpQn-µ >_ qo EP,, = goPn A_°

(n E N°)

IA=o

because Qn-o > qo (p E No,). Thus, for each v E No and n > v we get

0 < anv

--

Pn-vQv

n

Pn-vQv Pngo

E PtcQn-µ

0

(n

co )

A=O

since Np is regular. Therefore, A satisfies (Spo) too, so that A is regular. By the definition of A we have

Np,.gx = A(Ngx) = (ANq)x (x E w), which implies Np,q = ANq if we consider x := ek (k E NO). Thus, applying 2.6.1(b), we get CNq C cNp,q with consistency and, interchanging the roles of p and q (note p * q = q * p), CNp C CNq.p = CNp.q with consistency. We have now proved 3.3.8. To characterize when two regular positive Norlund matrices are comparable, we make use of the following lemma. For any sequence y = (Yk) E w we define

y(t)

E yntn it

where we consider only such t E C for which the power series converges.

Lemma 3.3.10. If Np is a Norlund matrix with (P) E co and if x = (Xk) E c,vF, then the following statements hold:

(a) p(t) = E. pnt' and P(t) :_ F_n Pntn converge for f tl < 1. (b) (p * x)(t) = E. (Ev_o pvxn_v) to converges for Iti < 1. (c) x(t) _ n xntn converges in a neighbourhood of zero.

Noriund methods

131

Proof. (a) Since (p) E co by hypothesis, we have

PP1

=1-P

P"

(n - oo)

which implies the convergence of the power series P(t) for Itl < 1 and also

of the power series p(t) for Itl < 1 since p(t) = (I - t)P(t). (b) Let x = (xk) E carp, that is NFx E c C m. Thus the power series (p * x)(t) converges for Itl < 1 because P(t) converges for Itl < 1 (cf. (a)) and n

n

E

E pvxn-v

pvxn-v

v=0

v=0

(n E N°) Pn < iiNPxiIooIPn!

holds.

(c) Because p(O) = p° $ 0, there exists an open neighbourhood of zero U C {t E C I Itl < 1} with p(t) ¢ 0 for each t E U. Hence, by (a) and

(b) we get that r, defined by r(t) :=

P*-) t)

if t E U, is a holomorphic function on U and, in particular, r is representable by a power series on some neighbourhood U C U. We denote the coefficients of this power series

by rk (k E N°) and also put r := (rk). Therefore, for each t E U we have n

(p * x)(t)

p(t)r(t) =

=

Ep"tvrn-

n

,tn-v

[by 2.1.9]

v=0

n

n

(vrn_u) E

to = (p * r) (t)

v=0

on the one hand, and (p * x)(t) = n

(Ex_)

to

(t E U)

v=0

on the other hand. Consequently, by the uniqueness of the coefficients in a power series, we obtain n

n

E pvxn-v = > pvrn-v v=0

(n E N°).

v=0

Since p0 0 0 this equality implies xn = rn (n E N°) by induction, that is 0 x(t) = r(t) for t E U and the series En xntn converges on U. Before stating a comparison theorem for regular positive Norlund methods, we observe the following.

132

Special summability methods

Remark 3.3.11. Let p = (pn), q = (qn) E w with po > 0, qo > 0 and pn, qn >- 0 (n E N) satisfy (P) E co and (

) E co. Then there exist

r=(rn) Ew and s = (sn) Ew with

r*p = q and r*P = Q and

s*q = p and s*Q = P

where P := (P,,) and Q:= (Q,,). Proof. We show that for the given sequences p, q E w there exists an r = (rn) E w such that r * p = q and r * P = Q. (The second part of the statement follows by interchanging p and q.) Since po = Po # 0 there exists a neighbourhood of zero, U C It E C I Iti < 11, with p(t) # 0 and P(t) 0 0, respectively, for each t E U. Therefore P M and QP(t) are well-defined on U and they are even equal since p(t) = (1 - t)P(t) and q(t) = (1 - t)Q(t) for Itl < 1, respectively. Thus, on some neighbourhood of zero, U C U, those quotients are representable by a power series r(t) = En rntn. That is, r(t) = I } = (t E U). Consequently, for every t E U we obtain the identities (r * p)(t) = r(t) p(t) = q(t) and (r * P)(t) _ r(t) P(t) = Q(t) which imply the equality of the corresponding coefficients of these power series. That is, r * p = q and r * P = Q. Q1

Theorem 3.3.12 (comparison). For regular positive Norlund matrices Np and Nq the following statements are equivalent: (a) N. is stronger than Np (with consistency, see 3.3.9).

(b) (q) Eco and 3M>0 VnEN° :

En0IrvIPn- < MQn

where r = (rn) is obtained as in 3.3.11. Proof. First we remark that, since N. is a triangle, we have Nq = CNp with C := NqNp-1. Thus (cf. 2.6.2(b)),

Nq is stronger than N.

4=*

C is regular.

(3.3:1)

(b) : Let Nq be stronger than Np and let x = (xn) E cNP C cNa (a) be given. We are going to determine C = NgNp-1. To that end let (yn) := Npx, (vn) := Nqx, and let r = (rn) with q = r*p be determined according to 3.3.11. Then, if Itl is sufficient small, we have (cf. 3.3.10(b)) Qn,Untn' = n

n` L L: gvxn- ,tn = (q * x) (t) = q(t)x(t) n v=0

and n

E Pnyntn = E E Pvxn-vtn = (p * x) (t) = n

n v=0

p(t)x(t)_

NSrlund methods

133

It follows that for Its small,

!(t) p(t)x(t) = r(t)p(t)x(t)

Qnuntn = q(t)x(t) =

p

n

n

_ n v=0

Therefore, comparing coefficients, n

Q'nun = E rn-vPvT./v

(3.3:2)

vv0

We define A =

by

r"-P

anv :=

0

Qn

if v < n if v > n

(v, n E N°).

(3.3:3)

Since A is row-finite (cf. 2.2.5, (3.3:2)) we get Nqx = A(Npx) = (ANp)x, so that Nq = ANp by inserting x := ek, k E NO. Thus A = NgNp-1 = C. The matrix C is regular on account of (3.3:1). This implies n

11C11 = sup

lrvlPn-v > J Qn

< Do

[C satisfies (Zn)]

n v-°

and

0 = lim co n= lim

Tn Qn

P° ,

so that

T"

Qn

-+ 0

(n -3 oo).

We have proved that (a) implies (b). (b) . (a) Let (b) be true and A be defined as in (3.3:3). Then

Nq = ANp ,

therefore A = C = NqNp-1 [note r * p = q and 3.3.11],

and (a) is established because of (3.3:1) if we show that A is regular. But this is clear by the Toeplitz-Silverman theorem since (Zsl) and (Zn) are implied by r * P = Q (cf. 3.3.11) and the second statement in (b). Also A enjoys (Spo) since anv =

Tn-vPv Qn

_

rn-v

Qn-v. Qn-1 P. -+ 0

Qn-v Qn-v+1

Qn

(n -4 00).

Here we made use of Q. -+ 0 and Qnt -> 1 (n -> oo). Theorem 3.3.12 enables us to give a sufficient condition for regular positive Norlund matrices to be stronger than C1 This result also follows from Exercise 3.1.19.

Special summability methods

134

Corollary 3.3.13 (Nq stronger than Ci ). Let N. be a regular positive Norlund matrix where q is increasing. Then N. is stronger than (and consistent with) C1.

Proof. We have C1 = Np with pn := I and Pn = n+ 1 (n E N°). Thus, using the notation in 3.3.10-3.3.12,

p(t) =

1 1

t

r(t) = (1 - t) q(t)

and

[if Itl is sufficiently small]

and consequently rn = qn - qn_1 (n E NO, q_1 := 0). Therefore we get (note rn > 0 since q is assumed to be increasing) n

n

n

IrvIPn-,, = Y',(n+1-v)(q,-q,-1) = 57 qv v=0

v=0

= Qn;

v=0

that is, we may put M := 1 in 3.3.12(b). Furthermore, rn

_

Q.

qn

Qn

qn-1 Qn

-+ 0

(n -* oo)

since Nq is regular (cf. 3.3.4) and q is increasing. Thus the other condition 0 in 3.3.12(b) is satisfied too, and the assertion follows from 3.3.12.

We close this section with a characterization of equivalence of regular positive Norlund matrices.

Theorem 3.3.14 (equivalence). If Np and Nq are regular positive Norlund matrices, then the following statements are equivalent: (a) Np and N. are equivalent (and consistent).

(b) r = (rn) E e and s = (sn) E P where r and s are defined according to 3.3.11.

Proof. (a)

(b) : Let (a) be true. Then, applying 3.3.12(b), we may choose an Al > 0 such that n

n

E

MQn and

O's.IQn_,, < MPn

(n E N°).

(3.3:4)

1-0

V=O

In particular (note r0 > 0 and s0 > 0)

0 < r0Pn < MQn and 0 < soQn < MP,, (n E N°), therefore

0< Pn

Q.

<

AI

r0

and

0
<

M (nEN°).

(3.3:5)

s0

Furthermore, we get from (3.3:4) and (3.3:5) for every n E N and fixed k E Nn the statement

Norlund methods

Irol+Iri I

Pn-1

+ ... +Irkl

P.

Pn-k

P.

<

MQn

P.

135

< MZ so

Thus, by the passage to the limit `n -* oo', the inequality k

EIrvl< V=o

M so

holds and p `` -> 1 (n -4 oo, v fixed and v <_ n). Hence Similarly, we obtain

E f.

E e.

(a) : Let (b) be true. Then we have necessarily (rn) E co and (b) (sn) E co, and therefore also

(Q) E co

and

(in-)

nE co

(3.3:6)

n

since 0 < qo < Qn and 0 < po < Pn (n E N). Because Qn < Qn+i this implies (cf. 3.3.11)

P. = Qosn + ... + Qnso < QnIISIIF Thus,

(n E N°).

n

E IrvlPn-v < llrlli IISIII Qn (n E N°).

(3.3:7)

v=0

Similarly we obtain n

E ISyIQn-. < IIsII1 IIrVI, Pn

(n E N°).

(3.3:8)

M=O

With (3.3:6), (3.3:7) and (3.3:8) statement (a) follows by applying 3.3.12. 0 Hence (b) implies (a). Exercise 3.3.15. Determine for a E K\ {0} the inverse of Z,,,, and show

that c = cze holds if 11 - al < Ial. Exercise 3.3.16. Prove the following statements: (a) The Norlund method (N,pn) with pn := n+z}z (n E N°) is regular. (b) The Norlund method (N, pn) with pn := a' (n E N°) is conservative if 0 < a, and regular if 0 < a < 1. Exercise 3.3.17. Verify that, for p = (pn) defined by pn := (n+«-1), the n sequence (DD ) is monotonically increasing (cf. 3.3.6). Exercise 3.3.18. Apply Theorem 3.3.12 to prove the monotonicity of the Ce-methods for a > 0.

Exercise 3.3.19. Let pn :=

n+1

(n E N). Show that the :Norlund

method (N,pn) is strictly weaker than every Cesaro method C, a > 0. Bibliography: [196], [108], [201]; [267]

136

Special summability methods

3.4 Hausdorff methods In connection with moment problems the matrix methods named after Hausdorff were examined first by Hurwitz and Silverman (1917) and later by Hausdorff (1921). The class of Hausdorff methods includes-as we will subsequently show-the Holder, Cesaro and Euler methods. A large number of other matrix methods which play an essential role in summability are Hausdorff methods too. We will introduce Hausdorff matrices based on differences (of higher order) of sequences and the so-called difference matrix which is connected with them. On account of the agreeable properties of the difference matrix and the resulting definition of the Hausdorff matrices, we may easily deduce the essential properties of Hausdorff matrices and the associated methods. In particular, we may establish the (pairwise) consistency of all regular Hausdorff methods and characterize the comparison of Hausdorff methods using transition matrices. However, the situation will become much more complicated when we apply the Toeplitz-Silverman theorem to characterize those Hausdorff methods which are conservative or regular. A Hausdorff matrix satisfies the row norm condition (Zn) if and only if the sequence generating the matrix is a so-called moment sequence. The representation of Holder matrices as Hausdorff matrices enables us to extend the definition of the Holder matrices H from a E N° to a E C. In the last part of the section we will characterize those conservative (real) Hausdorff methods which are potent; these are exactly the strongly regular Hausdorff methods which were characterized by Lorentz (148]. We begin by introducing the notions of differences of order k for k E NO, the difference matrix, and subsequently Hausdorff matrices and methods.

Definition and Remark 3.4.1 (differences of order k, k E N°). For any x = (xk) E w and k E NO we define inductively AoXk := xk and Onxk := An-ixk - An-'Xk+t

(n > 1).

By induction (on n) we get A'Xk =

n

(-1)" (n) v

"_°

Xk+v

(k,n E N°).

In particular n

tlnx° "-0

(n) X, V

(n E N°).

All

The last identity defines a matrix transformation which we wish to emphasize.

Hausdorff methods

137

Notation 3.4.2 (difference matrix). The matrix 1

0=

p

1

-1

1

-2

1

-3 3 -1

1

with coefficients Ant,

{ (-1)v (v) if 0 < v < n

ifv>n

0

is called the difference matrix. We now use the difference matrix to define Hausdorff matrices and methods.

Definition 3.4.3 (Hausdorff methods). Let p = (pa) E w be given. Then the matrix

HP := (H, p) := (H, pn) := d diag(pri) A is called the Hausdorff matrix (for p) and the associated matrix method is called the Hausdorff method (for p). (Note that Hp is well-defined since it is the product of row-finite matrices, see 2.2.6(i).) D Before we give examples of Hausdorff methods, we establish some properties of the difference matrix and general properties of Hausdorff matrices.

Properties 3.4.4 (of Hp and O). Let Hp and Hq be Hausdorff matrices. (a) Coefficients (of Hp): If Hp = (hnk), then

hnk = to [ln'kpk

if 0 <

k Sn

(k, nEN°).

if k > n

In particular hnn = pn (n E N°) and Hp is a lower triangular matrix. (b) Hp as a triangle: Hp is a triangle if and only if pn # 0 (n E N0). (c) Sum (of Hausdorff matrices): Ha + Hq = XP} q.

(d) Inverse (of A): ©A = I, that is A` = A. (e) Products (of Hausdorff matrices): HpHq = (H,pngn) = HgHp. (f) Inverse (of Hausdorff matrices): (Hp)-1 exists if and only if pn # 0 (n E N). If applicable, (Hp)-i = (H,pn 1) (g) Characterization (of Hausdorff matrices): Let p = (pn) with p, # pk (n # k) be given (for example, pn := n+1 ) and let A = (ank) be a lower triangular matrix. Then A is a Hausdorff matrix if and only if AHp = HPA. (h) Consistency (of regular Hausdorff methods): Regular Hausdorff

138

Special summability methods

methods are pairwise consistent.

(i) Comparison (of Hausdorff methods): If Hp and Hq are Hausdorff matrices and Hp is a triangle, then the following statements hold: (1) Hq is stronger than Hp if and only if (H, p) is conservative.

(2) Hq is stronger than and consistent with Hp if and only if (H, I-) is regular.

Proof. We only sketch the proofs. (a) The last statement in (a) is trivial because the product of lower triangular matrices is again a lower triangular matrix. Because A diag (pn) _ we get for all k < n the equalities Anvpv. vk =

hnk

E(-1)"

()P(_1)k (k )

v=k

//1

n

E(-1).+kp. (k) (v - k)

v=k

n-k

(k) (-1)°Pv+k (n I/

k)\

_ (k) an-kpk

(b) Note, pn is the coefficient of Hp in the nch position of its diagonal. (c) This is an immediate corollary of (a) since An-k is linear for n > k. (d) To prove this we apply 3.4.4(a) to determine the coefficients of (hnk)

He = aIA = AA. We have hnk

(n) ,n-k 1 if k < n 0

ifk>n

which forces AA = I because All = 1 and and = 0 if n > 1. (e) On account of (d), and since Hausdorff matrices are row-finite, we get

HpHq = (adiag (pn)A) (adiag (qn)A) = adiag (p,) diag (qn) A = Adiag(pngn)A = (H,pngn).

(f) Hp is a triangle if and only if pn 54 0 (n E N°) on account of (b). Thus, for q = (qn) with qn := pn 1 (n E NO), we get HpHq = (H, pngn) _ (H, e) = I by (e), which proves that H. is the inverse of Hp. (g) On account of (e) the condition AHp = HpA is necessary. To prove the sufficiency we assume AHp = HpA. Using the definition of Hp and multiplying this equation from both sides by A, we get AAA diag (pn) =

diag (pn) AAA. Therefore, if we put B = (bnk) := AAA, the identity bnkPk = pnbnk results. Thus bnk = 0 if k 36 n since pn 0 Pk (k 0 n).

Hausdorff methods

139

Hence B = diag (bnn) = AAA. Therefore, A = A diag (bnn) A. (h) To prove this let x E cH, fl cH, be given. Then

limes, x = lim Hx = limes, Hpx = lim (HqHp)x = lim (HpHH)x = limH, Hgx = lim Hqx = limes, x on account of (e), Hpx E c and Hqx E c. (i) Both (1) and (2) follow from 2.6.2 because

HgHp-I

= (H, p) (cf. (f),

(e)).

Cl

Now we show that both the Holder methods and the Cesaro methods are Hausdorff methods.

Example 3.4.5 (Holder methods). H° = (H, (n+l}a) for any a E N°.

Proof. Since H° = I = He and because H" _ (CI )" we see from 3.4.4(e) that it is sufficient to prove HI = (H, +I by induction on v the identity v!

1

To that end we show

(v, k E N°)

k+1

which obviously implies HI = (H, n+i) by 3.4.4(a). For v = 0 it is trivially true. If it holds for a given v E NO and each

kEN°, then Qv+l k+

Ov 1

k+ 1-

All

k

1

2

[compare 3.4.1] V!

V!

(k +

1)

(k + 2)...(k + v + 2)

[using the inductional assumption]

_

V!

(k+v+2)-(k+1) (k+1). .(k+v+2)

which completes the induction.

(v+1)! C]

The equality of HI and (H, (nil),) for each a E No suggests that we introduce Holder methods for any a E C by using the relation in 3.4.5.

Definition and Remark 3.4.6 (Holder methods). For any a E C the matrix H" :_ (H, (n+I e) is called the Holder matrix of order a, and the associated matrix method is called the Holder method of order a. (Note, by 3.4.5 this definition is consistent with that in 3.1.1.) Obviously,

H°H3 = Ha+a for all a,,3 E C in view of 3.4.4(e).

We will subsequently return to Holder methods.

LL

140

Special summability methods

Example 3.4.7 (CesarQ methods).(.;y). Ca = (H,p(a)) for each a E R with -a 0 N where p(a) _ (pna)) :_ n

Proof. Using 3.4.4(a) and noting that Al (a) = pk

1

1

k° -

k+i

= 3+1 a

T

= a+1 a pk(a+l)

1 1

(k E

`l ),

we obtain for k < n the identity

n An-k (a)

a n-k-1

a 1 (k) a+1 a+2 (n

pk

(k)

a+n-

n!

n-k)! k! (a+1) (a+2)

n-k+a-1

_ thus Ca =

(

/n+a)

ln

)

=

( pkn-k+a)

a1) (n-k+a-l) k!

(a+n-k) (n-k+a+1)

(n+a)

[cf. 3.1.7];

cnk)

(H). 71

O

As a further example of a Hausdorff method we present the EulerKnopp method Ea. For a = 2 it was introduced (with a slightly different notation) by Euler (1755) in an attempt to speed up convergence for the purpose of computation with series. For the case of real non-negative a it was introduced by Hausdorff (1921) and Hurwitz (1922) and was exhaustively examined by Knopp (1922/23). The case of negative and complex a was considered by Agnew (1944) in an extensive paper.

Definition and Properties 3.4.8 (Euler-Knopp methods). For any a E C the Hausdorff matrix

Ea := (e(a)) := (H, an)

is called the Euler matrix (of order a) and the associated matrix method is called the Euler-Knopp method or just the Euler method (of order a). For a, 8 E C we have the following properties of the Euler matrix (method): n-k / (1)

6nk)

- (k)

L_. (-1)v (nUk)

k+v

v=0

= (k) ak (1 - )n-k (k < n) [cf. 3.4.4(e)]. (ii) EaEE = EE0

(iii) (E,,,)-' = Ei if a # 0

[cf. 3.4.4(a)].

[cf. 3.4.4(f)].

[because F,k=° elk) _ (a + 1 - a)" = 1]. (iv) Ea satisfies (Zsl) (v) E. satisfies (Zn) if and only if 0 < a < 1 (because Ek_o Ienk'I = (Ial + I 1 - aI)" f o r each n E N O ].

(vi) Ea is conservative if and only if 0 < a < 1.

Hausdorff methods

141

[Referring to 2.3.71, the method EQ is conservative if and only if it satisfies (Zn), (Sp) and (Zs). Thus, on account of (iv) and (v) it remains to prove that E. satisfies (Sp) if 0 < a < I. The case a = 0 is trivial since e(o) = 1 (n E N° ) and e ;,°k = 0 (n E f , k > 0). In the case 0 < a < 1 the condition (Sp), in fact (Spo), follows from ienk>I

nk

k1 = (k)ak(1 - a)n-k < Qk(l-Q)-k

(n,kENO)

(l_Q)-n

since l } converges to zero if n tends to infinity by L'Hospital's rule.] (vii) Ea is regular if and only if 0 < a < 1 [cf. (vi) and its proof].

(viii) For all a, 0 E R with 0 < j3 < a the method E8 is stronger than and consistent with Ea. Hence, in this sense the Euler methods E, 0 < a < oo, are monotone. [From (ii) and (vii) we get E,e = EpEQ and that Ep is regular. Now apply a

3.4.4(1).]

o

Ll

Theorem 3.4.9 (type M). Each regular Euler matrix is of type M. Proof. Let Ea = (ank) be a regular Euler matrix which is unequal to the identity matrix, that is 0 < a < 1 by 3.4.8(vii). We now apply Theorem 2.7.5. To that end we consider the inverse E,,-' = (ank) = El (cf. 3.4.8(iii) and (i)). We have for every z E C :

E

ak"z" =

"=0

" =0

E (k)

(a)

a}k "z"

(1-

"

"=0

()--' {a - 1)k-" (a

}}

k

(a - (a - 1} } It is easy to verify that supk sup {Isk(z)I n

n

k

k=0

I

Iz - (1 - a) I < a} < 1 and that n

k

ak"z" = E z"

E anksk(z) = E ank k=0

= ak (z

"=0

"-0

ankak"z" = Zn "=0

for each z. Thus, if we choose a sequence (zi) with zi $ z? (i 11(zi)II. < 1 and Izi-(1-a)I < a (i E 1k°), then Xki := sk(zi) (i,k E N°) fit the conditions in 2.7.5. Hence, Ea is of type M.

0

We now aim to characterize conservative and regular Hausdorff matrices in terms of conditions on the sequence p which determines the matrix H. For example, in the case of the Euler method E. we have p = (an) and, see 3.4.8(vi) and (vii), the corresponding characterizations by 0 < a < 1 and 0 < a < 1, respectively. The Euler matrices are typical for that question.

142

Special summability methods

We will state that each Hausdorff matrix satisfies the row sum condition (Zs) and that it is conservative if and only if (Zn) is satisfied.

Theorem 3.4.10 (row sums). If Hr = (hnk) is any Hausdorff matrix, then F_k hnk = po (n E NO). In particular Hp satisfies (Zs) with limit po.

Proof . Obviously we have = d diag (pn)1 e = A diag (pn) e° = po1e° = poe

hnk) nEN°

k

since

n

n V=0

/n

1

Anv = 1(-1)L \() = (1-1)n = {o ll

v=0

if n = 0

ifn0

0 for every n E N°; thus Ae = e°. The row norm condition (Zn) is certainly satisfied in the case of any matrix with non-negative entries if the row sum condition (Zs) is satisfied. Therefore, 3.4.10 and 3.4.4(a) ensure that a Hausdorff matrix Hp has (Zn) if the differences A'Pk are non-negative. We single out sequences p with that property in the following definition. Definition 3.4.11 (totally monotone sequences). A real sequence

x = (xk) is called totally monotone if dnxk _> 0 for all n, k E N°. We call a complex sequence x = (xk) totally monotone if ( txk) and (xk) are totally monotone sequences. Now, the following remark is an immediate consequence of Hp+q = Hp + Hq and the discussion above. Remark 3.4.12. If p E w is representable as a difference of totally monoA tone sequences, then the Hausdorff matrix H. satisfies (Zn). Hausdorff proved in 1921 that the conservative Hausdorff matrices Hp are exactly those for which p is the difference of two totally monotone sequences.

Theorem 3.4.13 (conservative Hausdorff methods). If p = (pn) E w and Hp = (hnk) is the associated Hausdorff matrix, then the following statements are equivalent: (a) Hp is conservative. (b) There exist totally monotone sequences r, s E w with p = r - s. LIMIT FORMULA and (Sp) : In such a case h° := lim hno = lim An'po , hk := lim hnk = 0 (k > 1) n n

n

(3.4:1)

and limHy x = (po - lim tlnpo) lim x + x0 lim Onpo n

n

for each x = (xk) E C. (3.4:2)

Hausdorff methods

Proof. (b)

143

(a) : Let (b) hold. Since every Hausdorff matrix satisfies

(Zs) (cf. 3.4.10) and (Zn) holds if (b) is satisfied (cf. 3.4.12) the matrix

Hp is conservative if it enjoys (Sp) (cf. 2.3.71). To that end we establish (3.4:1). Then, as an immediate consequence of the limit formula in 2.3.71, we get (3.4:2). Assume that p is real and

totally monotone. (Otherwise, in the case of p = r - s we may consider the real and imaginary parts of r and s and note that [Hp}nk = hnk = [HRr}nk + i [Hc3r}nk - [HRs}nk - i as well as lim hno n

=

lim A'Rro + i lim A'nsro - lim AnQso - i lim Onsso

=

lim Onpo

n

n

n

n

n

and limn hnk = 0 for all k > 1.) We now verify the existence of

hk = lim (k) ,n-kPk

(k E N0)

and prove ho = limn tlnpo as well as hk = 0 (k > 1). By the definition of An we get An-kpk =

An-k+l pk

+ An-kpk+l

(n > k).

Multiplying this equation by (n + 1) /(k) we obtain from

(n+1) (n) _ (n-k+1) (nk 1) _ (k+1) (k+1) the identity

(n + 1) hnk = (n - k + 1) hn+l,k + (k + 1) hn+l.k+l and so

(n + 1) (hnk - hn+1,k) = (k + 1) hn+l,k+l - khn+l,k

for all 0 < k < n. Putting Hn

(n+l) (Hnt,-Hn+t,i) =

F_x=o hnk (n, v E N°) we get

(n+1) (hnk-hn+l,k) = (v+1) hn+1.i+1 > 0 k=O.

(3.4:3)

for all n > v. This proves that the sequence (Hnr,)nENo is non-negative and non-increasing for n > v. Thus it is convergent. Consequently, the sequence (hnv)nENo is also convergent for each v E No since nv

ifv=0

Hnv-Hn,v_1 ifv>1 {Ho=po

Therefore Hp satisfies (Sp) and ho = limn Anpo

(n

)

E 1Nl°.

144

Special summability methods

We still have to show hk = 0 (k > 1). For that it is sufficient to verify the convergence of the series

57 n

n + 1 hn+l,v+l

for each v E No

which obviously implies 0 (v E N°). Because of the convergence of this is an immediate consequence of N

N

1

1

n=a

7t + 1 hn+l,v+l

v+1 E (Hn,, - Hn+1,,,)

[because of (3.4:3)]

n=u

V+1

(Hp -

0

(N > it -+ oo)

and the Cauchy criterion. (a) (b) : Since the proof is very technical, we omit it and refer the reader to Wilansky's book [254, proof of Theorem 2.3.14 on pages 26/27] 0 or Hardy's book [108, Theorem 201].

Although Theorem 3.4.13 characterizes the conservative Hausdorff methods, it is usually very hard to verify directly from the definition whether a sequence is totally monotone or the difference of such sequences.

Therefore the problem now is to characterize those sequences which are representable as a difference of totally monotone sequences. The solution of this problem, which we now present, is due to Hausdorff.

Theorem 3.4.14 (moment sequence). (a) A real sequence p = (p,,) is totally monotone if and only if there exists a non-decreasing function

x:[0,1]-+R with

Pn = JI 1 tndf,(t)

(n E N0).

(3.4:4)

0

(b) A sequence p = (pn) E w is representable as the difference of two totally monotone sequences if and only if p is a moment sequence, that is5

3

E

BV([0,1]) : pn =

f tnd(t)

for every n E N°.

In both cases we say that I generates p.

Proof. (b) We reduce the statement of (b) to that of (a) by using the fact that a real function on [0, 1] has bounded variation if and only if it is representable as the difference of two non-decreasing functions on [0, 1]. 5 BV([0,1]) denotes the set of all functions 4' : [0, 1] - K with bounded variation. In connection with moment sequences we use the Riemann-Stieltjes integral.

Hausdorff methods

145

If p = (pn) - r - s where r = (rn) and s = (sn) are totally monotone then, in accordance with (3.4:4), for Rrn, rn, Rsn and !s, we may choose non-decreasing functions 4)1r, 4'2r, 41s and 12$i respectively. Obviously, 4i := 4lr - (11a + i(4i2r - 12s) E BV ([0,1]) and 1

pn = f tnd4i(t) for every n E No

(3.4:5)

d0

which is the desired representation.

Conversely, if p = (p,) E w has a representation of the form (3.4:5) with It E BV([0,1]), then we get R1 and s4i E BV([0,1]). Therefore, using the fact that a real function on [0,1] has bounded variation if and only if it is representable as the difference of two non-decreasing functions on [0, 1], there exist on [0, 1] non-decreasing functions 4ik (k = 1, ... , 4) such that 4>

_ R1+i&t '1- 2+i43-i4t4 =

Putting

rn = J tnd($1(t) + i' 3(t)) 1

and

s,, = fo tnd(4-2(t) + i'14(t)) 1

0

for each n E N° we get totally monotone sequences (rn) and (sn) with

p = (rn) - (sn). (a) First we assume that p = (pn) is a real sequence which is representable in accordance with (3.4:4) where 4? : [0, 1] ---+ 118 is a non-decreasing func-

tion. Then, using the binomial formula, we obtain for all n, k E NO (cf. 3.4.1)

QPk

(-1)v =

(:) Pk+v

(-1() f tvd(t) v=0

J tk(l - t)nd4i(t) > 0 1

0

(3.4:6)

0

since the integrand is non-negative and 4? is non-decreasing. Thus the sequence p is totally monotone. Conversely, let p = (pn) be a real, totally monotone sequence. For the construction of a non-decreasing function 4i with the desired property, we refer again to Wilansky's book [254, proof of Theorem 2.3.18] or Hardy's book [108, Theorem 204].

Combining 3.4.13 and 3.4.14 we have the following characterization of conservative Hausdorff methods.

146

Special summability methods

Theorem 3.4.15 (conservative Hausdorff methods). For every sequence p = (pn) E w the following statements are equivalent: (a) Hp is conservative. (b) p is representable by the difference of two totally monotone sequences. (c) p is a moment sequence.

Additional statements: If p is the moment sequence generated by 4i E BV([O,1]), then the matrix Hp = (hnk) enjoys the following properties:

(i) h.nk = (k) 4 tk(1 - t)n-kd4i(t) for 0 < k < n and n E N°. {ii} Ek hnk = po = 4i(1) - 41(0) for each n E No.

(iii) ho = limninpo = 4i(0+) - 4>(0) and hk = 0 (k > 1), where we put 41(0+) := limt-,o,t>o (1(t). As an immediate consequence of the preceding theorem and 2.3.711 we obtain a characterization of regular Hausdorff methods.

Corollary 3.4.16 (regular Hausdorff methods). For every p E w the following statements are equivalent:

(a) Hp is regular. (b) p is a moment sequence which is generated by some 4) E BV([0,1]) satisfying 4i(1) - 4i (0) = 1 and ' (0) = (1 (0 +).

Proof of 3.4.15. In view of 3.4.13 and 3.4.14 we need only prove the `additional statements': (i) This follows directly from 3.4.4(a) and (3.4:6) in the proof of 3.4.14.

(ii) Thanks to 3.4.10 we have only to verify that po = 1(l) - 1(0). This is true since

= fo do(t) = b(1) - 4i(0). l A = f t°d4i(t) 1

0

( iii)

Applying 3.4.13 we have only to show that ho = 4'(0 +) - 4i(0). By (i) we have for each n E NO the identity

hno = Lnpo = fo t°(1 - t)nd4i(t) = 1

f (1 l - t)n'd (4;(t) - 4i (0 0

If n > 1, then integration by parts gives us hno = 4i(0 +) - 4i(0) + In with In := n

f(i -t)1 ((t) 1

(0 +)) dt.

Now, let e > 0 be arbitrarily given. Using the definition of 4i(0 +) we may choose a 5 > 0 such that

4i(0+)) < 2 for every t E]0,5].

Hausdorff methods

147

Thus we obtain

III

<

n

J6(1-t)n-114?(t)-4?(0+)Idt 0

+n

f(i - t )1I(t) -

(O+)j dt

r

n f(1 - t)1dt + 6
<

fdo

2

n

f(i - t)1t + sup I(t) -

(0 +)1 (1-

o
+ K (1 - S)n where K := supo
Since (1 - b)n -+ 0 as n -4 oo, we may choose an no with 6) n

< 2(K + 1)

(1 -

for each n > no.

This gives 1In1 < s for each n > no, so that limn hno = (1 (0 +) - 'P (0),

0

which completes the proof.

Having developed a theoretical base, we now examine the moment sequences for the Euler-Knopp, Holder and CesAro methods. This gives an easy way to determine when these methods are conservative or regular.

Examples 3.4.17. (a) The non-decreasing function 4i : [0, 1] -+ R defined by 4i(t)

J0 if0
ift=1

provides

pn :_

I1

tnd4i(t) = 1

(n E N°).

Jo

Thus Hp = I. (b) If a E [0, 11 is given, and the non-decreasing function 4ia : [0, 1] --+ lI is defined by 0

(Do (t)

{

1

if t = 0

if0
and 45a(t) :_

0 1

if O
ifa
(a E ]0, 1]),

then

pn := J tnd4?a(t) = an 1

(n E 1O).

0

Therefore HD = Ea (cf. 3.4.8). It is now an easy exercise (cf. Exercise 3.4.24) to prove that Ea is conservative if a E [0, 1] and regular if a E 10, 1], as we proved in 3.4.8(vi) and

148

Special summability methods

3.4.8(vii).

(c) Let a > 0. If

(1 - t)° = a tf(1 - r)ldr

(t E [0,11),

then (D : [0,1] - R is non-decreasing and differentiable. Therefore, using a theorem on Riemann-Stieltjes integrals, we have

pn = j tnd4'(t) = fo t9'(t) dt = a foi t"(1- t)°-idt. 0

T hus along with standard representations of the beta function and the gamma function we may show that n+al-i

pn =

n

/)

for each n E No

Therefore, Hp is the Cesaro method C°. Applying 3.4.16 we get, since F(1) - 4t(O) = p0 = 1 and 40 is continuous (at t = 0), the regularity of C° (a > 0), as we proved in 3.1.9.

(d)Ifa>Oand

t0

fo (log u)

°-i

du

if 0 < t < 1

ift=0,

we have

p

f

td

(

t) =

t

(log

dt = (n +

1)

(n E

II

that is Hp = H. Using H° = I and 3.4.16 we see that H° is regular for any a > 0, which we proved for a E NO in 3.1.3.

A

We complete the characterization of the comparison of Hausdorff methods in 3.4.4(i) as follows.

Theorem 3.4.18 (comparison of Hausdorff methods). Let p = (pn), q = (qn) E w be given. If Hp is a triangle, then the following statements hold:

(a) Hq is stronger than Hp if and only if (P) is a moment sequence.

(b) Hq is stronger than and consistent with Hp if and only if (.9) is a moment sequence generated by a function E BV ([0,1]) which satisfies ,P(1) - 44(0) = 1 and is continuous at t = 0.

Proof. The statements follow immediately from 3.4.4(i) and 3.4.15 and 3.4.16.

0

Hausdorff methods

149

From 3.4.18 we may conclude that the Holder and Cesaro methods are equivalent, although the details are somewhat technical.

Corollary 3.4.19 (see also 3.1.16). For each a > -1 the methods Ca and H* are equivalent and consistentProof. For this we refer the reader to [108, Theorem 211]. The monotonicity of the Ca-methods (a > -1) may also be directly concluded from 3.4.18; however, the proof of the fact that () is a moment sequence, that is the determination of 1, is non-trivial. IL is much easier to prove the monotonicity if we make use of the equivalence of the Ca- and Ha-methods (cf. 3.4.19) and the monotonicity of the H°-methods (a E IIt), which follows immediately from H'+J3 = H"Ha (cf. 3.4.6), the regularity

of H' for a > 0 (cf. 3.4.17(d)) and the `comparison by the transition matrix' (cf. 2.6.2). We record these facts in

Remark 3.4.20 (monotonicity of Ca- and Ha-methods). For all a,,6 E R with a < 03 we have

cH. C cHs with consistency and consequently also

cc. C ccs with consistency for all a, ;Q E IR with -1 < a < 0. The details of the proof are left to the reader (cf. Exercise 3.4.25).

0

We have stated in 3.4.7 and 3.3.2(c) that each Cesaro matrix is both a Hausdorff matrix Hq and a Norlund matrix Np. If we restrict our interest to real sequences q and p then we may prove (for details see Exercise 3.4.26) that these are the only methods with this property. We now characterize the conservative (real) Hausdorff matrices which are potent and we state that these are exactly the strongly regular Hausdorff methods. The last class has been characterized by a simple condition by Lorentz (1948) and, based on that result, Kuttner and Parameswaran (1994) have characterized the Hausdorff methods which are potent.

Theorem 3.4.21. If p = (p,) is a real moment sequence, generated by E BV([0,1]), and Hp = (hnk) is the corresponding Hausdorff matrix with column limits hk, then the following statements are equivalent: (a) Hp is strongly conservative.

(b) TCcH,. (C) (SUpkIhnk - hkI)n E Co.

(d) is continuous at t = 1. (e) (hnn)n = (p,,)n E co. (f) X n CH,, is not countable.

150

Special summability methods

(g) Hp E KG. (h) Hp is potent, that is Hp E KP. Additional remarks: (f) implies X n cHy ¢ c since X fl c is countable. If p is a complex moment sequence, then-as one can prove-the conditions (a), (b), (c), (d) and (e) are pairwise equivalent, whereas there exists a potent regular Hausdorff matrix which is not strongly regular. For the proof of this theorem we need the following lemma.

Lemma 3.4.22. Let pnv be Bernstein's basis polynomials defined by pnv : [0 1] -> R, t ---+ Pnv (t) Then for any E ]0, 1[ the sums uniformly for t E [0, S] as n --> oo.

(fl)tv(l - t} "-" for n, v E

.

Ip(t) -pn,,,+1(t)I converge to zero

0

Proof. Since it is very technical we omit it (cf. [151]).

Proof of 3.4.21. We will verify `(a) #- (h)

(f)

(g)

(e)

(d)

(e)'. We first note a number of these implications that (h)' (because every strongly conservative are immediate. Namely, `(a) (g)' (cf. 2.9.1), `(g) (f)' (cf. 2.9.9), matrix is potent, cf. 2.9.5), `(h) (b)' (since T C fo, cf. Exercise `(f) . (e)' (cf. 2.9.10 and (p,,,) E c), `(a) (c)' (cf. 2.4.14), `(c) 1.2.23), `(b) (e)' (since (hk) E co). (e) = (d) : We are going to prove `(e) = (d)' by contraposition. To that end we assume that, without loss of generality, it is monotone increasing (a)

(b)

(c)

and has a jump in t = 1. Then 4'(1) # lim2,1_ I(x). We put a 4i(1) - lilnx +1_ 4i(x) and consider the function

r= ift<1

a ift=1.

Of course 4f E BV([0,1]) and 4? := 4? - 4t is continuous at t = 1. Now, we show that this leads to limn hnn = limn p,, $ 0. From our definition of 4' we have 1

pn =

(1

to d 4 (t) _ (

0

fo

f

(1

trz d T(t) + (

0

(1

t" d 4 (t) - (

todT(t)+ f tnd(t) = a+ J 0

to d IF (t)

0

tnd4>(t).

(3.4:7)

0

Now, since 4;(t) is continuous at t = 1, the total variation Vo (4?) of 4i on [0, t] is continuous at t = 1. Thus, for a given e > 0, we can choose a

E E10, 1[ such that V11 a(4?) = v,,(4?) - Va -a(4i) < z. Therefore we have fotnd4i(t)

<

fo

tnd4i(t) I +tnd1?(t) 1

a

Hausdorff methods

<

(1 - 6)n Vo -b(4:) + V'

6

(4t) < e

151

(3.4:8)

for sufficiently large n. Using (3.4:7) and (3.4:8) we conclude pn -4 o as n -4 oo. By our assumption o # 0, (e) does not hold. (a) : Let 4i be continuous at t = 1. Noting (d)

ho=,P(a+)-4i(0) and hk=0(kEN) for the column limits of Hp (cf. 3.4.15(iii)), Hp is strongly conservative (cf. 2.4.9) if n

E Ittnv - hn,v+1 I = T v=1

v=1

1

(pn,(t) - pn,v+1(t)) d 4,(t)

- 0 (n -3oo),

0

where pnv denote Bernstein's basis polynomials defined in Lemma 3.4.22. We have used that (Ihno - ho - hn1 )n E co. To that end, we remark that

v-1

0

(pnv(t) - pn,v+1(t)) d 4i(t) < f EI()v=1

holds, that the integrand of the last integral is less than two for all t E [0, 1], which is a known property of Bernstein's basis polynomials, and that Vo (11)

is continuous at t = 1 since 4i is. That is why it suffices to show that for every 0 < S < 1 the statement

f

1-6

n

57, pnv(t)

- p+(t)I dVo ()

0 (n

oo)

v=0

holds. But this is obviously true, because En o Ipnv(t) - pn,v+1(t)I converges to zero uniformly on [0, 1 - S[ as n tends to oc by Lemma 3.4.22. Therefore statement (a) follows.

0

Now, we apply Theorem 3.4.21 to special Hausdorff methods.

Examples 3.4.23. For each a > 0 the methods Ca and H« are strongly regular as well as potent. The Ha-matrix is, since (7n+1)° ) n E co, and Ca is, because it is equivalent to H. If 0 <_a < 1, then the Euler method Ea is strongly regular and potent since (an) E co, if 0 < a < 1 (cf. 3.4.8).

Exercise 3.4.24. Apply 3.4.15 and 3.4.16 to prove that Ea is conservative if a E [0, 1] and regular if 0 < a < 1 (cf. 3.4.17(b) and also 3.4.8(vi) as well as 3.4.8(vii)).

Exercise 3.4.25. (a) Prove the monotonicity of the Ha-methods for a E R and of the Ca-methods for a > -1 following the sketch of a proof in advance of Remark 3.4.20.

152

Special summability methods

(b) Verify that the sequence x = ((-1)k) is summable by each Cesaro method Ca of order a > 0. (c) Prove the strict monotonicity of the Ha- and Ca-methods for a E 1R and a > -1, respectively.

Exercise 3.4.26. In the sequel we denote by H the set of all Norlund methods Np with p = (p,) and pn E R (n E N°) and by 7{ the set of all Hausdorff methods Hp with p = (p,) and pn E R (n E N°). Prove the following statements: (a) For each Norlund method (N,pn) we may assume po = 1 whereby the coefficients of the corresponding matrix remain unaltered.

(b)11\H54 0.

(c)H\7{00. (d)Hn7i={CajaER and -agN}. Bibliography: [108], [196], [201]; [267], [148]

3.5 Methods of function theoretical type In our discussion of Norlund methods we used a connection with power series which is useful for the characterization of the comparison of Norlund

methods (cf. 3.3.10-3.3.12). In this section we discuss a further class of methods of function theoretical type, namely the class of all Sonnenschein methods. We restrict our interest mainly to some necessary and sufficient conditions for the regularity of Sonnenschein methods which we will apply to characterize the regularity of special Sonnenschein methods. As we will see in Sections 5.2 and 5.3, methods of function theoretical type are exceptionally appropriate for applications connected with the summation (of the sequence of partial sums) of power series.

Definition 3.5.1 (J. Sonnenschein, 1958). Let D be a domain in C with 0 E D, let f : D --3 C be holomorphic at z = 0 and for each n E NO let (ank)n,kENo be defined by the Taylor expansion

(f (z))n = E ankZk k

of f n about zo = 0. Then S f := (ank) is called a Sonnenschein matrix, and the associated matrix method is called a Sonnenschein method. In the following examples we verify that some of the matrix methods, which we have studied already, are Sonnenschein matrices and we define some new Sonnenschein methods.

Examples 3.5.2. (a) If f (z) := z (z E C), then Sf = I.

Methods of function theoretical type

(b) If f (z) :=

ez-1

(z E (C), then we get nkzk k!

153

(nEN° and zEC}.

k

Therefore Sf = (e-n k;) = Bi is the discrete Borel matrix. (c) If a E C and f (z) := az + (1 - a), z EC, then

(fl)ak(1_a)zk

(f (z))n =

(n E Nl° and z E C).

k=0

Thus Sf = ((k) ak(l - a),-k) = E,, is the Euler matrix of order a. the matrix Ta := S1 = (ank) is called a Taylor matrix and its coefficients are given (d) F o r any a E C and f o r f (z) by

k-1)

ank =

1-ii-aiz (z E C

n(1

-

)k-n

if 0 < n < k

1

ifk=0=n

0

otherwise.

In particular, Ta is an upper triangular matrix. (e) Let Q E C and f (z) := 1-4(3}z (z E C\ { 11/3 }). Then SR := Sf = (ank) is called a Laurent matrix with coefficients

ank =

Ck + k - 1) (1

- 83)kon

(k, n E N°).

Proof. We are going to verify the identities for the coefficients ank of the Sonnenschein matrices considered in (d) and (e). Since f°(z) = 1 we have (aok)k = (1,0,0,...) for each Sonnenschein

matrix. Thus we may assume n > 1 in the following. (d) The function f (z) := 1- i-a}-- is holomorphic on the neighbourhood

Q:= {z E C I Jz( < 1 iai } of zero and has the Taylor expansion

(1 - a)kzk = a 00E(1 -

f (z) = az

a)k-1 zk

(z E Q)

k=1

k

which implies aF0 = 0, alk = a(l - a)k-1 (k > 1). We assume now that we have already proved

(f (z))n =

k

1} an(1 kE (n -

a)k-nzk

CO

(z E Q)

for an n E N. Then for n + 1 and any z E Q we get (f(Z))`+1

=

tl

all -

a)k-1zk E

(I)a1_azk)

154

Special summability methods

k-1

(v - 1) (1

00

k=n+1 v=n 00 00

an+l

k

> zk(1 _

-

1

a)k-n-1

k=n+1

(v -

v=n

an+l ` (1 - a)k-(n+1) f k

l

k=n+l

a)k-v-1 zk-v

- a)v-nzv(1 -

n-1

1

)zk

n

[c£ 3.1.5(d)],

and thus the desired identity for an+l,k (k E N°).

1-(i (i)z

(e) The function f (z)

is holomorphic on the set Q

{z E C I IzI < 11,3 } and has the Taylor expansion

f (z) =01 :( 1-) 3 ) ' z '

(z E Q).

k

Therefore, we get alk = (1 - /3)k)3 (k E NO). Assuming that we have already proved the identity

(k +

(f (z))n = 1

- 1) (1 k

-

(z E Q)

3)k f nzk

for an n E N, we obtain for n + 1 and z E Q the statement Rn+1

(f(z))n+1

k+n-1

j:((1_)kZk) (1k: (v + n - 1) (1

an+1

F

V

k v=0

_

(1 -

v=0

a}k (k+n)On+1ZII

k

-

- $)kzk lJ

(1 N

N

#)k-vzk-v

(v+n-1'

3n+1zk(1-Q)k k

k

v [cf. 3.1.5(d)],

k

0

and thus the desired identity for an+l,k

We emphasize a subclass of Sonnenschein matrices due to J. Karamata and examined by B. M. Bajsanski [93 and W. T. Sledd [220) with respect to regularity.

Definition and Examples 3.5.3 (Karamata matrices). Let a,(3 E +i1= s)z (z E C\ { }). Then the SonnenC be given and f (z) schein matrix K[a, /3] S f = (ank) is called a Karamata matrix and 16

its coefficients are given by

Methods of function theoretical type

ank =

(vn {1-a-Q} va n_v (n+k_v1)sk_v k-v

155

(n,k E NO).

In particular, K[0, 0] = I and we obviously get by 3.5.2(c), (d) and (e) the

statements K[1 - a, 0] = Ea, K[0,1 - a] = Ta and K[(3,1 -,6] = Sa, respectively.

IL

In the following remarks we relate the Sonnenschein methods to other classes of matrix methods we have studied.

Remarks 3.5.4. If f : D -> C is holomorphic at z = 0, then the following statements hold:

(a) Sf is row-finite if and only if f is a polynomial function. (b) Sf is a lower triangular matrix if f is constant. (c) The identity matrix I is the only matrix which is both a Narlund matrix and a Sonnenschein matrix. (d) The class of Euler matrices Ea (a E C) is exactly the intersection of the class of Hausdorff matrices with the class of Sonnenschein matrices.

Proof. The statements (a) and (b) are obviously valid, whereas (c) and (d) are results due to M. S. Ramanujan [205] and which come from the special form of the matrix coefficients. We omit the proof and refer the reader to [205].

Using the Cauchy product of power series we may recursively calculate the coefficients of Sonnenschein matrices.

Remark 3.5.5 (recursion formula). Let Sf = (ank) be a Sonnenschein matrix. Then AZ) = Ek alkzk by definition of S f, aoo = 1 and aok = 0

(k # 0)

and k

ank = 1: ayvan-µ,k-v

(n, k E No with n > 2, p E N,_1 ).

v-0

Proof . Apply the Cauchy product formula to the product of the Taylor expansions of f u and f'-1 which is equal to the Taylor expansion of fn = fm fn-µ. To get a feeling for the conditions (Zn), (Zs) and (Sp) in the case of Sonnenschein matrices, we deduce first of all necessary conditions on f for the validity of (Zn) and then, assuming the validity of (Zn), necessary and sufficient conditions for the validity of (Zs) and (Sp).

Theorem 3.5.6 ((Zs) and (Sp) in the case of (Zn)). Let f be holomorphic at z = 0 and let Sf = (ank) satisfy (Zn), that is JISf11 < oo.

156

Special summability methods

Then the following statements hold:

(a) f is holomorphically continuable into the open unit disc and has a continuous extension to the closed unit disc of C. (We denote the contin-

uation by f too.) (b) If (z)I < 1 (z E IC with IzI < 1). (c) Sf satisfies (Sp) (d) Sf satisfies (Spo) (e) S f satisfies (Zs)

If (0)I < 1 or f (0) = 1. If (0)I < 1. If (1) I < 1 or f (1) = 1.

(f) Sf satisfies (Zsi) 4=:. f (1) = 1.

Proof. (a) Because Ek IalkI

IISfII < oo the power series f(z) _

E. alkzk converges uniformly for I z I < 1. (b) If(z)I < I for IzI < 1 since If(z)I" <- IISfII < oo (n E N°, IzI < 1)-

(c) If Sf satisfies (Sp), then `I f (0)I < 1 or f (O) = 1' follows from ano = (1(0))". On the other hand, if f(0) = 1, then by (b) and the maximum modulus principle we get f (z) = 1 (lzI < 1); thus ano = 1 (n E N°) and ank = 0 (n E NO, k > 1), therefore (Sp) holds. Now, let If (0)I < I and let 0 < r < 1 be given. Then we get 7 := max if(z)I < 1, IzI=r

since otherwise, as above, we would have f (z) = 1 (IzI < 1), in particular f (0) = 1. Thus we obtain

and = (AM" -'* 0

(n -+ oo)

and for k > 1, using the Cauchy integral formulae, Iankl C

21

(f (z))n dz < f, zl=r zk+l

1

2Tr

27rr

1

rk+l

n

1 n = rk 7'

Thus ank -k 0 (n - oo, k > 1) because -y < 1. (d) This equivalence is an immediate corollary of (c) and its proof. (e) and (f) are obviously true since F_k ank = (f (1))n (n E N°).

0

Theorem 3.5.6 implies that whether a Sonnenschein matrix is conservative or regular depends on the validity of (Zn) and on the values f (0) and f (1). However, even the determining of useful sufficient conditions for the validity of (Zn) is difficult in general. Thus we restrict our interest to the particular case of the regularity of Karamata methods which was examined by W. T. Sledd [220].

Theorem 3.5.7 (regularity of K[a, f3]). Let a,,6 E C be given. Then K[a, 0] is regular if and only if 1 - IaI2 > (1- a) (1- f3) > 0 or a = 0 = 0.

Summability methods defined by power series

157

Proof. We omit the proof and refer the reader to [220].

Corollary 3.5.8 (special cases). (a) Let a, J3 E R be arbitrarily given. Then K[a,0] is regular if and only if either a = J3 = 0 or a < 1, 0 < 1 and a +,6 > 0 are satisfied. (b) V a E C : E,,, is regular if and only if 0 < a < 1. (c) V a E C T,,, is regular if and only if 0 < a < 1. (d) V 0 E C : Sp is regular if and only if 0 < J3 < 1.

Proof. (a) If a = $ = 0, then K[0, 0] = I. If a < 1, 0 < 1 and a + 0 > 0, then 1 - a2 > (1- a)(1 - J3) > 0 which implies by 3.5.7 that K[a, a] is regular.

Conversely, if 1 - a2 > (1 - a) (1 - a) > 0, then a2 < 1, therefore 1 - a > 0; furthermore we obtain 1 + a > 1 - a > 0 and thus a < 1 and

a+0>0.

(b) For any a E C we have Ea = K[1 - a, 0]; thus, by 3.5.7 we get

Ea is regular

a=1 or 1-(1-a)2=a(2-a)>a>0 a=1 or (a>0 and 1-a>0) 0
The statements (c) and (d) can be proved similarly. For further matrix methods of function theoretical type we refer to the book [267] by Zeller and Beekmann and the bibliography cited therein.

Exercise 3.5.9. Prove the following statement about the product of Sonnenschein matrices. If f, g are functions holomorphic at zero, if f (z) = Ek aikzk converges for jzj < r f, where r f > 0, and if ig(0)I
Exercise 3.5.10. Let f be holomorphic at zero and f (1) = 1. Assume that the power series f (z) = Ek alkzk has radius of convergence R > 1. Then for the matrix Sf and the set M {z E C I +zi
:.

(b) If z341 andzOMU{wEC I IwI=R}, then (Ek_0zk)n0csr. Bibliography: [220], [9], [205]; [267]

3.6

Summability methods defined by power series

In 1.2.14 we considered the summability methods of Abel and Borel. Both methods have in common that their definitions are based on power series

and that, as we will subsequently show, they are not matrix methods.

158

Special summability methods

Proceeding from these observations we deal in the present section with summability methods which are defined by power series >k pktk. To that end we make the following assumption.

General assumption 3.6.1. Throughout this section let (pk) be a real sequence with po > 0 and pk > 0 (k E N), and such that the corresponding Ek o Pk tk has radius of convergence R with 0 < R < power series p(t)

0

00.

Now, we define so-called power series methods which are also members of the class of all continuous summability methods. (Those methods are called continuous since, in contrast with matrix methods, which are called discrete, the parameter of the limiting process is continuous.)

Definition 3.6.2 (power series methods). Let p = (pk) be given in accordance with 3.6.1. Let

If :1-R, R[--i I
C := Cp :=

I

lim

1

0
f (t) exists

and Cpp

EPkxk tk has

S x = (xk) E W ( P. (0

l

k

radius of convergence > R and px E Cp I . Let

P, Cpp-4 C.,x-4 px with p,:}-R,R[-*R, t-+px(t):=Epkxktk k

and PP-lim : Cpp -+ K, x = (xk)

+

Jim

1

pk:cktk.

0
k

Then (Pp,Cpp, Pp-lim), or Pp for short, is a summability method which is called a power series method. Remarks 3.6.3. (i) If 0 < R < oo, then we may replace the sequence (pk) by (pkRk) and get the value 1 for the radius of convergence of the

i

power series Ek pkRk tk. Consequently, it is sufficient to consider only the

cases R = 1 and R = oo.

(ii) If pk = 1 (k E No), then R = I and p(t) =

1

tt

(t E]-1,1[).

Therefore, Pp is just the Abel method Ai defined in 1.2.14(a). For that reason, in the case R = 1 we call Pp an Abel-type method and denote it by Jp.

Summability methods defined by power series

159

(iii) If pk = k, (k E N°), then R = oo and p(t) = et (t E R). Obviously, Pp is just the Borel method B1 introduced in 1.2.14(b). On that account, if R = oo, we call Pp a Borel-type method and denote it by Bp.

Example 3.6.4. (i) We stated in 1.2.8 and 1.2.14(b) that the sequence x :_ (1, 0, 1,...) is Abel- as well as Borel-summable to the value 2. Similarly, we may verify that the sequence y :_ ((-1)k) is both Abel- and Borel-summable to zero. These statements fail in general for arbitrarily given power series methods Pp. If P2k := 1 and p2k-1 := 0 (k E NO), then x, y E Cpp and Pp lim x = 1 = Pp- lim y, as we may easily check.

(ii) We now consider the sequence x = (xk) = (zk) for certain z E C and ask whether it is summable by the Abel method or the Borel method.

The Abel method is not applicable to x if JzJ > 1 because the series Ek xktk = > k(zt)k does not converge for t E } and z 0 1 hold, then we have

,1

[. However, if Iz < 1

(1-t) --+0 if t-+ 1-. (1-t)E(zt)k= (1 - zt) k=0

In contrast, the Borel method is applicable to x for every z E C since R = oo, and we get k

- - - + 0 if Rz < 1 and t -* oo.

e-'5 k=0

Consequently, the Abel method is not stronger than the Borel method. Moreover, by considering Fourier series one may also prove that the Borel A method is not stronger than the Abel method. As we have already stated in 1.3.6, the Abel and Borel methods are regular. Generalizing these observations, our next aim is to characterize those power series methods which are regular. To that end we verify first that the domain of any power series method is the intersection of the domains of matrix methods which are `discrete versions' of the power series method under consideration and which are stronger than and consistent with it.

Definition and Remarks 3.6.5. (a) Let p = (pk) be given in accordance with 3.6.1 and let

(t,,.)

with 0 < t, - R- be fixed. Then the

matrix method corresponding to the matrix Pp,t _ (%4) is called a discrete power series method (with respect to p and (tn) ). Obviously, Pp,t is stronger than and consistent with Pp. (b) Cp9 = n {cf'p.tn (t,) with 0 < t, -+ R- } . [This is an immediate consequence of (a) and the sequential criterion for the I

existence of a limit in a point.]

IL

160

Special summability methods

Applying the Toeplitz-Silverman theorem 2.3.7 11 to the discrete power series methods in 3.6.5(a) we get the following characterization of the regularity of power series methods.

Theorem 3.6.6. Let p = (pk) be given in accordance with 3.6.1. Then the following statements are equivalent:

(a) Pp is regular (that is, c C Cp and Pp-lim 1, = lim). (b) For each (tn) with 0 < to -+ R- the discrete method Pp,t is regular. (c) P. satisfies (Spo) : limo
Proof. (a) q (b) : For a proof apply 3.6.5(b). (c) If Pp is regular, then we have necessarily ek E CPp and 0 = Pp- lim ek = limo 0 and the matrix Pp,tn has row sum equal to one, the matrix Pp,t obviously enjoys the conditions (Zsi) and (Zn). Moreover, condition (c) restricted to the sequence (tn) is exactly the condition (Spo) for the matrix P,,t,,. Thus, by the Toeplitz-Silverman theorem 2.3.711 we have proved the regularity of Pp,t,,, that is statement (b).

(a)

If we distinguish the case of 'Abel-type methods' and that of `Boreltype methods', then we can characterize regularity in terms of the defining sequence p. Theorem 3.6.7. Let p = (Pk) be given in accordance with 3.6.1. (a) For an Abel-type method Jp (that is, R = 1) the following statements are equivalent:

(i) J. is regular. (ii) p(t) -a oo (t -+ 1-). (iii) Pn op., - + oo (n -+ oo). (b) A Borel-type method Bp is regular if and only if p(t) is not a polynomial; that is, pk # 0 holds for infinitely many k E No. In particular, we recover the result that both the Abel method Al and the Borel method Bz are regular (cf. 1.3.6). Proof of 3.6.7. (a) Let Jp be a regular Abel-type method. In particular, by 3.6.6 which forces statement (ii). 0 = 1imo
1< nPk Pr tr r=0

(kENo, 1-2
Summability methods defined by power series

161

we get limo
If p(t) is a polynomial with highest coefficient pn # 0 (note, po > 0), then we have o

Pn

pn

p{tt)

- n-L.s

Pn

pvty-n

= 1 (t -i oo).

v=0

Conversely, if p(t) is not a polynomial and k E NO is given, then there exists an n > k with pn # 0. Because pv > 0 for all v E NO we get tk

p(t)

-

tk

= pntn-k

pntn

--> 0 (t

p

oo).

For the example in 3.6.9(c) and to prove Theorem 3.6.11 we need the following result which is of independent mathematical interest.

Lemma 3.6.8. Let (ck) be a sequence with

0
and

limsupIckIr =1 k-oo

and (dk) be a sequence with -dk -> d (k --* oo) for some d E IL Then F_ dktk k

E Cktk

--*d (0
(3.6:1)

k

holds for the power series Ek Cktk and Ek dktk.

Proof. Obviously, 1 is the radius of convergence of the power series Ek Ck tk. Frthermore, if r denotes the radius of convergence of the power

series Ek dk tk, then we have r = 1 and r = oo, if d # 0 and d = 0, respectively. To prove (3.6:_1) let e > 0 be given. Then there exists a ko E N with for each k > ko

< 2

and, because 0 < Ck -+ oo as k tends to oo, a to = to(ko) E ]0, 1[ with

L(IdkI + IdlCk) < 2 k=0

Cktk k

for each t E [to,1[.

162

Special summability methods

Consequently, for each t E [to, 1[ we have

1Edktk k

F_ Ckt

k

-d =

1

Ckt

k

k

dktk

k k

- dECktkl k I

Z Idkl tk + IdI F_ Ck tk k=0

k=O

ECktk

sup I k>kp

+

k

- dl [

Ck tk

k=k0+1

L Cktk k

< 2+2 =F which proves (3.6:1).

Examples 3.6.9 (Abel-type methods). (a) If Pk := kk+1 (k E N), then we have p(t) = Ek pktk = -1 log(1- t) and the generated Abel-type

method is called the logarithmic method and is denoted by L. It is regular by 3.6.7(a).

(b) If a > 0 and pk := (k+k-1) (k E N°), then we have p(t) _ Ek pktk = (1-t)-a (with R

= 1) and the corresponding Abel-type method is called a

generalized Abel method and denoted by Aa. Obviously, Aa is regular by 3.6.7(a), and Al is identical with the Abel method defined in 1.2.14. (c) For any a E R an Abel-type method Jy is given by Pk := (k + 1)a; it is regular for each a > -1 by 3.6.7(a) since the corresponding power series p(t) satisfies p(t) (t -+ 1-) as we now verify: Using Exercise 3.1.18 we get Pk = k° N r(a + 1)(kka) and thus by 3.6.8 we have p(t) - r(a + 1) Ek (k °)tk = I'(a + 1)(1 - t)-a-1 A F(a+1)(1-t)-a-1

Example 3.6.10 (Borel-type methods). (a) Let a > 0 and,3 > 0 be given, and let pk := r(ak+)3+1) (k E N°). The power series method given by (pk) is called a generalized Borel method and is denoted by B(a, (3). The method B(a,13) is regular by 3.6.7(b) since p(t) = Ek (ak+s+1) is not a polynomial. (b) Let a E C, Ra > 0, and let pk := (k!)-a (k E No). The power series method given by (Pk) is (also) called a generalized Borel method and

is denoted by Ba. The method Ba is regular by 3.6.7(b) since p(t) _ Ek k, a is not a polynomial. Note, B1 = B(1,0) is the Borel method. With the aim of gaining insight into the large class of all power series methods, we compare the Abel method and the Borel method with the class of all regular Cesaro methods and of all regular Euler methods, respectively.

Theorem 3.6.11. (a) The Abel method Al is strictly stronger than and consistent with each Cesaro method Ca with a > 0, and, moreover, Ua>0 cce C CA, holds.

163

Summabifity methods defined by power series

(b) The Borel method BL is strictly stronger than and consistent with each Euler method Ea with 0 < a. Furthermore, Uo 0) and of Ea (0 < a < 1), the domains cc. and CE, are monotonically increasing if a tends to oo and to 0, respectively (see 3.1.10 and 3.4.8(viii)). (b) The classical analytic proof, that the inclusions in 3.6.11(a) and (b) are strict, is relatively hard. In contrast to this, in Chapter 6 we will make available functional analytic tools such that the proof of these results becomes A simple (cf. Theorem 8.1.14=1. (nn a) Proof of Theorem 3.6.11. (a) For convenience we put An (n E NoAk-Ltk , a > 0). Using the power series expansion (1 - t)-a = for Iti < 1 and the Cauchy product of power series, we Ek0

get for every sequence x = (xk) E cc. (first by formal calculations)

x

_

(1-t)aIL

00

xktk kk=O co

ak

(1 -

tk k=0

`LxLAa-L k-v

Aktk

k=0

Aatk

v=0

k

k=O

where ak denotes the kth term of the sequence Cax. Starting with the last term in the above equalities, we may easily verify that all of the power series

converge for Itl < 1. Since x E cc., we obviously have °A -> limca x as k -+ oo. Applying Lemma 3.6.8 with Ck := Ak and dk := ak Ak and noting the fact that ( kka) k` r{«+ii (k -* oo) for any a > 0 (cf. Exercise 3.1.18), we get, by the previous equalities, 00

xktk -+ limca x

(1 - t)

(t - 1-),

k=0

that is x E CA, and AL- lim x = limca x. Altogether we have proved Ua>o CC. C CA, (including consistency). Now, we are going to sketch a proof that the last inclusion is strict. Let x = (xk) with xk :_ Fk=° a,, (k E 1\l°) defined by the power series expansion exp(L4. t) = Yk o aktk. Obviously, x is Abel-summable to the value e a . To get a contradiction we assume that x is also Ca-summable

for some a > 0. Then we would have (k ) E co, as one may prove, and therefore (k) E Co. Then, applying Lemma 3.6.8 (with Ck := ka and dk := ak), we would get

(1-t)a+'Eaktk -+0 if t-+ 1k

164

Special summability methods

and therefore (1

-

!tl)a+i

P1

1+

t) = (1 - Itir+1 r` aktk _+ 0 if Itj -> 1-, kk

=J0

which contradicts the fact that exp(h) as h tends to 0o diverges to co faster than each positive power of h. (b) Also in this case the proof is based on identities for power series and on the Cauchy product of power series. Since Ea, a > 0, are monotone (cf. 3.4.8(viii)) and because B1 is regular, it is sufficient to consider a with 1, that is a = 1+a . If x E cEo , then we get (first 0 < a < 1. Let A by formal calculations) 00

xk k

t

1.1

k=0

=

e-(x+1)t

00` xk tk

E

100:

'x

k=0

=

00

e-(.\+1)t

k=0 k

tk E v=0

k =0

=

00

e-(.\+1)t

k tk

Ak-v

V! (k - V)I xv

((1 + A)t)k

((1

e-(x+1)t

+ (k) (rx) v (rx) 1 k_v

v=o v

k=O

ka)t)k

ak -

e_T

xY

1

00

k' ak ,

(3.6:2)

kk O

where r := (1 + \)t and ak := (Ea]k denotes the kth term of the sequence Eax. Now, since Eax converges to limEd x and the Borel method B1 is regular, we have that Eax E CB1 and B1- lim Eax = limE, x. Consequently, by the above equalities, going back to the first term, we see that x is Borel-summable with B1- lim x = limEd x. Thus we have shown that B1 is stronger than and consistent with Ea. To prove that the Borel method is strictly stronger than each Euler method Ea (0 < a < 1), we consider the sequence x = (zk) with z E C. By 3.6.4(ii) we know that x E CB1 if Rz < 1. However, x V cEa if z := 1 - 2 , since then we have for E,,, = (enk)) the equalities

En e(a) nk k=0

(2)/c

(k ) k(i - a)n-k [ I - a)

a

k

I n// f (1 - a)n-k(a -2) k

k=0\

Notes on Chapter 3

165

= (1-a+a-2)" _ (-1)1. Note that the last considerations do not prove that the inclusion UO
{z E C I Rz < 1} = U°
Proposition 3.6.13. Let a > 0, and let x E w be given. Then x E CB1 if and only if Eax E CBi (with consistency).

Proof. Note that the calculation in (3.6:2) holds for a 36 1. For a = 1 the statement is trivially true since El is the identity matrix.

Exercise 3.6.14. Let 0 < 0 < 1 and p = (pk) with pk :=

eka

(k E N°) be given. Prove that the corresponding power series method PP is a regular Abel-type method. Hint: Prove first p(e-Y) - d(J3)y'l? exp(c($)y') (y -> 0+) for certain constants d(13) and c($), and make the substitution t = e-y.

Exercise 3.6.15. Prove that power series methods are either regular or coercive. (In your proof distinguish the cases R = 1 and R = oo.) Exercise 3.6.16. Prove, for the logarithmic method L, that the following statement holds: if x = (xk) E CpL then x = (xk+l) E CPL and L- lim x = L- lim Y.

Bibliography: [108], [216], [267]; [134], [228]

3.7

Notes on Chapter 3

In Theorems 3.2.13 and 3.4.21 we characterized the potent Riesz matrices and Hausdorff matrices, respectively. It is of mathematical interest (see Examples 11.5.2(d) and (e) concerning the Riesz matrices) to characterize also the potency of other classes of summability methods such as the class of all conservative N6rlund methods and-extending the notion of potent matrix methods to non-matrix methods-the class of all conservative power series methods.

Every infinite matrix A gives rise to matrix operators A : E -* w where E is any sequence space with E C WA. So it is an interesting problem to study the operator in the case that A(E) C E; that is, A is a mapping in itself, and we have the following problems for a given matrix A and sequence space E satisfying E C wA :

Give necessary and sufficient conditions such that A : E -p w is a mapping in itself. Determine or estimate the operator norm of A if E is a Banach space or even a BK-space (cf. 7.3.1).

166

Special summability methods

. Determine or enclose the spectrum of A if A is a map in itself. Hausdorff started the investigation of this problem area in his paper [112] where he characterized those Hausdorff matrices which map c and co into c and co, respectively. Hardy gave in [107], in the case of positive real Hausdorff matrix H, a sufficient condition for H to map e" in itself. In both cases the operator norm can be verified. About 20 years later Brown, Halmos and Shields (cf. [51]) continued this investigation and solved all three problems in the case of the special Hausdorff matrix Ci and the sequence space P. Further papers of different authors were devoted to Cl and the sequence spaces c, co, by,..., and to Hardy spaces 1i and the Bergman spaces AP of holomorphic functions. Motivated by the paper of Brown, Halmos and Shields, Rhoades started in [207] (see also the later papers of Leibowitz [143] and Rhoades and Sharma [208]) to deal with the above problems in the case of certain classes of Hausdorff matrices instead

of single special matrices. The papers mentioned have in common that the methods of proof are closely adapted to the matrix and the sequence space under consideration, that is the methods of proof are ad hoc. New ideas have been brought into play by Siskakis in the papers [217] and [218] where he examined the Cesaro operators on ?-LP and AP by considering

a certain semi-group of linear operators. Motivated by this and Hardy's paper, Rudolf, a doctoral student of Beekmann's, developed in [213] a very general theory for Hausdorff matrices on (general) BK-spaces. This theory, applied to the spaces 1P, co, c, W' and AP, gives a great variety of results including the known one. The reader can get an overview of the extensive literature on the spectrum of matrix operators by applying MathSciNet to `Classification: (40G OR 40H)' AND `Anywhere: ((spectr* AND oper*) OR (spectr* AND matrix*) OR (spectr* AND Hausd*) OR (spectr* AND Ces*))'.

4

Tauberian theorems In Chapter 2 we dealt mainly with inclusion theorems and with ToeplitzSilverman-type theorems and, in Chapter 3, with their application to special matrix methods. In contrast to that, in this chapter we turn to Taube-

rian theorems. For given matrix methods A and B with CB C CA we seek a Tauberian condition, which determines a subset L of w, such that x E L n CA implies x E CB. Essentially, we consider the case that B = I; that is, we aim to conclude from x E L n CA that x E C. We should emphasize that our main concern is-besides the presentation of Tauberian theorems in the case of special summability methods-to put, by way of examples, different methods in the hand of the reader to prove Tauberian theorems in the case of special summability methods. In Section 4.1 we deal with Tauberian conditions for the Cesaro method C1, and more generally for C. Why we start with C1 is due to both historical reasons and the fact that in the case of C1 the proofs are technically easier than in the case of other methods like (general) Hausdorff methods. For the same reasons we give, in the case of C1, different types of Tauberian conditions as a '(one-sided) oscillation Tauberian condition', a '(onesided) local Tauberian condition' and a 'gap Tauberian condition'. The highlight of Section 4.1 is Theorem 4.1.13, which gives a Tauberian condition enabling us to conclude C_,+S-summability from Ca-summability where a > -1 and b > 0. A consequence of this theorem is the `convexity' (cf. 4.1.14) and the 'b-equivalence' (cf. 4.1.16) of the Cesaro methods. The class of all Riesz methods contains the Cesaro method C1 and the handling of Riesz methods is just as convenient as that of C1. So we try in Section 4.2 to generalize the Tauberian theorems, which we proved in Section 4.1, to the case of Riesz methods. In particular, we generalize in this sense the `(one-sided) oscillation Tauberian theorem' (cf. 4.2.2) and the 'gap Tauberian theorem' (cf. 4.2.8). However, we do not get a 'local Tauberian theorem' as a corollary of the '(one-sided) oscillation Tauberian theorem'. Its proof requires new methods (cf. 4.2.5). In proving a Tauberian theorem for regular non-negative triangles satisfying a mean value property (cf. 4.2.10) we get as an application a one-sided oscillation Tauberian theorem for Riesz matrices which is, in the case of real sequences, more general than the above-mentioned Theorem 4.2.2.

168

Tauberian theorems

In Section 4.3 we present an `o-Tauberian theorem' for power series methods and apply it to both the Abel method and the Borel method. We do not discuss the corresponding and more general `0-Tauberian theorem' because the proof of it requires deep-seated and very technical tools. However, we prove an 0-Tauberian theorem in the special cases of the Abel method and Borel method in Sections 4.4 and 4.5, respectively. As immediate consequences of these theorems we get that the Cesaro methods Cam, a > 0, and the Abel method are b-equivalent, as are the Euler methods

E, 0 < a < 1, and the Borel method.

4.1

Tauberian theorems for Cesaro methods

Since the Cesaro methods are (strictly) monotone (cf. 3.1.10 and Exercise 3.4.25) the problem arises naturally to find the weakest possible condition

(Tauberian condition) on sequences x such that for -1 < a <)3 the Cfi-summability of x implies its C,,-summability. First we consider the

particular case a = 0 (note, Co = I) and

= 1; that is, we look for

conditions on sequences x such that C,-summability of x implies the convergence of x. We hope, of course, to find a Tauberian condition as weak as possible. To formulate some of the Tauberian conditions we make use of Lan-

dau order symbols which we now introduce. For readers not familiar with them we suggest trying Exercise 4.1.18 immediately after reading the definition.

Notation 4.1.1 (Landau order symbols). For any gk > 0 (k > K) and z = (zk) E w we put

zk = 0(9k) :-* S M > 0 : IzkI < Mgk for each k > K and

zk = i?L(gk) :4=: 3M > 0 Zk = i?R(gk) :4=: 3 M > 0

zk > -Mgk for each k > K, zk < Mgk for each k > K.

Here `L' and `R' stand for `Left' and `Right', respectively. We put

4 = 0(9k) :4->

zk

9k

--+ 0 (k > K and k -+ oo).

In the case 9k := 1 (k > K) we use the notation i0(1), £JL(1), OR(1), and o(1), respectively. Furthermore, in the case of o(gk) and 0(g0 we speak in general about an o- and 0-condition, respectively. We give a series of Tauberian theorems for the Cesaro method C, starting with the earliest and best known.

Theorem 4.1.2 (0-C1 -+ K).1. Let x = (xk) E w be given. Then2 xk - xk-1 = 0(1) and x E cc, implies x E C. 1 `K' stands for the German word Konvergenz which means `convergence'. 2 For x = (xk) we always assume x_1 := 0.

Tauberian theorems for Ces3ro methods

169

The easier case Xk - xk-1 = 0(1) and the present case xk - xk-1 = D (I.) in this theorem goes back to Tauber (1897) (hence the adjective

'Tauberian') and Littlewood (1910), respectively. After the proof of 4.1.8, we will deduce Theorem 4.1.2 from a more general Tauberian theorem of Hardy and Littlewood (cf. 4.1.7).

Remarks 4.1.3. (a) Note that in the above Tauberian theorem the limit of x coincides with limc1 x because C1 is regular. So we do not subsequently mention the limits in the case of Tauberian theorems for regular methods.

(b) Recall that the additional condition on the sequence (xk)-in Theorem 4.1.2 it is the condition xk - xk-1 = D (k) -is called a Tauberian condition. There are various Tauberian conditions in use. We consider a number of them in what follows. A We now introduce the notion of a slowly decreasing sequence and, more specially, of a slowly oscillating sequence. We should mention that the last notion is not consistent with that introduced in 2.5.3.

Definition 4.1.4 (slowly oscillating, slowly decreasing). (a) A complex sequence (xk) is said to be slowly oscillating if it satisfies lim

1
(xr - xn) = 0;

that is, for each e > 0 there exist a 6 > 0 and an no E No such that

ixT -x,,<e holds for allr,nEN° with r>n>n° and n <1+b. (b) A real sequence x = (xk) is said to be slowly decreasing if lim inf (xr - xn) > 0; 1
that is, for every e > 0 there exist a 5 > 0 and an no E NO such that

x,-x,, >-e holds for all r,nEN° with r>n>n° and

<1+6. A

real sequence x is said to be slowly increasing if -x is slowlyndecreasing.

Remarks and Examples 4.1.5. (a) Each convergent sequence is slowly oscillating, which follows directly from the Cauchy criterion. On the other hand, the sequence (xk) = (Fn =1 is not convergent, but it is slowly n) oscillating since 1x1.-x,,


n

<1+e.

Moreover, every increasing sequence is obviously slowly decreasing. (b) If a sequence is both slowly decreasing and slowly increasing, then it is obviously slowly oscillating.

(c) The sequence (xk) = ((-1)k) is neither slowly oscillating nor slowly

170

Tauberian theorems

decreasing since lim inf?EN0 (xr - xr_1) _ -2.

is slowly decreasing since it is in(d) The sequence (xk) = CF-n=1 } creasing. However, it is not slowly oscillating because for each n E N° and S > 0 we get for r :_ [(1 + S)nl the statements L. < 1 + S and

= xr - xn > r_n > (1+6)n-n-1 (1+6)n

6

1+6

vrn-

-

-4ooas n-ao0

1

(1+6)n

0

hold.

Theorem 4.1.6 (one-sided oscillation Tauberian theorem for C1). If x E cc, and if x is slowly decreasing, then x E c; that is, each slowly decreasing C1-summable sequence is convergent.

Proof. Because C1 is regular we may suppose x E coc, since otherwise we consider the sequence z = x - (limc, x)e noting that z is slowly decreasing

if x is. Now suppose x V c so that a fortiori x

co. By this we get 0 < lim supk.. xk =: p or 0 > lim infk.oo xk. Without loss of generality we may consider the first case. (In the second case we apply the first case to the sequence -x.) Therefore we can choose an index sequence (kj) with

xk, -4 p as j -+ oo and xk, > z for all j E No where a := p if p < 00 and a := 2 if p = oo. For e := a there are, since x is slowly decreasing,

a S > 0 and a j1 E N° such that Zr - xk, > - 4 for all r > kj > kj, with kJ < 1 + S. Thus Zr > a whenever r > kj _> kj, and k; < 1 + S. Putting rj :_ [(1 + S)kj] and y = (yn) := C1x we get for each j > j1 the statement

kj+1

yr'-rj+1ykJ

xkJ+1+ ..+xr, > a rj - kj

rj+l

a

6- 1

4 (1 + S)kj + 1

- 4 rj+1 Sa

j-aoo

4(1 + S) > 0

contradicting y = C1x E co since y E co implies yr, -

ykj ---Y 0 when

j tends to oo. We have the following corollary.

Corollary 4.1.7 (slow oscillation Tauberian theorem for C1). Each slowly oscillating C1-summable sequence is convergent.

Proof. Let x = (xk) E cc, be slowly oscillating. Then it is easy to verify that the sequences (Yk) :_ (RXk) and (zk) := (sxk) are also slowly oscillating and therefore slowly decreasing. Since (Xk) E cc, is equivalent to (yk), (zk) E cc,, we have (yk), (zk) E c by 4.1.6, and thus x E c. Next we discuss various Tauberian conditions which allow a Tauberian conclusion from C1-summability to convergence.

Tauberian theorems for Cesi ro methods

171

Proposition 4.1.8. (a) In the case of complex sequences (xk) the local

Tauberian condition xn - xn_1 = £ (n) implies that (xk) is slowly oscillating.

(b) For real sequences (xk) the one-sided local Tauberian condition xn - xn-1 = DL (,1-,) implies that (xk) is slowly decreasing.

(c) A complex sequence (xk) which satisfies the local Tauberian condition xn - xn_ i = o (n) is slowly oscillating. Moreover, it satisfies lim

1oo

I xr - xn l = 0

for all li > 1;

that is, for each e > 0 there exists an of E No such that Ix,. - xnI < e for

allr>n>ne with n<µ.

Proof. (a) Let x = (xk) with xn - xn_1 = £ (n) ; that is, with Ixn xn_11 < M for n E N and a suitably chosen M > 0. Then we have for all

n < r E N the inequalities

Ix - xv_iI < M

Ixr - xnl < v=n+l

and consequently

r-n n

iim (x,. - xn) = 0, that is x is slowly oscillating. 11,n-aoo

(b) Let (xk) be a real sequence which satisfies xn - xn-1 = OL (1-n) ; that

is, xn - xn_1 > - M for n E N where K > 1 and M > 0 are suitably chosen. For all K < n < r we obtain r

xv - xv-1 > -M

xr - xn = v=n+1

v=n+1

L

>

-Mr-n. n

Hence, for each e > 0 and S := tit we get for all K < n < r E N and n < 1 + S the statement xr - xn > -M (n - 1) > -M S which proves that x is slowly decreasing. (c) The proof is left to the reader (cf. Exercise 4.1.19).

Proof of 4.1.2. Apply Corollary 4.1.7 and Proposition 4.1.8(a). Theorem 4.1.9 (JDL-Cl -+ K). Each real Cl-summable sequence (xk) which satisfies x,, - xn_1 = OL (ri) is convergent. Proof. Apply Theorem 4.1.6 and Proposition 4.1.8(b). Besides theorems like 4.1.6, in which the Tauberian condition offers a certain regularity of the oscillation of the sequence (xk) under consideration, Tauberian theorems play a role in which the sequence (xk) has (infinitely many) constant pieces and no further restriction on the oscillation between the constant pieces. Since xn - xn_1 = 0 within these constant pieces, we use for the sequence (xn - xn-1) the designation gap sequence which we now define precisely.

172

Tauberian theorems

Definition 4.1.10. A sequence (an) is called a gap sequence if there exist a A > 0 and index sequences (nk), (rk) such that 0 < no < ... <

rk < nk+1 < rk+1 < nk+2 < ..., n > 1+A (k E N°) and an = 0 for nk < n < rk (k E NO).

In the case of gap sequences we have the following Tauberian theorem. The corresponding Tauberian condition is called a gap Tauberian

condition.

Theorem 4.1.11 (gap Tauberian theorem for C1). Ifx = (xk) E coo, is a complex sequence such that (ak) :_ (Xk - xk_1) is a gap sequence, then (xnk) E co where (nk) is chosen according to Definition 4.1.10. In particular, if nk+1 = rk + 1 (k E No), then we have x E co.

Proof. Let x = (xk) E coo, be given such that (ak) :_ (xk - xk_1) is a gap sequence, and let y = (yn) := C1x. By the definition of a gap sequence we have xn = xnk for nk < n < rk and k E NO. Thus nk-1 Yrk

rk

xv+r r +1 E v=° k

rk - nk + l

nk

rk +

v_nk

ly"k-1

+

nk

rk+lYnk_1+

rk + 1

X.

xnk

nk (1rk +1)xnk \

Consequently, since y E co, < 1+x and for each k E No, we conclude that (xnk) E co.

(1 -

rk+1 }

> 1 - 1+.` > 0

0

Next we consider whether Tauberian theorems for C1 can be extended to Ca. We treat the case of the local Tauberian condition in 4.1.8(a). First, for 0 < a < 1, by 3.1.10, we have cc. C cc,. Therefore, xk - xk-1 = JD Gk) is a Tauberian condition from the CQ-summability to convergence for 0 <

a < 1. This remains true for a > 1 as we will show subsequently. We start with a Tauberian theorem from convergence to C_1+6-summability where S > 0. Then we establish a Tauberian theorem from C, -summability to C_1t5-summability where a > 0 and S > 0 are arbitrarily given. On account of the monotonicity (and the consistency) of the Cesaro methods we have that x E cc_,+s implies x E c (= cc.) for each 0 < S < 1 which is, from the view point of Tauberian theorems, a theorem of 'C_ 1+6 - K'type (without any Tauberian condition). In general, the converse is not true since C, is not conservative if a < 0 (cf. 3.1.9(d)). However, the following theorem holds.

Theorem 4.1.12 (ID-K -* C_l+S). If x = (xk) E c and xk - xk_1 = il(k) , then x E cc_1+a for each S > 0.

Tauberian theorems for Cesaro methods

173

Proof . In view of the monotonicity of Ca and the fact that Co = I, the theorem is trivial when b > 1. Now assume 0 < 6 < 1. Because e E cc, for a > -1 (cf. 3.1.9(a)) we may also assume that x E co. Put a := b - 1, ak := xk - xk-1 (k E No ) and consider, for n > 3, the terms

S"

=

(n_k+a_1)

=

(k+a-1)E

k

n_x

N Ck+a 1 k

n

Xn-k + E ... + E .... k=N+1 k=n-K+1

k=0

=:Sn3

where N, K E N, 0 < N < n - K
Sn = o(n"), that is n-"Sn -> 0 (n -+ oo). Let e > 0. We prove that

,Sn1 <

3

n"

(n sufficiently large)

[An] with a A E ]0, 1[ fixed subsequently. Then we where we put N choose a K E N with n - K> N + 1 and prove that (n sufficiently large)

jSn2 I < 3 n"

and verify, using (Spo), that ISr3I <

3

n"

(n sufficiently large).

Applying Abel's partial summation formula, we get for all n, N E N with N < n, by 3.1.5(d), SnI

=k

(k+al

k J a6n_k +

N+a N J xn_N-1.

(4.1:1)

By 3.1.18(a) and since ak = 0 (k) , we may choose an M > 1 with Ck

k

a) < Mk" and IakI < k (k E N).

From this and (4.1:1) we obtain, for n > 2 and N the inequalities Sn1

<

0<&, max

N Ian-,,I

[An] with A E ] 0, 2

. k+a) k + MN"Ixn-N-1 k

=0

(4.1:2)

174

Tauberian theorems

M /N+a+1\

n- N M2

<

r1N

<

M2

<

M2

N

+ MN" I xn-N-1 I

Na+1 + MN" x n_N_1

-

1A "+1

I

1

[see 3.1.5(d)]

[c f. (4 . 1 : 2)]

n" + M[an]" I xn-N-1I

[since 0 < a + 1 ]

A"+1

1-A

n" + M(2-"a"n")Ixn-N-1I

when n > 2. Now, we choose a A E ] 0,1 [ small enough that

M2"+1 < E 1-A

6

holds. Furthermore, because x E co, we may find, for this A, a ko such that

Ixn_N-I I < 6 (2-"MA")-1

(n E N with n - N - 1 > ko)

(4.1:3)

is satisfied. Hence, for the chosen A and ko the desired inequality

ISn1I < ',n- + sn" = 3n" when n >

(4.1:4)

follows. Now, we estimate ISn2I. For each K E N and n E N with N < n - K we get n-Y,K

- N
ISn2I

(k

k

- 1) J

k=N+1

nnote ( k'

=

man

K<

) < 0 for k 0] K -(N+a\I N J

N

[because -1 < a < 0 and .1.5(d)) < <

2MN"

max [cf. (4.1:2)] K K
Now, we choose a K > ko such that IxvI <

3(2-"+1MA")-1

(v > K)

[note x E co].

Then we have ISn2I <

3

n" f o r all n E N with N < n - K and n>

A.

(4.1:5)

Tauberian theorems for Ces3ro methods

175

For this K we have

K-1 n-k+a-1 (f+a) = n

(n+a)

K-1

n

k=0

Cnk xk -4 0 as n -4 oo,

xk = k=0

because x E m and Ca satisfies (Spo). Hence (cf. Exercise 3.1.18(a)), n-aSn3 -+ 0 as (n -+ oo). Therefore, for the given e > 0 and the chosen A and K, there exists an no E N such that, for n > no,

n-N-l>ko, n-K>N, n> 2 and

(4.1:6)

ISn3I < 3 n

Therefore we have (4.1:4), (4.1:5) and (4.1:6) for n > no. Thus S, = o(na), 0 so that x E coca .

From the Tauberian theorem for C1 (cf. 4.1.2) as well as the `.i7-K -* C-1+6-theorem' (cf. 4.1.12) we can now deduce the following Tauberian theorem for Ca-methods (a > -1) which is due to Landau.

Theorem 4.1.13 ()D-Ca -+ C_l+s). If a > -1 and x = (xk) E cc, with xk - xk-1 = D(D) , then x E cc_,+a for each 6 > 0. In particular

xEc. Proof. It is sufficient to consider the case a E No since for any a > -1

x E cc. implies x E cc for each v E No with v > a. Therefore, let a E No and x = (xk) E cc, with xk - xk_1 = 1)(!) be given. We make use of the equivalence of the Cesaro and Holder methods (cf. 3.4.19) and show that x = (xk) E CJJ. = Cc, and Xk -xk_1 =

ik1

imply

x E CH-,+a = cc-1+s

for every 6 > 0.

(4.1:7)

First we prove that Xk - xk_1 = D(k) implies for y = (y,) := Hx that Yn - yn-1 = i7(,1-,) . It is easy to check (by induction) that it is sufficient

to consider the case a = 1. So let a = 1 and let lt'I > 0 satisfy

Ixk - xk-1 I < k

(k > 1).

Then, for y = (yn) = C1x and all n > 1, we have 1

Iyn - yn-1I = n

n

n-1

v=o

'=o

n-1

n E xv - (n + 1) E xv n+1

1

E (xn - xv) v=0

n

n+1

176

Tauberian theorems n-1

n

n-1

E E Ixk - xk-1I 1 v=0 k=v+1 n+1

n

n k-1

=

M= n

1

k

<

n

Mn

n

EE 1 v=0 k=v+1M n+1

< M

n n+1 - n'

n+1

thus yn - Yn-1 = D(n) . Continuing the proof of (4.1:7), for any fixed a > 1 we put y = (yk)

Ha-1 x and get C1 y = Hax E c and yk - yk-1 = JD (l) by the above argument. Therefore Ha-1x = y E c by Theorem 4.1.2. Proceeding in this way we finally obtain HOx = x E c. Since xk - xk-1 = 0(,-t) is satisfied, by hypothesis, we have x E cc-,+8 for each 6 > 0 by Theorem

0

4.1.12.

Since the H. and Ca-methods are equivalent, Theorem 4.1.13 is also true with Ha replacing C. The convenient behaviour of the Holder methods whereby Ha+a = H°HR enables us to deduce easily from Theorem 4.1.13 the following well-known convexity theorem due to Hardy and Littlewood (1912), Andersen (1921) and Riesz (1923).

Theorem 4.1.14 (convexity of Ca , a > -1). Let a, y E R with -1 < a < y be given. Then each x E w with Cax E m and x E cc satisfies x E cc,, for all 3 E R with a <,8 < y. In brief, mca fl cc,. C cc0 for -1 < a < 3 < y. First we establish a proposition which is of independent interest.

Proposition 4.1.15. Let A be a triangle with IIAII < oo and let B be a row-finite matrix. If B is stronger than A, then mA C MBProof. Let x E MA be given. Since A is a triangle there exists a y E m with x = A-1y. Hence, we get Bx = B(A-1y) = (BA-1)y by 2.2.5. Because B = (BA-1)A, the matrix BA-1 is the transition matrix between the triangle A and the stronger matrix B. Therefore, by 2.6.2, the transition matrix is conservative. In particular, BA-1 satisfies the condition (Zn). This implies (BA-1)(m) C m by 2.3.5. Therefore, because y E m, 0 we get Bx E m. Proof of 4.1.14. From the equivalence of the Holder and Cesaro methods

(cf. 3.4.19), and since Ca and Ha are triangles, we get mC, = mHa by 4.1.15. So we can replace Ca with H° in Theorem 4.1.14. Let -1 < a < y, x E cH-, and Hax E m. If we put p := y + 1, then we also have x E CHP because CH, C CHP. Now, let z = (zk) := Hax and y = (yk) := C,z {= Ha}1x). Then yn - Yn-1 = 0 (1-n) since for every n E N (note z E m) n-1

rn

Iyn - yn-1l

-

I

n+1

k=0

zk

n E Zk k=0

Tauberian theorems for Cesero methods

e

Zn

1

n(n-I-1)k=0 -

n+1 ll±ii O

n+1

177

+

nIIzIIoo

n(n+1)

=

2

n+1

PHoo-

Because HPx = HP-"-1(H1+1 x) and p - a - 1 = y - a > 0, we obtain from 4.1.13 the identities

y= H"+1x E c and H-1+ay = H°+ax E c (b > 0).

That is, H'3x E c for all 0 a R with a< Q< y. 0 As an immediate corollary of this convexity theorem we get the bequivalence of the C,,,-methods (a > 0).

Corollary 4.1.16 (b-equivalence of C,,,, a > 0). All Cesaro methods C,, with a > 0 are b-equivalent. That is, m n cc, = m n cco for

a,/3>0.

Proof. On account of the monotonicity of the C,-methods (cf. 3.1.10), it is sufficient to show m n ccs C m n cC, for all 0 < a < 83. Let X E m n cc3 be given, that is Cox E m and x E cca. Applying the convexity theorem 4.1.14 (in the case y 33 := a and a := 0) we get x E cc.. Thus,

xEmncC,.

0

In closing this section we discuss briefly whether the Tauberian conditions used in the Tauberian theorems are best possible in the sense that the theorems fail when we weaken the corresponding Tauberian condition. We illustrate this with an example in the case of the local Tauberian condition and of Theorem 4.1.2. A more general method to check whether a Tauberian condition is best possible in the above sense is based on the notion of a summability function and is mainly due to Lorentz (cf. [148], [149], [150], [152] and [151]). For an introductory discussion we refer the reader to [201, Section 10 of Chapter II].

Example 4.1.17. More precisely, the question is whether there exists a (non-negative) sequence (.fin) with An ---+ 00 and with the property that xn - xn_1 = O(n) is still a Tauberian condition for the Tauberian conclusion from C1-summability to convergence. The answer is 'NO' as the interested reader may show. (For a given (An) with 0 < An --* 00 one may construct a sequence x = (xk) E coc, with xn - xn-1 = O( ) and which oscillates between +1 and -1; consequently, x c.)

Exercise 4.1.18. Let (Xk), (yk) E w and fk > 0, gk > 0 for k > K be given. Prove that the following statements hold:

(a) If fk = O(gk), then xk = 0(fk) implies xk = 0(9k) and xk = O(fk) implies Xk = 0(90-

(b) If Xk = O(fk) and yk = O(gk), then Xk + Yk = O(fk + 9k) and

178

Tauberian theorems

Xkyk = vtJk9k) (c) If xk = 0(1) and yk = 0(1), then xk+yk = JD(1) and xk+yk = o(k). (d) k = 0(k), k = o(k2) and 10k2 + 100k = 0(k2). Exercise 4.1.19. Prove that

Ix,n - xn I = 0 for some p° > I

lim sup 1< ZZL <µo,n-+oo

n

implies

lim sup

I< n <µ,n-+oo

I Xm - xn I = 0 for all Ii > 1.

Exercise 4.1.20. Let ao = 1, ai = -1, a2 = 21

_ an

if

n

1n

22k-1 <

n < 22k

if 22k < n < 22k+1

and xn = En=o a,,, n E N°. Prove that (xn)

(k E N, n > 2),

ce,.

Bibliography: [108], [196], [201], [227], [244], [148], [149], [150], [152], [151]; [267], [123]

4.2

Tauberian theorems for Riesz methods

In the first part of this section we generalize the Tauberian theorems for Cesaro methods to the case of regular Riesz methods. After that we will prove a Tauberian theorem for regular triangles which enjoy the mean value property and apply this result to regular Riesz methods.

General assumption 4.2.1. Throughout this section let p = (pk) be a sequence with po > 0 and Pk > 0 (k E N), and Pn := >k_° pk -4 00 (n-3oo). In particular, the Riesz method Rp is regular (cf. 3.2.7). We now prove a Tauberian theorem for a large class of Riesz methods. It is obviously a generalization of Theorem 4.1.6.

Theorem 4.2.2 (one-sided oscillation Tauberian theorem for Rp ). Let p = (pn) be a sequence which has, besides 4.2.1, the property P.

-4 1 (n -4 oo).

(4.2:1)

Pn+1

Then each x = (xk) E CRv which satisfies the Tauberian condition3

liminf

-pn"-+I,r>n-roo

(Zr - xn) > 0

(4.2:2)

is convergent. 3 By definition, this means that for all e > 0 there exist 5 > 0 and no E No with

x,-xn> -e for all r,nEN° with r>n>no and p <1+5.

Tauberian theorems for Riesz methods

179

Remarks 4.2.3. (a) For p = e Theorem 4.2.2 is just Theorem 4.1.6. The Tauberian condition (4.2:2) depends on P. Sequences satisfying (4.2:2)

are said to be slowly oscillating relative to (Pa). (b) The examples 3.2.12(a)-(d) satisfy (4.2:1), whereas 3.2.12(e) does not, as we may verify from the asymptotic behaviour given in 3.2.12. (c) In the case that (pk) satisfies 4.2.1 and the condition

Pr = 1 lim 1
(4.2:3)

the local Tauberian condition xn - xn_ 1 = i7 (p ) implies the two-sided oscillation condition

Ixr - xn I = 0

lim sup

--+I, r>n->o0

which implies, as the one-sided local Tauberian condition xn - xn-1 = DL { P the one-sided oscillation condition (4.2:2) in Theorem 4.2.2. [Note that, assuming Ixn - xn-1l < cyn, we have I xr - x,,, I < c Ev_n}1 P. Pn) since (Pn) is monotone.]

< P (Pr -

The examples in 3.2.12(a), (b) and (c) satisfy (4.2:3) as we may deduce from the asymptotic behaviour given in 3.2.12. (d) The slow oscillating Tauberian theorem is again valid in the case of complex sequences (cf. proof of 4.1.7).

Proof of 4.2.2. As in the proof of 4.1.6, since Rp is regular, we may consider x = (xk) E coR,. We assume x V c; thus x 0 co since Rp is regular. Without loss of generality we may assume (otherwise we consider

-x) 0 < a := lim supk.,,,, xk < oo. Thus, there exists an index sequence

(nk) with xnk --a a (k -+ cc). We put p := min{1,a}. On account of the choice of (nk) and the Tauberian condition (4.2:2), there exists a b > 0

and an N E N such that

(Zr - xnk) > -

and

xn, >

2

for all4 r > nk > N with Pnk < Pr < Pnk (1 + 6). In particular, we have xr > P- for all such r. Putting

rk:=min (nENI Pn>Pnk(1+2)} (kEN°) we get rk > nk and Prk -I < Pnk (1 + 1) . Furthermore, on account of (4.2:1), we have P1 < 1 + a for sufficiently large n. Thus, Prk = Prk

Prk_1

(1+)Pflk(1+)
180

Tauberian theorems

< for sufficiently large k. Hence, the sequence (rk) satisfies Pn,k (1 + 2) Prk <_ Pn1b (1 + 5) for sufficiently large k. Similarly as in the proof of Theorem 4.1.6, we put y = (yn) = Rpx and obtain

* Pv xv > p Prk - P 1. > P

Pnk

yr1. - P ynk rk

v=nk+1

Prk

4

Prk

b 2

4 1+ 32

This contradicts y E C0 which requires (yrk - P Ynk) k

> 0.

E C°.

Examples 4.2.4 (cf. 3.2.12). We quote here the particular local Tauberian conditions which we can deduce from Remark 4.2.3(c) taking into account the asymptotic behaviour of (Pn) given in 3.2.12. We note that (4.2:1) is satisfied in all cases. pn = 1 : xn - xn_ 1 = JD(n)

xn - xn_ 1 = Pn = n+l log n 1 . - nxn_1 = Dln) (n+,) xn , CL > -1 : Pn = {

xn - xn_1 = JD(nO-1). pn = en , 0 < Q < 1 : The question now arises, what happens in the case of the last example in 3.2.12? In this case the condition (4.2:1) fails and the results so far are

not applicable.

Theorem 4.2.5 (0-Tauberian theorem for Rp ). Let p = (pn) satisfy 4.2.1. Then each Rp-summable sequence x = (xk) E w which satisfies the Tauberian condition xn+1 - xn = AD (P) is convergent.

Remark 4.2.6. Without additional assumptions on p or x Theorem 4.2.5 falls in general if we replace the two-sided Tauberian condition by a onesided Tauberian condition. To show this in the case of the one-sided Tauberian condition xn+1 -xn = DL (*), the interested reader may consider Example 3.2.12(e). The sequence x = (xk) defined by x21 = 0 and x21+1 = -1

(l E N°) satisfies xn+i - xn > -4 P. and x E COR, \ C. Proof of 4.2.5. Without loss of generality we may assume x = (xk) E CORP. Putting P_1 := 0 and

ifu=0

0 A(u)

(xv - xv_1)

:=

0

{ xv we have

n

n

T'Pvxv = E v=O

if u > 0


1

ifu=0 if u E ]Pv_1, Pv] and v E N1°

p

vc0 P.- 1

xvdu = J 0

A(u) du.

Tauberian theorems for Riesz methods

181

P

Therefore

Pn1 (

A(u) du --> 0 (n -* oo).

(4.2:4)

A(u) du -> 0 (0 < - co)

(4.2:5)

dddo

From (4.2:4), we will deduce F

1

o

which implies, for every y > 0, the statement 1

f

(1+7)1'n

(1 + y)Pn and consequently r(1+71P Pn 1 J

Jo

A(u) du - 0 (n -+ oo)

A(u) du --4 0 (n -+ oo).

(4.2:6)

First we note that, because x E coRD and Pn < Pn+1, we have Pn+lxn+1

1

Pn+1

Pn+

Pn

n+1

to

pvxv -

1

1

A(u) du

f0 P.

A(u) du +

1

1

1

+

Pn

(4.2:7)

r A(u) du

J

P

Pn

+

Pn

0

> Po an n E NO such that

Pn < < Pn+1 holds. We obtain £

n°

Pn+1 Pn v_0

as is stated in 3.2.6. Now, we choose for 1

n

>2 pvxv

ixn+1I

Pn+i - Pn Pn+1

(xn+11

[the function C ->p- is monotonically increasing] ---+ 0 ( -> oo, thus n -+ oo) [by (4.2:4) and (4.2:7)]. Thus we proved (4.2:5) and consequently (4.2:6). Now, we choose t > u > 0 sufficiently large. Then there exist n, r E NO

with Pn < u < Pn+1 and Pn+r < t < Pn+r+1. Without loss of generality we may assume that x satisfies the Tauberian condition I xj+l - xj 1 < c P with c = 1. Then, because (Pn) is monotonically increasing, we get, with av = xv - X"-1 IA(t) - A(u) I = Ian+2 + . . + an+r+11 pn+1

Pn+1

+

+

pn+r

Pn+r

<

Pn+r - P.

P.

<

t - Pn

Pn

{4.2:8}

Assuming x 0 co, that is A(t) -f4 0 as t -* oo, we derive a contradiction. Choose 6 > 0 and a real sequence (tk) diverging to oo such that, without

182

Tauberian theorems

loss of generality (otherwise we consider -x), A(tk) > 6 > 0 for each k E No. For each k E No there exists an nk with Pnk < tk < Pnk+l . Thus, by the definition of A(t), we have A(t) > d for Pn, < t < Pnk+1 Applying (4.2:8) with u = tk and t satisfying Pnk < t < (1 + Pnk, we 2) > ? . Consequently, get the inequality A(t) > d - t

(6)2

(1'Fz)p -< Pnk1 / 2 fJP k

k

(1+2)P^k Pnk1

fo

A(u)du rF k A(u)du k A(u)du - P.;k1 (

0

0

0

by (4.2:6) and (4.2:4). This contradicts 8 > 0.

We now generalize the gap Tauberian theorem for C1 (cf. 4.1.11) to the more general case of regular Riesz methods.

Definition and Remark 4.2.7. Let p = (pn) satisfy 4.2.1. A sequence (ak) is called a gap sequence relative to p if there exist a A > 0 and < rk < nk+1 < rk+1 < index sequences (nk), (rk) such that 0 < no <

with P > 1 + A (k E N°) and an = 0 for nk < n < rk nk+2 < (k E NO). If p := e, then we have exactly the notion of a gap sequence introduced in Definition 4.1.10. IL Theorem 4.2.8 (gap Tauberian theorem for Rn ). Let p = (pn) satisfy 4.2.1. If x = (xk) E CORE is a complex sequence such that (xk - xk-1) is a gap sequence relative to p, then (xnk) E co where (nk) is chosen according to Definition 4.2.7. In particular, if nk+l = rk + 1 (k E N°) then we can conclude x E co.

Proof. We adapt the proof of the corresponding gap Tauberian theorem in the case of C1 (cf. 4.1.11) to the more general case of Riesz methods. Let x = (xk) E CORE be given where (xk -xk_1) is a gap sequence relative to p, and let y = (yn) := Rpx. By the definition of a gap sequence we have xn = xnk for nk : n < rk and k E NO. Thus, nk-1 Yrk

rk

57

pvxv + Prk E pv v=0 k v=nk

Pr

Pnk-1

Prb

yak-1 +

Prk - Pnk-1

/

Pak-1 ynk-1 + [

Prk

\

Prk

`{

1 - Pak-11 xnk.

Hence, since y E co, P k+t < 1+x and E1 k E No, we conclude (xnk) E co.

xnk

Prk

,'

p

> 1+ > 0 for each 0

Tauberian theorems for Riesz methods

183

Examples 4.2.9. Continuing the examples in 3.2.12 and 4.2.4, we consider especially gap sequences with rk = nk+1 - 1 where (nk) is any index sequence. Then we can apply the gap Tauberian theorem 4.2.8 to Rp for the following sequences p = (pk) and certain index sequences (nk) : For example, pk = k+1 1 i : For it > 0, choose (nk) with nk+1 ? nk = [exp((1 + 2p)k+r)] where r > 0. Note no = [exp((1 + 2µ)r)]µ > 2. ng+µ.

pk = (k + 1)a, ct > -1 : For p > 0, choose (nk) with nk+1 > (1 + p)nk. For example, nk = [(2 + 11)k].

pk = exp(kl) , 0 < 8 < 1 : For p > 0, choose (nk) with nk+1 - nk > p. For example, nk = [((1 + p)k) a ]. The proofs that the (nk) satisfy the stated conditions are left to the reader. A In the remaining part of this section we generalize the oscillation Tauberian theorem 4.2.2 to a large class of regular matrix methods which have the mean value property.

Theorem 4.2.10 (Tauberian theorem for A with MK (A) ).

Let

A = (ank) be a regular triangle with ank > 0 (n, k E N°) which satisfies the mean value property MK(A). Assume that for all integers 0 < r < n there exist non-negative real numbers O(n, r) such that ((k(n, r))'=, is rnonotonically decreasing for each n E NO and such that

anv =: 41(5) > 0

liminf n->co

(4.2:9)

O
holds for each S > 0. Then each real sequence x = (xk) E CA which satisfies the Tauberian condition4 lim inf n-ioo

(xr - xn) > 0

(4.2:10)

converges.

Proof. We assume that the conclusion of the theorem fails. Thus, there exists-without loss of generality--an x = (Xk) E COA (that is, (yn) := Ax E c°) with lim supk,QO Xk =: 2a > 0 and satisfying the Tauberian condition (4.2:10). The Tauberian condition gives us a S > 0 and an no E N

such that

xr - xn > - 2 for all n > no and r < n with 46(n, r) < S.

(4.2:11)

Now, choose an index sequence (kn) with k° > no and a <_ xk -> 2a. for all r < kn with O(kn, r) < S. We then have, in view of (4.2:11), xr > It follows that

22

4 By definition, this means that for all e > 0 there exist a S > 0 and an no E No such that xr - xn > -s for all r, n E No with n > no, r < n and 45(n, r) < S.

184

Tauberian theorems

a E

2 (8)

NO. Now, if we can show that

E

Rn :_

0 (n -+ oo),

0b

then the inequality

E

Yk _

E

44;(8)+Rn

Ob

O
contradicts (yn) = Ax E co. By hypothesis, (¢:(n, r))r 1 is monotonically decreasing. Thus, for each n E No we may choose an rn E NO with 0 < rn < kn and with 0(kn, v) > 6

equivalent to v < rn, where 0 < v < kn. Considering the matrix B with nth row (ak,,o, ak.,1, ... , ak r , 0, ...) we obviously have (Rn) = Bx. Furthermore, B satisfies (Spo) since A does. Now, applying MK(A) we get ru

< K sup

E bnvzv

µ
v=0

v=1

aµvzv

< K IlAzj

for each z = (zv) E CA, that is CA C TDB. Noting that A is of type M, since it enjoys the mean value property (cf. 2.8.3), we now get COA C COB by Theorem 2.7.6. This yields (Rn) = Bx E co. The theorem is proved. 0

We now apply Theorem 4.2.10 to Riesz methods R, where p = (pk)

satisfies 4.2.1 and pn # 0 (n E No). We put O(n, r) := P--P- for all 0 < r < n. Then ci(n,r) > 0 and (¢(n,r))r 1 is monotonically decreasing for each n E No. Let 8 > 0 be given and let no be chosen such that O(no, 0) = P - 1 > 8 holds. For each n > no we choose an rn < n with ¢(n,rn) > 8 > 0(n,rn + 1).

(4.2:12)

This gives n P.,

;PL.

v=

0
>

1

8 prn

__

Pn

Pn - Prn Pn

[first inequality in (4.2:12)]

n

(1 + 8)Pr.+1 Now, if p also satisfies the condition

[second inequality in (4.2:12)].

Tauberian theorems for Riesz methods

pn > 0, n- oo pn+1

lim inf

185

(4.2:13)

then 4(b) > 0 (cf. 4.2:9). For example, (4.2:13) holds if (4.2:1) in Theorem 4.2.2 is satisfied. Theorem 4.2.10 may be applied to A = Rp if (4.2:13) is satisfied in addition to 4.2.1. Next we verify that-in this particular casethe Tauberian condition in Theorem 4.2.10 is equivalent to that in Theorem 4.2.2. For each index sequence (nv) and every sequence in NO with

r < n, (v E N°) we have n,, --+ ) = p-'" -1-40 (v-+o--+ oo and (f1,,, r

pr

if and only if

rv ---+ oa and PT"

-

1 (v _00).

Consequently, the Tauberian condition in 4.2.10 is equivalent to

.

lim inf

(Zr - xn) > 0

which is exactly the Tauberian condition (4.2:10) if we exchange the roles

of r and n and replace x by -x. Thus, we have deduced from Theorem 4.2.10 the following generalization of Theorem 4.2.2.

Theorem 4.2.11 (one-sided oscillation Tauberian theorem for Rp). Let p = (pn) be a sequence that satisfies 4.2.1, pn > 0 (n E N°) and pn

lim inf > 0. n->oo pn+l

Then each real x = (xk) E CRP satisfying the Tauberian condition lim inf

n

->1,r>n-loo

(Zr - xn) > 0

is convergent.

Remark 4.2.12. Example 3.2.12(e) satisfies the assumptions of 4.2.11. But, naturally, the sequence x = (xn) with x21 = 0 and x21+1 = -1 (cf. 4.2.6) fails to satisfy the Tauberian condition in Theorem 4.2.11 since, for

r=2j+1 and n=2j, we have

2j+1 + 1

.+1 1 < p2p2j < 2i+1 - 1

1 (j

00),

but x2J+1 - x2j = -1 for each j E N°.

In closing this section we remark that one can also deduce Tauberian theorems for Norlund methods from Theorem 4.2.10 (cf. Exercise 4.2.14).

186

Tauberian theorems

Exercise 4.2.13. Prove the statements in Examples 4.2.9. Exercise 4.2.14. Apply Theorem 4.2.10 to prove the following Tauberian theorem for the N6rlund matrix Np where p = (+i)- If x = (xk) E CN is a real sequence satisfying the Tauberian condition

(xr - xn) > 0

liminf n-+oo

r
for some y > 0, then it is convergent. Bibliography: [201], [227]; [239], [176]

4.3 Tauberian theorems for power series methods In this section we establish Tauberian theorems for power series methods and apply them to both the Abel method and the Borel method. Throughout this section we consider real sequences (pk) satisfying 3.6.1, that is po > 0 and pk > 0 (k E N), and the corresponding power series p(t) 0 pktk has radius of convergence R with 0 < R < oo. In the case of matrix methods, Tauberian conditions are in general connected with the largest absolute value in the kth column of the matrix.

El

For power series methods Pp, this value corresponds to supo
Definition 4.3.1. For a power series method Pp where p = (Pk) satisfies 3.6.1, we define the values

On :=

inf p(t)t-'

(n E N°).

0
Examples 4.3.2. (a) In the case of the Abel method, that is Pk = 1 (k E N°) and p(t) _ I 't (t E ] -1, 1[), we have tt

An

o
-nt=(n+l)!1-n}1)-n1

e- n.

(4.3:1)

Clearly Pn = EV=o p = n + 1 < An. To prove (4.3:1), note that the function t - i-t (t E]0,1[) has a minimum at to = 1 - n+i, and, for the asymptotic behaviour of On, that ((1 + n)n) converges to e. (b) In the case of the Borel method, that is pk = k, (k E N°) and p(t) et (t E ] - co, Co[ ), we have

On = inf

t>o

ett-n

=

enn-n

n!

2irn.

Here, the function t -> ett-n (t E ]0, oo[) has a minimum at t = n, and the asymptotic behaviour follows from Stirling's formula (which we used in A the proof of 2.4.13(c)).

Tauberian theorems for power series methods

187

In the following lemma we study some properties of the values An which

enable us to generalize certain calculations, used in the case of the Abel method and the Borel method, to a general class of power series methods.

Lemma 4.3.3. Let a regular power series method P,, be given such that p = (pk) satisfies 3.6.1. Then there exists a sequence (tn) in 10, R[ with An = p(tn)tnn for all n E No.

(4.3:2)

Moreover, each sequence (tn) in ]0, R[ which satisfies (4.3:2) has the following properties: (a) 0 < to < to+1 -> R (n -* oo).

(b) trn-r < n < to -r for all n,r E No. ifR=1 (Pn (C) An > = Ek=n Pk

if R = oo and to > 1

(n E N`0).

Proof. First we note that p(O) = po > 0 and, on account of the regularity of Pp, that p(t) diverges monotonically to oo as t tends to R- (cf. 3.6.7(a) and 3.6.7(b) if R = 1 and R = oo, respectively). We note also that p(t) is not a polynomial (cf. 3.6.7(b) in the case of R = oo). Consequently, for

any n E N we get p(t)t-' -> oo for t -4 R- or t -4 0+. Because the function g : ]0, R[--+ R, t -a p(t)t-n is continuous, we can choose for each n E N a number to E ]0, R[ with An = p(tn)tn n. Moreover, we have :,o = p(to)t0° for to = 0 because p(t) is strictly increasing. Thus we get, by the definition of An, for all n, r E No the statements An < p(tr)tT n and Ar < p(tn)tn r which yield to-r = p(tr)tr r < Qr < p(tn)tn r r

p(tr)tr n - A. - p(tn)tn-n

= t n-r n

If we now consider n E N and r = n - 1, then we get tn_1 < tn, that is (tn) is monotonically increasing. To prove that to -4 R (n -p oo) we assume that to < a < R (n E N) for some a. Then, by the definition of An, the function g has a minimum for each n E N which implies

p'(tn)tn n - np(tn)tn -n- I = 0 and therefore P (t )tn = n -+ o0 R)

This contradicts p(t) > po > 0 (t E ]0, R[) and suptE 1o,,,] p' (t) < oo (the power series p'(t) also has radius of convergence R). Thus, we have proved the existence of (tn) and, independently of the particular (tn), statements (a) and (b). Concerning (c), for any (tn) which satisfies (4.3:2), we have w

n

An = p(tn)tn n = E Pt-k-n > > pktn k=0

k=0

_n

n

> E pk = Pn k=o

188

Tauberian theorems

if R = 1, and 00

On =

00

00

k=n

k=n

pktn -n > E pktn n >

k=0

pk

for to > 1

0

if R=oo.

Examples 4.3.4. From the discussion in 4.3.2, we see that a sequence (tn)

with p(tn)tn-n = An is given by to = 1 - n+1 for the Abel method and by to = n for the Borel method. A We now state a general Tauberian theorem for power series which uses an o-Tauberian condition and is due to Kratz and Stadtmuller (cf. [134]).

Theorem 4.3.5 (o-Tauberian theorem for Pp). Let Pp be a regular power series method such that p = (pk) satisfies 3.6.1. Then each x = (xk) E Cp, which satisfies the Tauberian condition xk - xk_1 = o(o) is convergent.

Remarks 4.3.6. (a) The Tauberian condition in Theorem 4.3.5 is best possible; that is, if (Ak) with 0 < Ak < Ak}1 --) 00 (k -+ oo), then xk - xk-1 = 0('n '.) is not a Tauberian condition for Pp. For a proof of this fact we refer the reader to [134]. (b) By analogy with the Abel-Dini theorem (cf. [128, p. 299]), we have = oo (cf. 4.3.15(c)). Therefore, the Tauberian condition in 4.3.5 does not trivially imply convergence.

E0 0

Proof of 4.3.5. Let x = (Xk) E Cp, satisfy the Tauberian condition xk - xk_1 = o(o ). Without loss of generality we may assume 0 = Pp lim x = iim°k pkxktk. (Otherwise, on account of the assumed regularity of Pp, we may consider the sequence x - (Pp-limx)e instead of x.) The Tauberian condition gives us for each e > 0 a number k, E No with l xk - xk-1 I

<- C

k-

for all Jr > kE.

(4.3:3)

We now choose (tn) in accordance with 4.3.3 and obtain (below, we verify the convergence of the series that occur) pz to n) P( W

00

- xn =

00

to

p(

)

pkxktn - xn E Pktn

k=0

k=0

1

p(tn

15?-, Pk(xk - xn)tn

k=0

1

p (tn ) {

n-1

00

pkIxk - xnltnk +

pklxk - xnltn k=n+1

k=0 =:Si(n)

=:S2(n)

Tauberian theorems for power series methods

189

We consider first S2(n). For n > kg we get 00

S2(n)

<

00

k

Ixv - xv-11tH =

Pk

v=n+1

v=n+i

k=n+l

k=V

00

00

E E pktn k=v

(applying (4.3:3)]

v

v=n+l 00

e

CO

E Ixv - xv_lIE Pktn

Lr

=n+l

tyv

pv t v E pkt y

(to

00

)

tv

E pvtn

k-v

.

v=n+1

< 1 (since k > v > n and (tn) is increasing) and G1vty = p(tv) > E00v pktY for each Note, in the last inequality we used the fact that

k

v E N1. Moreover, since the last series is convergent, the above calculations

are justified. We now estimate Sl (n). For any n > k, we have n

n

n-1 S, (n)

E

Pk

k Ixv - xv-lit n

IX- - xy-3I E Pktnk v=1

v=k+1

k=0

v-1

=

v-1

ke-1

V-1

E I xv - xv-1I E Pk tok + v=1

k=0

n

I xv - xv-3 I v=kt

k=0

E Pk to k=0

=:S3(n)

v-1

n

S3 (n) + e EOp E Pk to v=k, n

S3(n)

v

(cf. (4.3:3)]

k=0

t y v-1

t

k-v

+ev=k. E Av tv k=0 >pktY \tv)

+6 S3(n)

n

L1 Pv tn

v=k,

where we again make use of ( )k < 1 (since k < v < n and (tn) is increasing) and of av tv = p(tv) > Ek=o Pk t,, for each v E NO. Noting 1 P(t)

pk tk -> 0 - (v fixed and t -+ R-) k=°

we obviously get (S3(n))n E co (cf. 4.3.3(a)). Thus, we have shown lira sup IxnI n--4o0

<

lim sup i px/(tn } + lim sup n->oo

I p(tn)

n--+oo

Px (tn)

p(tn)

- xn

190

Tauberian theorems

0 + lim sup

n-roo p(tn)

n

00

1

<

pt+e

e v-n+1

p to + S3 (n)

< 2e.

v=k,

This proves x E co. We now apply Theorem 4.3.5 to the particular cases of the Abel method and the Borel method. In the case of the Abel method this gives Tauber's original result of 1897.

Corollary 4.3.7 (o-Tauberian theorem for the Abel method). Each sequence x = (xk) E CAI which satisfies the Tauberian condition xn - xn_1 = o (n) converges.

Proof. In the case of the Abel method we have pk = 1 (k E N°) and An - e n (cf. 4.3.2(a)). Therefore, applying Theorem 4.3.5, we get the Tauberian condition xn - xn_1 = 0 (n) for Abel-summability.

Corollary 4.3.8 (o-Tauberian theorem for the Borel method). Every sequence x = (Xk) E CB1 which satisfies the Tauberian condition

xn - xn_1 = °(*) is convergent. (k E 1J°) and An J. 31 n, k (cf. 4.3.2(b)). Thus, applying 4.3.5, we get that xn - xn_1 = o(*) is a

Proof. For the Borel method we have Pk =

,

Tauberian condition for Borel-summability.

Remark 4.3.9. The preceding Tauberian theorems 4.3.7 and 4.3.8 remain true if we replace the o-Tauberian condition by the corresponding £-Tauberian condition xn - xn_1 = k7(n) and xn - xn_1 = D(3), respectively. We will return to this question in Sections 4.4 and 4.5.

A

We now give an O-D-theorem which allows the conclusion from boundedness of the Pp transform of a sequence x to the boundedness of x.

Theorem 4.3.10 (0-0-theorem for Pp). Let Pp be a regular power series method such that p = (pt) satisfies 3.6.1 and let x = (xk) E w be such that the power series p2 (t) has radius of convergence > R. Then P(e}) = D(1) (t -+ R-) and the Tauberian condition xn - xn_1 = D (a }

implies xn = 0(1), that is x E m.

Proof. By hypothesis, we have jxn - xn_11 < Ma for some M > 0. Now, to prove the theorem we can follow the lines of the proof of Theorem

4.3.5 where we have to replace a with M and put kf = 0. The details are left to the reader. As a corollary of Theorem 4.3.10 we consider the special case of the Borel method.

Corollary 4.3.11 (1J-B1 -+0). Every sequence x = (xk) E CB, which satisfies the condition xn - xn_1 = 0(1 ) is bounded.

Hardy-Llttlewood's D-theorems for the Abel method

191

Proof. Let X E CB. Then vt? = 17(1) (t -+ oo) where (Pk) = (k,). (cf. 4.3.2(b)) we may apply Theorem 4.3.10 to this Noting An - Z Fsituation and get x E m. O

Exercise 4.3.12. Let p = (pk) satisfy 3.6.1 and R denote the radius of convergence of the power series p(t). Show that the values An (n E NO) have the following properties: (a) (A, ) is monotonically increasing and o }1 -+ R when n - oo. (b) If R < 1, then (An) is (strictly) monotonically increasing. (c) If 1 < R < oo, then there is an no E NO with (An)n>no is (strictly) monotonically decreasing.

Exercise 4.3.13. Let p = (Pk) satisfy pk > 0 (k ENO) in addition to 3.6.1. Prove that x = (xk) E Cop, and xk > 0 (k E N°) implies (xo ) E co. That is, xn = o (p ) . (So, in the sense of 2.7.10 the sequence ( Pn } is a `summability order' of the power series method Pp for non-

negative sequences.)

Exercise 4.3.14. Assume p = (Pk) satisfies 3.6.1. Prove the following statements concerning the corresponding 1n. (a) If p = (kk+1 } , then An ^- log n (n -4 oc). (b) If a > 0 and p = ((k+k-t)), then An --- n°a (n -a oo).

(c) If (3 E]0,1[ and p = {eke), then On ^ pn 8(il3) n'-4 (n -+ oo). Exercise 4.3.15. Let Pp be a regular power series method such that p= (pk) satisfies 3.6.1, and let (ak) E 2. Prove the following statements: (a) There exists a sequence (3k) which increases monotonically to 00 and satisfies E' o kaki Qk < 00. Hint: Consider 16k = rk # with rk = E', for k E N. (Note that this is a special case of Dini's theorem (cf. [128, p. 302].)

(b)

Ek

o

ak1ktk = o(p(t)) (t -+ R-).

(c) E1 o = 00.

Bibliography: [227], [134], [244]; [135], [137], [136]

4.4

Hardy-Littlewood's ID-theorems for the Abel method

In Corollary 4.3.7 we stated Tauber's result (cf. [238]) that if x = (xn) is Abel-summable and xn -.x,,-, = o (-L) , then x is convergent. Moreover, in 4.1.16 we proved that the Cesaro methods Cam, a > 0, are all equivalent for bounded sequences. In this section we improve both results with two Tauberian theorems for the Abel method due to Hardy and Littlewood (cf. [110, Theorems 11 and 9], see also [145]). The first says that D(n)

192

Tauberian theorems

is a Tauberian condition for Abel-summability and an immediate consequence of the second tells us that the Cesaro methods Cn, a > 0, and the Abel method are equivalent for bounded sequences. The shortest proofs are due to Wielandt [246] and Karamata [124] and, by modifying Wielandt's proof, to Tietz and Zeller [240]. We follow the lines of Tietz and Zeller's proof which avoids integrals and is based on the Weierstrass approximation theorem. First we extend Tauber's theorem for Abel-summability.

Theorem 4.4.1 (.t?-Tauberian theorem for the Abel method). Every sequence x = (xk) E CA, which satisfies the Tauberian condition xn - xn-1 = i7(n) is convergent. Proof. Considering both the real and the imaginary parts and noting that for real sequences the Tauberian condition xn - 2n-1 = 17 (n) implies the corresponding 17R-Tauberian condition, the theorem is an immediate

0 Theorem 4.4.2 (DR-Tauberian theorem for the Abel method). consequence of Theorem 4.4.2 which we now give.

Each real sequence x = (xk) E CA, which satisfies the Tauberian condition

xn - xn-1 = f7R(n) is convergent. We precede the proof of this theorem with some notation and a technical lemma.

Notation 4.4.3. Throughout this section let a := z and let F be the set of all functions f : [0,1] -+ R. Let id denote the identity map on [0,1] and s : [0, 1] --> R be the step function with

for 0< t< a s(t) :- jl10 fora
:= 3(t) if t E [0,a[U[a+s,1] linear on [a, a + e] ,

r2 (t}

s(t) if t E [0, a - e[U [a,1] linear on [a - s, a].

Obviously, ri < s < r2.

A

Lemma 4.4.4. Let0<(3< y<1 and0
foreverytE]0,1[. n=1

(b) If g(t) < t(1 - t) for each t E [3, ry[ and g(t) = 0 otherwise, then 5 C[0,1] denotes the set of all continuous real functions on [0,11.

Hardy-Littlewood's 0-theorems for the Abel method 193 co

En-1g(tn)
Proof. First we assume /3 > 0. If 0 < t < 0, there is nothing to prove, since to < 8 and therefore g(tn) = 0 (n E N). If 0 < 0 < t, then we can choose k, l E N with k < 1 such that tk < y < tk-1 and t1 < /3 < t1-1 Then we get, under the assumptions of (a), that 1-1

1-1

00

(1- t)Eg(tn)=(1-t)Eg(tn) <(1-t)Etn =tk-t1
n-k

ft=k

and, under those of (b), that 00

En- ig(tn) =

!-1

1-1

E n-1g(tn) < E n-1tn(1 - tn) n=k

n=1

n=k 1-1

< (1-t)1: tn
That n-1 (1- tn) < (1- t) is a simple corollary of the mean value theorem. If S = 0, then for each t E ]0,1[ we have /3 = 0 < to (n E N) and we can choose k E N with tk < ry < tk-1. Then we get the desired inequalities when we replace 1-1 by oo and t' by 0 in the above calculations. 0

Proof of 4.4.2. Let x = (xk) E CA, be a given real sequence satisfying xn - xn-I = OR (n) . Obviously we may assume without loss of generality

that nan < 1 with an := xn -xn-1 (n E N°) and, since Al is regular, x E COA, (otherwise we may consider the sequence x -(A1- lim x)e ). Moreover,

it is sufficient to consider the case a° = x° = 0 (otherwise we may change a° and a1 ). Our hypothesis is then that F,- 1 antn converges for t E [0, 1[ and En_1 ant' = 0. We have to conclude x E co. Put an f (tn) converges for each t E [0, 1[ n=1

and

lim

00

t-f1- n=1

an f (tn) = 0

.

Obvious, id E W by our assumptions on x. To prove x E co, that is 1imK F_k=1 an = 0, it suffices to show s E W (cf. 4.4.3). Note that for each

K E N there exists a t E ]0, 1[ with tK = a and to < a (n > K) which yields F°O_I ans(tn) = En 1 an. Further, because limt+l- E' antn 0, we obtain gk E W for every k E N where gk : [0,1] -+ R, t -+ tk. Thus for each non-zero polynomial p the pointwise product id p, that is a polynomial of degree greater than zero, belongs to W.

194

Tauberian theorems

Next we prove

f := id(1 - id)h E W for each h E C[O, 1].

(4.4:1)

Let h E C[0,1] and E E ]0, a[ be given. Applying the Weierstrass approximation theorem to both the function h + i and h - i we get polynomials

P1, p2 such that pi < h < p2 and 0 < p2 - p1 < e. Consequently, 0 < f - id(1 - id)PI < Eid(l - id) and 0 < id(l - id)p2 - f < Eid(1 - id). Applying Lemma 4.4.4(b) with S = 0 and y = 1 we obtain 00

CO

n-I(f

L an(f - id(l - id)P1)(tn) < E n=1 n=1

- id(l - id)PI)(tn) < E

and CO

CO

LS an(id(1 - id)P2 - .f)(tn) < E n-1(id(1 - id)p2 - f)(tn) < E. n=1

n=1

Thus, because id(1 - id)p1, id(1 - id)F2 E W, we obtain the inequality limsupt-.1_ JEOO I an f (tn)l S E. Hence f E W. The relation id(1 - id)hi = Ti - id (cf. 4.4.3) uniquely defines a continuous function hi on 10, 1[ which can be extended continuously to [0,1].

Applying (4.4:1) to h1 we get r1 - id E W. Similarly we define h2 and obtain r2 - id E W. Since id E W we now have rI, r2 E W. Furthermore, we have

(s - ri)(t) { < 4t(1 - t) if t E (a, a + E[ - 0

and

(r2 - s)(t)

otherwise

< 4t(1 - t) if t E [a - E, a[ {=

0

otherwise

and therefore again by Lemma 4.4.4(b), cc

00

57an(s-rI)(tn) < 4(a+e-at) and E an(r2-s)(tn) < 4(a-(a-e)t). n=I

n=1

Hence, noting r1 , r2 E W, we obtain limsupfi1_ IF_O°_1 ans(tn)1 < 4E; o thus s E W.

In the second part of this section we prove that the Cesaro methods Cam, a > 0, and the Abel method are equivalent for bounded sequences.

Theorem 4.4.5 (JD(1)-Al-+Ci-theorem). (a) Every sequence x = (Xk) E CA, which satisfies the Tauberian condition xn = 0(1) is summable by the Cesaro method C1. (b) For each a > 0 the Abel method is b-equivalent (and consistent) with the Cesaro method C¢, that is m fl CA, = m fl cc. (with consistency).

Hardy-Littlewood's i7-theorems for the Abel method 195

Proof. (a) Considering both the real and the imaginary parts and noting that for real sequences the Tauberian condition xn = 0(1) implies the corresponding £'R-Tauberian condition, the theorem is an immediate consequence of the subsequent Theorem 4.4.6. (b) By (a) and Theorem 3.6.11(a) we have m !1 CA, C cc, and CC, C CA, (with consistency). That is, Cl and Al are b-equivalent (with consistency). With that the statement follows, since Cam, a > 0, are pairwise b-equivalent 0 (with consistency) by Corollary 4.1.16.

Theorem 4.4.6 (k?R(1)-Al -+ Cl-theorem). If a real sequence x = (xk) E CA, satisfies the Tauberian condition xn = i?R(1), then x E cc,. Proof. The proof proceeds essentially as the proof of Theorem 4.4.2. We may again assume without loss of generality that x = (xk) E COA, , xo = 0 x) and xn < 1. In particular, we have xntn = 0. We have to conclude x E coc, . Put

f E F ' E xn f (tn) converges for each t E [0,1[ n=1

and

lim t->1-

(1-t)7xnf(0) =0 00

.

n=1

Obviously, id E K by our assumptions on x. To prove x E coc, it suffices

to show s E K where-in addition to the corresponding remark in the proof of 4.4.2-we note E,°°_1 xns(tn) = En=j xn if t = 2-N and that limn (1 - 2- ) (n + 1) = In 2. As in the proof of 4.4.2 we get id p E K for each polynomial p. In the next step we prove

f := id h E K for each h E C[0,1].

(4.4:2)

For that let h E C[0,1] and E E JO, a[ be given. By the Weierstrass approx-

imation theorem there exist polynomials p1 , P2 such that p1 < h < p2 and 0 < p2 - p1 < E. Consequently, 0<_ f - id pl < eid

and

0 < id p2 -f :56d-

Applying Lemma 4.4.4(a) with 6 = 0 and 'y = 1 we obtain 00

00

n=1

n=1

(1-t)Exn(f -idpl)(tn) < (1-t)E(.f -idPI)(tn) < e and

(1-t)Y' xn(idP2- f)(tn) < (1-t)1: (idp2-.f)(tn) < e. n=1

n=1

196

Tauberian theorems

Hence, since id pl , id p2 E K, we obtain 00

limsup(1 - t) E xnf (t') t->1-

< E.

n=1

Thus f E K. Applying (4.4:2) to r1 = id h for a certain h E C[0,1] we get rl E IC and similarly r2 E K. Furthermore we have

(s-r1)(t)

< 2t iftE[a,a+e[ = 0 otherwise

and

2t if t E [a - -, a[ otherwise and therefore, again by Lemma 4.4.4(a),

(r2 - s) (t)

=0

(1-t)Exn(s-r1)(t') < 2(a+e-at) and n=1

xn(r2 - s)(tn) < 2(a - (a - e)t).

(1 - t) n=1

From this and rl , r2 E IC we obtain 00

lim sup (1 - t) E xns(tn) < 2e. t-+1n=1

Thus s E K.

0

Exercise 4.4.7. Prove that the Cesfiro methods Cam, a > -1, and the Abel method are convex, that is mca fl Ca, C cc for -1 < a < i. Bibliography: [145], [110], [246], [240], [132], [244]

4.5

Hardy-Littlewood's 0-theorem for the Borel method

As a corollary of the o-Tauberian theorem 4.3.5 for power series methods we got in 4.3.8 the result that xn - xn_ 1 = o (1) is a Tauberian condition for the Borel method B1. In this section we prove that even the weaker

condition xn - x._1 = f7 (1) is a Tauberian condition for the Borel method. This result is due to Hardy and Littlewood (cf. [111]). For a proof we go on from Corollary 4.3.11 which tells us that each Borel-summable

sequence (Xk), which satisfies xn - xn_1 = D(1), is bounded. Then we will use, as we will show in Theorem 4.5.2, that the generalized Borel methods B,, (0 < a < 1) are b-stronger than the Borel method. This

elementary proof was given to the author-for use in this book-by W. Kratz; he adapted it from a more general situation in a joint paper with U.

Hardy-Littlewood's JD-theorem for the Borel method 197

Stadtmuller (cf. [134]). For other methods of proof we refer the interested reader, for example, to [216], [108] and [267]. Finally, the 1-Tauberian theorem for the Borel method enables us to prove a convexity theorem (and consequently a b-equivalence theorem) for the family of summability methods consisting of the Euler methods E, a > 0, and the Borel method. We start with the formulation of the 1)-theorem for the Borel method which we are aiming to prove.

Theorem 4.5.1 (0-Tauberian theorem for the Borel method). Every sequence6 (xk) E CB; satisfying the condition x - x,-1 = O(=) is convergent.

On the way to a proof of this theorem we first derive the following 'Abelian result'.

Theorem 4.5.2 (c'-Bi -+ Be). If 0 < a < 1, then m fl CB, C CB, holds (with consistency).

For convenience we recall and introduce some notation which goes on from that in Sections 3.6 and 4.3.

Notation 4.5.3. Let a E C, to > 0, pk :=

(k!)-a

(k E N°), and let

x = (xk) E CB,. Then we put for t E R (see Section 3.6) xk k, t E (k!)«

pa,x (t)

p. (t)

pa,e(t) _

E(k!)-«tk,

k

oa y(t)

.-

pa.x{t) pa{t)

1 Ek(k!)-atk

[ xk tk. (k!) k

In the case of the Borel method (a = 1) we use the settings p. (t) := p1,y(t), p(t) := p1(t) and ox(t) := o1,z(t) which are the (general) notations in the case of power series methods.

IL

Proof of 4.5.2. Let 0 < a < 1. Obviously, by the regularity of B1 and B, it is sufficient to prove m fl C°B, C COB, . So let x = (xk) E m fl C°B, be given. By Stirling's formula we get, as k -+ oo,

-

(k!)1-a

,ak(a)

-

(21r)-1.

(i -

k(1-«)(k+-)e-(1-a)k(27r)2 a)-(1-«}(k+) J27r ((1 V

(k(1 -

a))-I+(1-a)k

1

a

- a)k - 2) e a

- 2(1 - a)k)

(1-a}k

6 Concerning the definition of C we refer to 3.6.2 in connection with 3.6.3(iii).

198

Tauberian theorems (21r)-2

(1 -

a)-(1-a)(k+2') I'(1 p00

(2ir)-2(1-a)-

/ \

I 1

t

)

- 2 + (1 - a)k) (i-a)k-

l

*), that is dt =

[by the substitution r = (ItC') 00

(27r)-1(1-a)-I' f01

e-tdt

r' dr}

e-(1-a)rT-'° dT

Tk T2P

fTkdXa(T),

for some non-decreasing function X,,,. Let

J0

lAk (a)

Tk dXa (r)

,

pa (t)

(a) {k }a

k

and

a'x(t)

/ k(a)

:= pa(t)

k

(k!)-

tk.

Then, by the preceding calculation, by the regularity of Ba, and since

x E m, we obtain that j5,, (t) - pa(t), and Ua,x(t) - Qa,x(t) _

Ftk(a) pa(t)) tk xk 1 1 E (k!)AA; (a) pa t)

P. (t)

-+0

\\

k

as t -> oo. Hence, x E COB, if Oa,x(t) -a 0 as t -3 oo. Finally, let a be an arbitrary positive number. Since x E COB,, there is a to > 0 such that I ax (t) I < e for t > to. Hence, for some C > 0, and every t > to, Ra.x(t)i =

uk(a)

pa(t}

(k 1}a

k

<

--

1 pa(t)

(

k

o

1

pa (t)

-+C

I

dX0(r)

IUx(Tt)I ert dXa(r)

1

p`a(t} Jo

<e+

xktk

ly (kx'.} a tkrk

ao

J

ax(Tt)I ert dXa(r)

o

eto

A. (t)

< 2e (for sufficiently large t).

-

Hence, Qa,x (t) -i 0 as t -* oo, so that x E COB. for 0 < a < 1.

Hardy-Littlewood's 0-theorem for the Bore! method 199

The proof of Theorem 4.5.1 is based on the following asymptotic result, which we shall derive below.

Lemma 4.5.4. Let a > 0 and rn(a) := nae2 . Then, if f denotes the principal branch of the square root, lim

(k (a)

pa(rn(a))(rn{a})_n (n!)a

1

= limps{rn(a}}

e

{ten)a

._ n

and the convergence is uniformly on compact subsets of R

{aEC I to>0}.

Moreover, we need an inequality, whose proof is similarly structured as that of Theorem 4.3.5.

Lemma 4.5.5. Assume that xn - xn_1 = 0(*), that is

Ixn-xn-1I<

n

forallnEN

(4.5:1)

with some c > 0. Then

limsup IOa,y(rn(a)) - xnI < c4?(a) for each a> 0, n-+00

where rn(a) and (D(a) are defined as in Lemma 4.5.4.

Proof. Let a > 0 and rn := rn/(a). Then (rn) is increasing since log

rL1

= a log I 1 +

-

1n )

2n (n + I) > 0.

We proceed similarly as in the proof of Theorem 4.3.5: 1

IQsz(Tn)-xnI < palm )

00

x(k1)xnI rn

k

E I k=O

where (using (4.5:1) and the fact that (Tn

Sal (n) :_ <

1

n-1

r yrv [= n

pa(Tn) v_1 c

pa(wn)

n

(v 1)a

< 1 for 0 < k < v
(xv - xv-1) v=k+1

_v v-1 k=O

rnv

)v-k

n

r nk

pa(wn) k_0 (k I)a C

= S i(n)+ Sa2(n),

11

rk

(7"

Q

(k .}

f

v-k

(

`rn v-1 rvk

pa(rv)rv-v(y 1}a

1

a(rv) Pa

E (k r)a

k=O

200

Tauberian theorems c

<

P. (r.)

Tnv

n

E ( v! V=1

pa(Tv)Tv-v(v (ta

Yv

) )k-V

and (using (4.5:1) again and (

<1 for 1 < n < v < k) k

k

E }

Sat{n} :=

E (xv

pa(Tn) k=n+1 00 C

n+1

rvT-v

c

Pa(Tn)

00

of

pa(Tn) Y_I 00

Tnv

(v 1)a

=n+1

- x"-1)

v=n+1

(.rn-r) k-v

k

(k !)a

v

=v

1 pa(Ti,)Ti-"(v!)a V" Pa(Tv)

00 C

Pc(Tn) v-n+l

Hence, CO

lim up Ia.,.(Tn) - xnI < lim n

n-+00

p(T) n

v-1 (v

pa(Tv)T

v{v !)a =

)a

and the regularity of B,,,. by the definition of The previous results enable us to prove Theorem 4.5.1.

13

Proof of Theorem 4.5.1. Without less of generality, since B1 is regular, we may assume x = (xk) E COB,. First, by Lemma 4.5.4 we have that, for

a > 0, lnm

r pa(Tf(a))(Tf(a))-n(n!)a = /21 =
uniformly on compact subsets of R = {a E C I Ra > 0}. Furthermore, the functions fn(a) := va,x(Tn(a)) are holomorphic in R, fn(a) -* 0 as n -> oo for 0 < a < 1 by Theorem 4.5.2, and since x E m by Corollary 4.3.11, we have that I1n(a)I

C

p(3(Tn(/3))

Ipa(Tn(a))I

4

C 13

uniformly on compact subsets of R, where fi = R a. Hence, the sequence (fn(a)) is uniformly bounded on compact subsets of R. Therefore, by

Vitali's theorem, fn (a) -+ 0 as n -+ oo for all a E R. Now, it follows from 4.5.4 and 4.5.5 that, for all a > 0, limsup IxnI = urn sup Ixn - Ta(Tn(a))I < n-a0o

n-+0o

Letting a -> oo, we deduce from this that x E co.

Hardy-Littlewood's JD-theorem for the Bore! method 201

Altogether, for a proof of Theorem 4.5.1, it remains to give the

Proof of Lemma 4.5.4. By our definitions and Stirling's formula, we have uniformly for R a /> 0, as n -+ oo, Vnpa(Tn(a))(Tn(a))

pa{Tn(a))

e

-an

21rn

)

n

Yn

f

a-an

) 00

Hake 3k 2

(kk e-k

k -1

fE CO

1

2,k)a =

eag(k),

where := g(n,k)

g(k)

klogn+2n -klogk+k-2logk+2logn-n-2 k2nn-ik+2jlog(n) +(k-n) for k>1. This notation implies that, for /t > 1, g(n) = 0, g'(t) = an - log (n) - Zt

i + - _< --Lt < 0. Hence, g(t)

so that g' (n) = 0, and g" (t)

is

increasing on 11, n[, it is decreasing on In, oo(, and g(t) < g(n) = 0 for all t > 1. Using the inequality )ft

feat - eanl = J aeav dv <

laliept

I

-

ean!

n

t

for J3 := R a > 0, and t E R and n E N, we obtain that 1

eag(k) _

I00 eag(t) I

k=1

lal

°°

2lal _<

v

e09{t} dt

fk

)I(

/

k+1

e(39(k) _

a7n

dt

I e'39(k1 k=1

n

\

1) - e139(k)1 + [ f 9{k} - e(j 9(k F1))) J kLn t

0

uniformly on compact subsets of R.

Next, let h(n, T) = h(T) := g(t) for T :_ 77 t-n >- 77 (t >- 1), and h(r) := -oo for -r < . Then by the definition of g(t),

h(r) = 2 - (n + T

+ 0 log ( 1 +

]+Tf

202

Tauberian theorems

< 2, fn-

( \\n+r

< -7,

< -2n-1-

+2

r > (e2 - 1) f, that is log (1 + '} > 2. But, if r < (e2 -

whenever

1)/, then h"(7-)

= n g"O T

n

2(n+rf)

--

so that

_

1

2(1+*} <

72

1

2e2'

72

h(r) = h(0) + h`(0)r + h"(e) 2

:5

2e2

for all r < (e2 - 1) vfn- and some , r < < (e2 - 1) Vfn-. (Observe that h(O) = h`(0) = 0.) Therefore, f (eoh(n,r) } dr converges for n 00 uniformly on compact subsets of R.

Finally, substituting T = t-n, we obtain that 1

[ 1

eog(n,t) dt =

eah(n,r)

dr

00

00

-(f+r+2)log(1+

=1

n( fff

exp (_a

r2) dr

f f11dr

=

00

uniformly on compact subsets of R.

0

As we stated in 3.6.11(b) the Borel method Bl is stronger than and consistent with the Euler methods E,,,, a > 0. So we obviously have the following corollary of Theorem 4.5.1.

Corollary 4.5.6 (.0-Tauberian theorem for the Euler methods). Let 0 < a. Then each (xk) E cEa which satisfies xn - xn_1 = D(') is convergent.

In comparison with the family of summability methods, consisting of the

Cesaro methods and the Abel method, we are in a quite similar situation in the case of the Euler methods Ea, a > 0, and the Borel method. Both families are monotone and in both cases we have available an 0-Tauberian theorem. In the first case these results led us to the b-equivalence (cf. 4.4.5) and the convexity (cf. 4.1.14 and 4.4.7) of the family of summability methods under consideration. We now aim to prove the following convexity theorem for the family of summability methods, consisting of the Euler methods E,,, a > 0, and the Borel method Bl (cf. [48]).

Hardy-Littlewood's D-theorem for the Borel method 203

Theorem 4.5.7 (convexity of E. and B1). The Euler methods E,

,

a > 0, and the Bore] method B1 are convex, that is ME. f1 CB1 C cEa for

all 0
Corollary 4.5.8 (b-equivalence of Ea and B1). The Euler methods E,,, 0 < a < 1, and the Bore] method B1 are b-equivalent.

Proof. Put a := 1 in 4.5.7 and note El = I. We prepare the proof of the convexity theorem by the following lemma.

Lemma 4.5.9. Let a > 0 and x = (xk) E w be given. (a) If y := (k(Xk -xk_1))k, then [Eax]n - [EEx]n-1 = n[Eay]n (n E N). (b) If 0 < a < 1 and x E m, then (Vln-([E,,x]n - [EEx]n_1)),, E m. Proof. The statements are straightforward calculations and are left to the reader in Exercise 4.5.11.

Theorem 4.5.10. Let 3 > 0, x = (Xk) E w and y := (k(Xk - xk_1))k. Then X E CB, and [E,3y]n = O(/i) imply x E cER (with consistency).

Proof. From x E CB, we get Ej3x E CB1 (with consistency) by 3.6.13. With 4.5.9(a) and the assumptions on x we get Vrn([E$x]n - [Ei3x1n_1) =

[En VIn_

= f7(1).

Thus X E CEP follows from 4.5.1 applied to E0x.

Proof of Theorem 4.5.7. Let 0 <,8 < a, and let x = (xk) E ME, f1CB, be given. Applying Theorem 4.5.10 for a proof of x E cE,,, it is sufficient to prove

[E5y]r: = D(f).

(4.5:2)

Let and z = (zn) with [E,,x]n. Then, since E,,E13 = EQ,3, we get [Ea zJ n. Thus on account of Lemma 4.5.9(a),

yn = n([Eaz]n - [Eaz]n-1) = [E. (v(z from which we obtain, again with Lemma 4.5.9(a), the equalities [EAy]n = [E-,EEy]n = [E7(v(z,, -

n([Eyz]n - [Eyz]n_1).

From this, together with z E m and Lemma 4.5.9(b), applied to z, the desired statement (4.5:2) follows.

Exercise 4.5.11. Prove the statements in Lemma 4.5.9. Bibliography: [134], [111], [216], [108], [170], [149], [48], [267], [59]

204

Tauberian theorems

4.6 Notes on Chapter 4 In Section 4.3 we entered into a discussion of Tauberian theorems in the case of power series methods and stated in 4.3.5 that Xk - xk_1 = 0 (2A-) is a Tauberian condition for regular power series methods (satisfying 3.6.1). As we proved in Section 4.4 and Section 4.5 even the corresponding JDcondition is a Tauberian condition for both the Abel and the Borel methods. We used methods of proof which are designed for the Abel and Borel methods, respectively.

The examination of what extent Xk - xk-1 = (o) is a Tauberian condition is deep and beyond the scope of this book. Tkhus we just make some remarks.

In order to prove that xk - xk_ 1 = L7 (o) is a Tauberian condition for Abel-type methods Jp we may use the fact that the property of a sequence, to be Jp-summable, can be expressed by means of the asymptotic behaviour of a certain Laplace transform. Then the application of

Karamata's Tauberian theorem for Laplace transforms to the special situation of Jp-methods gives us Tauberian theorems of the desired type. (Note, a `Tauberian theorem for a Laplace transform' yields the conclusion from the asymptotic behaviour of the Laplace transform of a function, which satisfies a `Tauberian condition', to the asymptotic behaviour of that function.) For details on Tauberian theorems for Laplace transforms we refer the reader to the book [29] of N. H. Bingham, C. M. Goldie and J. L. Teugels. Concerning the applications to Jp-methods we refer to [134] and [135] and to the references in these sources. The proof that xk - xk_1 =1D is a Tauberian condition for certain Borel-type methods Bp, for example the Borel method, can also be done by the use of integral transformations (defined by the convolution of integrable functions) and is based on the so-called Wiener theory founded by N. Wiener in his famous paper [247]. More details on Wiener's Tauberian theorems are contained in the book [29] (cf. also [137], [244], [202], [108] and [216]).

Concerning remarks on Tauberian theorems for parts of summability domains (with certain properties), more general for FK-spaces, see the remarks in Section 7.7. Regarding an overview of several kinds of Tauberian conditions (general and application to special (classes of) summability methods), the interested reader should consult the `report of results' by Zeller and Beekmann [267, Chapter 6] and also the corresponding chapters in [29], [108], [216], [88] and [89].

5

Application of matrix methods In Section 1.1 and also in the Introduction of the book we remarked that, in attempting to formulate a notion of convergence which is application oriented, summability methods have been (and will further be) developed. On the other hand, by the investigation of these summability methods and the concomitant development of summability, the application of summability methods has been made possible in various fields of mathematics. For instance, summability methods are applied in function theory in connection with the analytic continuation of holomorphic functions and the boundary behaviour of a power series, in applied analysis for the generation of iteration methods for the solution of a linear system of equations, and for the acceleration of convergence in approximation theory,

in the theory of Fourier series for both the creation and acceleration of convergence of a Fourier series, and

in other fields of mathematics like probability theory (law of large numbers) and number theory (prime number theorem). As the section titles show, in this chapter we deal with some of the applications listed above. In the short Section 5.1 we deal with the boundary behaviour of a power series. There we establish a generalization of Abel's theorem (cf. 1.2.9)

which states that the theorem remains true on approach to a boundary point of the convergence domain in a 'Stolz angle'. After that we study a `converse' of that theorem. We conclude from the existence of a limit, when we approach the boundary in a Stolz angle, the convergence of the power series at the boundary point under consideration. Moreover, we deal with the boundary behaviour of power series obtained when we expand the function defined by a given power series about a point in the interior of the convergence domain of that power series. Section 5.2 is devoted to the analytic continuation of functions defined by power series. We essentially prove Okada's theorem which states that a matrix method A, satisfying certain assumptions, transforms the sequence

206

Application of matrix methods

sf (z) of partial sums of a power series f (z) to a sequence Asf (z) (of functions) which converges compactly in a domain, namely in a substar of the 'Mittag-Leffler star' of f, to an analytic continuation of f. Further, we apply Okada's theorem to certain examples of matrix methods. The applicability of matrix methods to analytic continuation is closely related to the generation of iteration methods for the solution of a linear

system of equations. In Section 5.3 we restrict our consideration to the generation of such methods and to the investigation of their convergence behaviour. In particular, we investigate neither the rate of convergence nor the complexity and stability of those methods. In Section 5.4 we deal with one of the most important applications of matrix methods, namely with the transformation of the Fourier series of 27rperiodic continuous functions. The aim is to give necessary and sufficient conditions for matrix methods to sum the Fourier series of 27r-periodic continuous functions f to a sequence of functions pointwise convergent to f. Matrix methods having that property are called `FC-effective'. Section 5.5 contains a survey of more applications of summability methods to other fields of mathematics.

5.1 Boundary behaviour of power series Let f (z) :_ Ek akzk be a power series with a complex sequence (ak) and

a radius of convergence r (0 < r < oo). In general, by the boundary behaviour of a power series we mean the convergence behaviour on the boundary of the convergence domain which is-as is well known-for r < oo a disc with centre 0 and radius r, the analytic behaviour of the function f (z) given by the power series when z approaches the boundary of the convergence domain. In connection with the first question we sketch under what assumptions on the convergence behaviour of a given power series f (z) we can conclude (uniform) convergence of the power series, which we get by expanding f (z) about a point z, with I7 < r, on the boundary of its convergence domain. The second problem contains for example the question of the existence of a limit of f (z) if z approaches zo along a path y in the convergence domain, and possibly the question whether f (zo) = limz,=o, zEy f (z) is true; that is, whether f, restricted to y, is continuous at zo. Because f is continuous in the interior of the convergence domain the stated problem is only of interest if zo is a boundary point of the convergence domain. Abel's theorem (cf. 1.2.9) tells us that if zo := r < oo, Ek akzo converges, and y = [0, r[, then

akzk ->

f (z) _ k

akzok as z tends to zo along y. k

Boundary behaviour of power series

207

We establish a generalization of Abel's theorem which states that if zo is an arbitrary point on the boundary of the convergence domain of Ek akzk, and converges, >k akzo

y is a path in an angular sector with

vertex zo

as

shown in Figure 5.1.1, then lim_.iZO,ZE'Y AZ) = Ek akzo independently of y. Such an angular sector is called Stolz angle. We may assume without loss of generality that Lk akz k has the radius of convergence r = 1 and that zo = 1. (Otherwise we consider the power seand a ries Ek akzokzk corresponding angular domain.)

V

__ ; a°`'°'i

(

I)

Fig. 5.1.1: Stolz angle

Theorem 5.1.1 (Abel (1826) and Stolz (1875)). Let (ak) E cs, Vo E [0,

z

[, and let W,,0 :_ I z = 1 - retlo I r > 0 and 'p E [-Wo, Wo] }

be the open angular sector with vertex 1 and semi-angle Wo which contains 0 and is symmetric with respect to the real axis (cf. Figure 5.1.1). Then

f (z) := 57 akzk where Dl

ak

(z -3 1, z E R,,, fl D,)

k

k

{z E C I f zj < 1} denotes the open unit disc.

Proof. Ek ak is assumed to be convergent, that is the sequence x = (xk) of its partial sums Xk :_ _ Fn=° a,, (k E N°) converges. Using Abel's partial summation formula (cf. also the proof of 1.2.9) we get

akzk = "(1 - z)

f (z) _ k

xkzk

(z E Dl).

k

In aiming to prove

f (Z) -4 E ak = l km xk (z -a 1, z E W p. fl D,), k

we choose zn E WP0 fl Di (n E N°) with

208

Application of matrix methods

z,, -+ 1

(n -+ oo)

and M2(1 - lJ zn) < Rzn < 1,

where M := tan cpo, and then we verify

f (zn) _ E(1 - zn)zn xk -3 lira xk = k

ak

(n -4 oo).

k

For this it is obviously sufficient to show the regularity of the matrix

A = (ank) defined by ank := (1 - zn)znk

(n, k E N°).

For this we use the Toeplitz-Silverman theorem (cf. 2.3.711). We see that

(Spo) and (Zsj) hold since zn --+ 1 and Ek(1 - zn)zn = 1 (n E N°), respectively. Further, (Zn) holds because 5I1-znllznIk =

k

II-znI (nEN°) 1-Iznl

and, as we are going to show, 11

zI

1

IzI

< 2 l + M2 (z E W,,0 fl D1 with M2 (1 - lRz) < Rz < 1) (5.1:1)

holds. To prove (5.1:1) we consider an arbitrary z E D1, z = x + iy = 1 - re"' ((p E [-Wo, Wo] and r > 0). Obviously,

IyI = (1 - x)Itan spI < (1 - x)M [because x < 1 and IVI <- wo]. Therefore

Z=

- x)2 + y2 < (1 - x) 1 + M2. If we assume also that M2 (I - x) < x < 1, then

(5.1:2)

1-IzI2 = 1-x2-y2 > (1-x)(1+x-M2(1-x)) > 1-x which implies with (5.1:2) the inequality 11 - zI

1-IzI thus (5.1:1) holds.

11 - zI(1 +IzI)

IZI2

< 2

1 + M2;

o

The theorem of Abel and Stolz tells us, that-under the assumptions of the theorem-the convergence of Ek ak implies that the function f given by the corresponding power series, is continuous at zo (= 1). Now, we will prove a theorem due to Landau which says that the existence of a limit of the function f Iw,,u\{zo} at zo implies the convergence of the series

Ek ak and therefore-applying Theorem 5.1.1-the continuity of f Iw,,, at zo. As an essential tool in the proof we use the regularity of the Cesaro method C1. As we will show, it is even sufficient to prove the convergence of (f (zn)) where (zn) is a suitable sequence in W,,, (n E N°) converging to zo. Further, as in 5.1.1, we may assume zo = 1.

Boundary behaviour of power series 209

Theorem 5.1.2 (of Landau (1907)). Let r = 1 be the radius of convergence of the power series f (z) := Ek akzk and let ak = o (k) . If there exists a c'o E [0, [ and a sequence (z,,) satisfying 2

zn E WponD1 (nEN°), zn-+1 [I1 - znI-1] = n for n > no and some no,

(5.1:3)

and

f (zn) --p a (n -+ oo) for some a E C, (5.1:4) then (ak) E cs and a = Ek gk. Proof. Without loss of generality we may assume a = 0 since otherwise we may consider f (z) - a. Furthermore, no E N may be chosen large enough, such that besides (5.1:3)

M2(1 - Rzn) < tzn < 1 with M := tan coo is also satisfied. Thus-as we have shown in the proof of 5.1.1-we get 11 - Zn

1 - Iznl

< 2,/_1 _+M1 (n > no).

(5.1:5)

If we put sup klak I k>n

En

(n E N° ),

then (En) E co because ak = o (k) . So, in view of (5.1:4), the statement

ak = 0

(ak) E cs and k

is proved if we verify n

(ak_f(zn)) E k=0

E CO. n

We remark that k-1 11

- zkl =

< I1-zlk

{1 - z) E z" v=0

(Izl<1 and kEN°)

holds and estimate as follows: n

E ak - AZ.) k=0

_

n

00

k=1

k=n+1

Eak(1-zn) -

akzn

n

Iakl 11 - znIk + 00E k=1

k=n+1

k Iakl Iznik

210

Application of matrix methods n

<

E klakf

6n+1

k=1

nl - zn(

1

+ n lznl The first term in the last sum converges to zero since n1 l - zn1 -+ 1 (n -+ oo) on account of (5.1:3), and because (klakl)kEN E coc, since n

ak = o (k) and C1 is regular. For the second term we get En+IM

n

[cf. (5.1:5)]

nl - znl

n(1

0

(n -3 oo)

[because of (5.1:3) and (En) E co}.

O This proves the theorem. Remark 5.1.3. As one may easily verify, Theorem 5.1.2 contains the oA1 -} K-Tauberian theorem for the Abel method (cf. 4.3.7). In the following investigations we sketch a further application of summability in function theory concerning the boundary behaviour of power series.

Let

(izI < 1)

f (Z) := E akzk

(5.1:6)

k

be a power series with radius of convergence equal to one. If p E )0, 1[ is given, then we can expand f in a power series about p to obtain a power series with radius of convergence equal to or greater than 1 - p and of the form

f (z) = E

(1

ap)k (z - p)k

where

(lz - p I < 1 - p)

(5.1:7)

(k E N°).

(5.1:8)

00

ak = (1 - p)k

(k) pv-ka,. v=k

To prove the last statement we recall that f(k) (p) _

00

av v(v - 1) ... (v - k + 1)pv-k v=k

and, if iz - p I <1-p, that f(z)

=

E L) p k!

k

E(z _p)k k

E k

(z - p)k

Ek v=k

(")P,-k k

(1 a p)k (z - p)k.

(k E r`lo )

Boundary behaviour of power series

211

It is not possible to get a general statement about the convergence of the power series (5.1:7) on the circle {z E C I Iz - pI = 1 -p} (cf. Fig-

ure 5.1.2). Moreover, there is a power series which converges, but not uniformly on that circle. three The following

theorems give us necessarq conditions for a positive answer to the questions which we have just raised. Fig. 5.1.2: Expansion of f about p

Theorem 5.1.4 (P. Fatou, 1906). If (ak) E cs and p E ]0, 1[, then the power series in (5.1:7) converges for each z with Iz - p I = 1 - p. Theorem 5.1.4 is obviously a corollary of the following theorem due to M. Riesz.

Theorem 5.1.5 (M. Riesz, 1911). If (ak) E cs and p E]0,1[, then for all -Go E ]0, ir] the power series in (5.1:7) converges uniformly on

(p + (1-p)e" I V)o
The following example due to Jakimovski and Meyer-Kong (cf. [118, p. 133]) shows that Theorem 5.1.5 fails in general for tyo = 0; that is, if we consider the whole circle {p + (1 - p)e"t' ib E [0, 2ir[ } . I

Example 5.1.6 (A. Jakimovski and W. Meyer-K6nig, 1980). Let p E ]0, 1[ be given. Then we put

zn := p+(1-p)aKi+ for nEN° and consider

(1 - zn) E00xk znk (x = (xk) E Co) k-0

and-as in the proof of 5.1.1-the matrix A = (ank) defined by ank := (1 - zn)z,k (n, k E N°). Since ZnJ 1

11-zn1 F, lzn ik k=O

(z n

-+ oo as n

oo,

the matrix A does not satisfy (Zn). Thus, by 2.3.5 there exists an x = (Xk) E co with

212

Application of matrix methods

Ax =

((1_zn)xkz)m. 00

n

k=0

For this x we consider 00

z) E0" xk zk = E ak zk (IzI < 1),

f (z)

k=0

k=0

where ak = xk - xk_1 and x_1 = 0. Since x E co we have

f(1) _ Eak = 0. 0" k=0

Now, we expand f (z) in a power series about p and get (cf. (5.1:7) and (5.1:8)) 00

f (z) = E(1--

g)k(z-p)k

for

(z-pj < 1-p.

k=O

If this series were uniformly convergent for Iz - p ( = 1 - p, then it would be uniformly convergent for Iz - p (< 1 - p and f (z) would be continuous on the disc Iz - p 1 < 1- p. This contradicts (f (zn)) = Ax m. IL In the light of Example 5.1.6, Jakimovski and Meyer-Kong proved the following theorem.

Theorem 5.1.7 (A. Jakimovski and W. Meyer-Konig, 1980). Let p E ]0, 1[ and assume the existence of a Wo E 10, ir[ such that the power series Ek akZ k converges uniformly on {e2`° - cpo < cp < cpo} . Then the power series in (5.1:7) converges uniformly on {z I Iz - p I = 1 - p} . I

The proof of this theorem is very technical, so we restrict ourselves to giving a rough sketch of the proof.

Basic idea of the proofs of 5.1.4-5.1.7. Let u = e2w with co E [0, 2ir[, and

v = p + (1 - p)(3 with 6 = e'''1' and 0 E [0, 2ir[ (5.1:9) be given. We have 0 = 1. For each n E N°, we denote the nth partial sum of the power series (5.1:6) at z = u and that of the power series (5.1:7)

at z = v by

n

= tn(u) := E akuk

[cf. (5.1:6)]

k=0 and

n

Tn

Tn(v) :_ E akI3k k=0

[cf. (5.1:7)],

Boundary behaviour of power series

213

respectively. The basic idea is to find an infinite matrix which transforms

t = (t,,) into r = (rn). Under the assumption that Ek ak converges (cf. (5.1:7)) one gets a matrix

A = A(p, u, v) with r = At

(5.1:10)

depending on the parameters p, u and v. Moreover, the coefficients of A can be determined explicitly (cf. [118, Theorem 3]). (In Exercise 5.1.9 the reader is asked to verify that A(p, 1, 1) is just the modification T1_P of the Taylor matrix Ti-p.) The matrix A, specifically A(p, 1, v), turns out to be conservative (cf. [118, Theorem 4]) if v is given arbitrarily in accordance with (5.1:9).

Consequently, if (ak) E cs and u := 1, then t E c. Thus, r E c since A(p, 1, v) is conservative. However, r E c means that the power series (5.1:7) is convergent at z = v. With that, Theorem 5.1.4 is proved. To prove Theorem 5.1.5 one considers also the matrices A(p, 1, v) where

the assumption Oo # 0 and the following properties of A(p, 1, v) are essentially used: A(p, 1, v) is conull (for each v in accordance with (5.1:9)), the row norm satisfies IIA(p,1,v)II

<-

5+11-1v1 < 5+M(tro)

VI -

for each v E { p + (1 - p) e"I I Vo < 0.-5 2ir - ipo } and some M (V7p) > 0 that depends only on i/io. Aiming to fix M(Oo) one can proceed similarly as in the proof of (5.1:1).

Concerning the proof of 5.1.7 we remark only that in this case the matrices A(p, u, v) where u $ 1 are applied.

0

In closing this section we remark that-for obvious reasons-regular methods play an essential role in almost all applications of summability to other fields of mathematics; for example, in the applications presented in 5.1.1 and 5.1.2. By contrast we note that in the proofs of 5.1.4-5.1.7, conull matrices are applied. This motivates the detailed investigation of conull matrices and their domains in subsequent chapters.

Exercise 5.1.8. Let 0 < r < 1 and let

K := K,.(1 - r) := {z 111 - r - z` < r} be the closed disc in C with centre 1-r and radius r. Show that (ak) E cs does not imply in general that

f(z) := E akzk -a k

ak

(z -4 1, z E K \ {1}).

k

Hint: Find where the proof of 5.1.1 fails if we consider sequences z --> 1 on the boundary of K.

with

214

Application of matrix methods

Exercise 5.1.9. Let p E JO, 1[ and let A = A(p,1,1) be the matrix in accordance with (5.1:10). Verify that A = Ti _p where B := (bn+l,k+l) for any matrix B = (bnk) Bibliography: [127], [142], [118], [267]

5.2 Analytic continuation Let D be a domain (in C), that is D is a non-empty, connected open subset of C. Let f : D -a C be holomorphic, that is f is differentiable for each z E D. Then it is natural to ask for the largest possible domain G with D C G and a holomorphic function g: G --3 C with 91D = f.

In such a case g is called an analytic continuation of f (from D to G). We remark that in general an analytic continuation is not uniquely determined. In summability, methods were developed to generate analytic continuations of functions (holomorphic at zero). A short overview is given in the `report of results' by Zeller and Beekmann [267, § 69]. In the present section we deal only with one method, namely the generation of analytic continuations by means of matrix methods. To this end we make some preliminary remarks, which may already be familiar from the study of function theory.

Remark 5.2.1 (expansion of holomorphic functions). Let D be a domain (in C), f : D -+ C be holomorphic, and-without loss of generality-let 0 E D. (Otherwise we consider for an arbitrary zfl E D the domain zo - D and the function defined by f (zfl - z) on za - D.) Now, f has a Taylor series centered at 0:

f (z) = E avz" for each z E D with Izj 0 and Q"

f

(") vt (0) 27ri

r J y_z-+W dz

(v E N°)

(5.2:1)

(y is a simple, closed rectifiable curve in D, for example a circle, with 0 in its interior and f is holomorphic in the interior of -y). Note, the formulae in (5.2:1) are the well-known Cauchy integral formulae.

An analytic continuation of a power series may be generated by repeatedly expanding f in power series about points along some curve (cf. Figure 5.2.2). This leads to the method of analytic continuation along a sequence of circles. We do not study this here since it needs a large amount of calculation. The basis for an essentially more effective method is given by the following theorem. We first recall the definition of compact convergence.

Analytic continuation

Fig. 5.2.1

215

Fig. 5.2.2

Definition 5.2.2 (compact convergence). Let 0 $ D C C and let fn : D -4 {C (n E N1°) and f : D -* C be functions. Then F := (fn) is said to be compactly convergent to f on D (briefly, fn,(z) f (z) D or F(z) f (z)) if F(z) (z) on each D compact subset of D. We say that the series n fn converges compactly (to f) if the sequence of its partial sums converges compactly (to f ).

Note, an example of a compactly convergent sequence of functions will be considered in Exercise 5.2.16. A further example is given by the partial sums of any power series which is compactly convergent on the interior of its convergence domain.

Theorem 5.2.3 (compact convergence). Let D be a domain (in C), let f : D -+ C be a function, and let fn : D -a C (n E N°) be holomorphic functions compactly convergent to f on D. Then f is holomorphic. Moreover, fnki --3 f(k) compactly on D for each k E N. Now, if we transform the sequence of the partial sums of a power series (for example, of the Taylor series of a function holomorphic at 0) by means

of a tow-finite matrix A, we get a sequence of polynomials (which are obviously holomorphic). We will prove that under suitable assumptions on .4 this sequence of polynomials converges compactly on a certain domain to an analytic continuation of the function under consideration. To get an idea of how to determine such domains we make a formal calculation which we carry out without the assumption that A is row-finite.

Preliminary considerations 5.2.4. Let D, f and a, (v E N°) be chosen as in 5.2.1, let A = (ank)n,kENo be an infinite matrix, and let

216

Application of matrix methods k

E avz"

with sk(z)

sf(z) := (sk(z))

(z E C)

"=o

be the sequence of the partial sums of the power series F_" a"z". Then k

sk(z)

v=0

Xi

(7 as in 5.2.1)

J C(+i dC zv Y

f C) "

(kEN°)

2702

S!

dS

and under certain additional assumptions (note again, this is only a formal calculation!) [Asf (z)]n

=

anksf (z) k

_

k

f {C)

1

(Z)'

(5.2:2) d( v=0 ` S [if interchange of integration and limit is allowed] ank

21ri

k

for each n E No. In the case of sf(z) E CA we thus obtain limA sf (z)

=

lim[Asf (z)]n n

f

Y

f (C)

k

Jim E ank C

k

v_0

(Z)' dC

(5.2:3)

[if interchange of integration and limit is allowed] 1

27ri

J

f(C) limA l

` v=a \\ Z

Y

dC k

1 / f {() d( 27ri f7 C - z

[if lima (

=

f (z)

k_ow } k = 1i holds for each

(5.2:4) to

and C E -t]

[using Cauchy's integral theorem].

These preliminary considerations show roughly which assumption on A, f and D we need, if for certain z E C we wish to conclude the summability

of sf (z) by A to `the right value', namely to f (z). To understand the formulation of those assumptions we introduce the so-called Mittag-Leffier

star (determined alone by f) and a substar of it which depends on f and a given domain G.

Analytic continuation

217

Definitions 5.2.5 (Mittag-Leffler star). A subset D of C is called star-like (from 0) if {pz 0 < p:5 11 C D holds for every z E D. That is, if for any z E D, each point of the connecting line from 0 to z lies in D. A star-like subset of C is also called a star and a star-like subset of a star is called a substar. Let f be a function holomorphic at 0. (a) Then

S[f] := {z E C I f is analytically continuable along the segment from 0 through z }

is called the Mittag-Leffier star (of f ). (b) Let G be a star-like domain with 1

G. (In particular, we have

[1, oo[ fl G = 0.) We use the notation

GS[f] := C \ ((C \ G) - (C \ S[f])),

where, as usual, A B := AB :_ lab I a E A and b E B} for A, B c C. We supplement the last definition and notation with some remarks which will be useful subsequently.

Remarks 5.2.6. If f is a function holomorphic at zero and G is a starlike domain with 10 G, then the following statements hold: (a) S[f] is star-like (by definition) and open in C.

(b) Dr := {z E C I Izi < r} C S[f], where r denotes (as in 5.2.1) the radius of convergence of the Taylor series of f (at zero).

(c) GS[f] C S[f]. (d) GS[ f ] = S[f] for each f holomorphic at zero if and only if G = C\[1,oo[. (e) GS[f] _ ()Zsif1 zG where zG :_ {zy I y E G} . (f) GS[f] = C if r = oo and, if r < oo, GS[f] = A Tg
re := sup{r>0 1 z=reie ES[f]} , zg := rge'o, if ro < oo, and {z=wze I wEG}.

GO

Furthermore, if r < oo, then r < re for all 0 E [fl; 21r[ and there exists a (E [0, 27r[ with r = r(. (g) GS[f] is star-like, thus a substar of S[ f ], and open in C.

Proof. (a) To prove that S[f] is open we consider any given z E S[f]. Since f is holomorphic at each point of the compact set K :_ {pz I p E [0, 1] } , there exists a b > 0 such that f is also holomorphic on M := {w E C 3 x E K : jw - xI < 51. Obviously, U6(z) :_

218

Application of matrix methods

{w E C 11w - zI < a} C S[f] since for each point w E U5(z) the connecting line from 0 to w is a subset of M. (b) Dr C S[f] because Dr is star-like and f is holomorphic on Dr. (c) Since 1 E C \ G the inclusion C \ S[f] C ((C \ G) (C \ S[f])) holds.

(d) If G = C \ [1,oo[, then C \ GS[f] _ ([1,oo[) (C \ S[f]) = C \ S[f] because S[f] is star-like. Conversely, let GS[f] = S[f] be true for each function f holomorphic at zero; in particular, we get [1, oo[ = C \ S[g] for the function g defined by g(z) := lZ i (z E C\ {1}). Therefore, because G is star-like, we obtain [1, oo[= C \ GS[g] _ (C \ G) [1, oo[= C \ G. Thus, G = C \ [1, oo[. (e) By de Morgan's rules we get

GS[f] =

C\

C \ (z(C

\ G)) = n zG. zfs[f]

ZV S(f]

(f) We assume r < oo since the case r = oo is obvious. Let 9 E [0, 21r[ be arbitrarily given, and let re := sup {r > 0 1 z = reie E S[f]} . If re < oo, then ze := reels S[f] because S[f] is open (cf. (a)). Thus, since Go = zIG, the inclusion

n GoD n zG re <00

0<0<2.

ZOS[f1

holds. On the other hand, for each z 0 S[f] there exists a 9 E [0, 27r[ and a p E [0, 1[ with zp = zo. Because G is star-like, we obtain z9G = zpG C zG, which implies

n zG D n Go. z¢S(f1

ra
It now follows from (e) that GS[f] = n re r which contradicts the definition of the radius of convergence of a power series.

(g) The domain G is assumed to be star-like. Consequently, zG is also star-like for every z E C and therefore GS[f] is star-like as an intersection of star-like sets (cf. (e)). Thus GS[f] is a substar of S[f] by (c).

Analytic continuation

219

The statement that GS[f] is open follows immediately from its definition since the

pointwise product of closed subsets is closed

too (note, G and S[f] are open).

0

Fig. 5.2.3

To illustrate the definitions of S[f] and GS[f] we give some examples which will play a role in the subsequent considerations.

Examples 5.2.7. Let f be a function holomorphic at zero and r be the radius of convergence of its Taylor series centred at 0.

(a) If G:=D1 := {zEC I Jz! <1}, then GS[f]

_ fCDr

ifr=oo, ifr < oo.

The case r = oo is trivial (cf. the first statement in 5.2.6(f)) whereas the case r < oo follows from the last statement in 5.2.6(f) which tells us that re > r for all 8 E [0, 27r[ and r = r< for some { E [0, 2ir[.

(b) We consider the set

G:= {zECIRz<1}. Then Go = reeiOG (8 E [0,27r[, re < oo), where re is defined as

in 5.2.6(f). Thus Go is the open (infinite) halfplane in C, containing 0,

-1 + i.

-,.

.1 + i

and with boundary

the straight line through

and or= re a thogonal to the straight ze

line through 0 and ze (cf. Figure 5.2.3). Hence,

GS[f] contains the disc

D. Moreover, GS[f] = Dr is true if and only if

re = r for every 8 E

Fig. 5.2.4

[0, 21r[; that is, if f is not analytically continuable beyond the convergence domain of its Taylor series.

220

Application of matrix met c is

In the case that f has only finitely many singularities, and if each quadrant of the complex plane contains at least one singularity of f, GS[f]

is the interior of a polygon. Figure 5.2.4 illustrates the case that f has

exactly the singularities 1. 1 - i. -I= i and -i. Now, we prove a well-kti%wu theorem due to Y. Okada (1925) which allows us, under additional a tunptions on a given matrix A, to conclude the summability of the (p:t-titl sums of the) Taylor series EY_o a"z" of any function f holomorphic at zero to the value f (z) for each z E GS[f]. Here G is a domain such that A sums the (partial sums of the) geometric series Ek zk to the (right) value : i : for each z E G. Consequently, as the proof will show, A, and thus G. are chosen in such a way that our formal calculations in 5.2.4 are justititd.

Theorem 5.2.8 (of Okada (1925)). Let G be a star-like domain with (w 0 1), and let A = (ank) be a 10 G and Dl e G, let matrix with {s9(w) I u; E G}

c..

and .4s°(w) 0 g(w)

(5.2:5)

(cf. also Exercise 5.2.17). T {sf (z)

I

z E G-

for each function f

c..

and Asf (z)

f (z)

hiC at zero. (Here, 0 (w) and sf (z) are de-

fined as in 5.2.4.)

Remarks 5.2.9. (a) sum of the geometric summability of (F-,, k =c,

Ek=o w" which is the kth partial rherefore. (5.2:5) includes for w E Dl the 5v .1 to the right value Ekwk =

is a sequence of polynomials and, in particular, Okada's theore::: 5.22._S tells us that functions holomorphic at zero can be approximated :::: rtts:y on compact subsets of GS[f ] by polyus an analytic continuation of f into GS[f] nomials. Consequently. it beyond the convergence .:,'z::.un D, of its Taylor series. Furthermore, the determined since GS[f] is star-like and analytic continuation is IL thus simply connected. to prove 5.2.8, we may follow the way As we have already u sketched in 5.2.4 if, for =y arbitrarily given compact subset K of GS[ f ], rectifiable integration curve 7 that lies in we may choose a simple.

(b) If A is row-finite. the-

S[f], such that K is

aed in the interior of 7 and such that S E G

for each z E K and i E where G satisfies the conditions in 5.2.8). The following lemma enables tz choice of such a curve. L, f'r' given as in 5.2.8, and let K be a compact subset of GS[f 1. Then r re eu is a simple, closed rectifiable curve -y which satisfies the following ct' .:ioms:

Lemma 5.2.10. Let f :%

Analytic continuation

221

(i) y C S[f]. (ii) y contains K in its interior. (iii) VzEK YCEy : S EG.

Proof. The proof is totally function theoretical in nature and does not contribute any new ideas to summability. So we refer the reader to the

0

book of E. Hille [116, p. 70, Lemma 11.4.1].

Proof of 5.2.8. Let A = (ank) and G be given according to 5.2.8 and let f be a function holomorphic at zero. Furthermore, let K be a compact

subset of GS[f] such that, without loss of generality, 0 E K. (If 0 0 K we may consider K U 10).) Vow, if we choose y in accordance with 5.2.10, then

I zEK and (Ey}

k:= JC Z

is compact, which follows, noting inf { n 1 c E y) rectifiable, from the compactness of K and y.

1

> 0 and that y is

To prove 5.2.8, it is obviously sufficient to show that in the preliminary considerations 5.2.4, the identities (5.2:2), (5.2:3) and (5.2:4) hold for each z E K. (The uniformity of the limit `limA' in (5.2:3) and (5.2:4) is then an immediate consequence of

As-' (w) 0

1 a 1-w

(cf. (5.2:5))

because k C G is satisfied since 5.2.10(iii) holds and k is compact.) Proof of (5.2:2): Because g(w) = 1 w (w # 1) we get ak(w)

ank 4gklw) k

towv

=

1 - wk+i

1-w

(w 54 1 and k E N° ).

Ek 1-1 W (anic

E Qnkwk+l

(5.2:6)

k

(wEGandnEN°), where we have to observe the following points. The series under consider-

ation converge since sg(w) E CA for every w E G and e = 39(0) E CA (cf. (5.2:5)). Furthermore, as an open set, G is a subset of the interior of the convergence domain of the power series Ek ankwk+r, which implies, in view of the compactness of K, the uniform convergence of that series on K. That also involves the uniform convergence of >k anksk(w) on k since 111W I w E K} is a bounded subset of C because 10 G. Thus, we

222

Application of matrix methods

can interchange the order of integration and summation in (5.2:2).

Proof of (5.2:3): Because K C G, the compactness of { S I S E 7} for any fixed z E K and As9(w) lw (cf. (5.2:5)) enables us to interG

change the order of integration and the passage to the limit in (5.2:3). Proof of (5.2:4): This identity follows immediately from K C G, since O As9 (w) 0 111w includes the pointwise convergence on G. G

Now, if we wish to apply Okada's theorem to analytically continuable holomorphic functions, we have to have suitable matrices A and star-like domains G which satisfy the assumptions in Okada's theorem. (For ex-

ample, 'suitable' may have the meaning 'row-finite' A and 'big' G.) In particular, if the matrix A is given, then we are interested in the set

G :=

k

WECI

= s9(w) E CA and limA s9(w) _ (F-W-) V=0

1

1-w

and a star-like domain G, as large as possible, with

GCG, 1¢G and As-g(w) ia

1 G 1-w

Furthermore, if As- '(w) _= 1w , which holds for example in the case of Dl

regular matrices (cf. 5.2.18), then we may choose a domain G with the additional property D1 C G. In the light of these considerations we now determine certain domains G associated with some particular matrices studied earlier in Chapter 3.

Example 5.2.11 ((discrete) Borel method). If g(w) :=lw (w # 1), then for the discrete Bore] matrix Bi = (bnk) = Ce-n, } the following statements hold:

(a) VwEC\{1} : s9(w)ECB, if and only if Rw<1. (b) Bls9(w) 11w where G := {w E C j tw < 1}

.

G

Furthermore, for that G and each function f holomorphic at zero, the star GS[ f] is determined by 5.2.7(b).

Proof. (a) Let w E C 1 {1} be given. Then for every n E N' we get (cf. also (5.2:6)) bnksk(w) k

_

1

1-w (Ebflk k

bnkwk+1 k

k

Analytic continuation 223

f 1 - we-n(1-w) ) .

1

(5.2:7)

1-w\ Thus, we obtain s9(w) E

(e-n(1-w))

CB7

!`r

4=

E co

(Ie-n(1-"')I)

E Co

((e92w-1)n)

a-n(1-R w) ) E co

eRw-1 < 1

E

co

Rw - 1 < 0

which proves (a). (b) The proof of part (a) establishes the convergence of BF s9 (w) to

1w 11

for each w E G. To show that the convergence is also compact on G we choose an arbitrary compact subset K of G. Then we have

a := sup {$2wI wEK} <1 ww (w E G )' is continuous,

and, since 'w

wEKI < co.

M:=sup

Consequently, for each w E K and n E No we obtain (cf. 5.2:7)

1I=

w

1-w

en(Rw-1) < Men(®-1).

Thus, on account of a < 1, statement (b) is proved.

0

In the next example we deal with regular Hausdorff methods. We will, however, answer the question of the summability of the geometric series only for a small class of regular Hausdorff methods.

Example 5.2.12 (regular Hausdorff methods). Let p = (p,,) be a moment sequence, that is

pn = J

1

to d k(t)

(5.2:8)

0

for some 4 E BV([O,1]) with P(1) - 4)(0) = 1 and 4)(0+). Then the Hausdorff matrix Hp = (hnk) is regular and (cf. 3.4.15) 1

hnk = (n f tk (1 - t)n-k df(t)

(0 < k < n; k, n E N°). (5.2:9)

o

Thus, for w E C \ {1} and g(w) := lw we obtain for the nth coefficient of Hps9(w) the identity 1

hnksk (w) k

=

1 1

w

(

hnk k

hnkwk+l k

224

Application of matrix methods n

1 1

__

w

[ 1- w

E

(n) k

k=O

(F wktk(1

Jo

-

t)"-k dt(t)

[because (5.2:9) and Ek hnk = I(1) - CO) = 1] 1w ( [ 1-w 1(1-t(1-w))nd4b(t))

1

fo

[interchange of integration and summation, application of the binomial formula].

This gives the following result for the summability of the geometric series by regular Hausdorff methods: (a) b w E C \ {1} : 8 (w) E cHv and limHP s°(w) = liw

fI (1 - t(1 - w))n d-P(t)) E

cc.

(b) If G C {w E C `{1} ( 8°(w) E cHp and limes, s9(w) then 1

a 1-w -#

Hps9(w) 0

1

J1

(1 - t(1- w))n do(t)

0.

G To determine a concrete G, we assume that 4P is normalized at zero, that is P(0) = 0 and 4'(0) = P (O+) hold. (This is possible on account of the above assumptions.) Then
Ga :_ (- EC with centre 1 - I and radius

that 1

Hps9(w) c

1

-w

(5.2:11)

Note, GaCGp for 0<,8
w=

a

a

holds for some p E [0, 1[ and cp E [0, 21r[. Thus, for every t E [0, a[ we obtain 2

11-t(1-w)12

=

I1`

t (1-pes`°)I a

Analytic continuation

(l_!)2+2p(1_.) cosrp+(t) a

a

a

225 2

a

(l_!(1_P)) <1. a2

In view of (5.2:10), for the integral in (a) with an arbitrarily given b E 10, a[ we have I

J (1 - (1 - w)t)' d(t)I

=

1

I

J o (1 - (1 - w)t)' d4)(t)

0

J <

... d-P (t) +

Vb(,I) +

If

1-a(l-p)

... d4)(t) I ri Vol (4))

where V'6 (1) denotes the total variation of 4) in the interval [0, 0]. Because 4) is assumed continuous at zero. and 11 - (1 - p) I < 1 we obtain

Hps9(w) "

w

1

for each w E Ga.

Now, we prove that (5.2:11) holds. For that let K be a compact subset of Ga. As a compact subset, K has a positive distance to the boundary of Ga; that is, there exists a p0 E [0, 1[ with

(1 - a1)I

Iw-

-

P < Po a a

1

a

This is because for each w E K we have w = 1 - a + « et'' for some p E [0, 1[ and cp E [0, 21r[. Thus, for each w E K we obtain [Hpss (w)].R

<M

-1

I

fo

1

wI

(1 - t(1 - w))" d4)(t)I

[because sup (I 1 wW I J w E K} < oo ]

< M (V.6(t) + 1 - all - p)IR

<M --3 0

1

-all - po)

Vol (It) ) Vo

(1))

((n i oo).

Since the last term is independent of w, (5.2:11) is proved.

0

226

Application of matrix methods

In particular, we apply the results of 5.2.12 to regular Euler and Cesaro methods.

Example 5.2.13 (regular Euler methods). If a E ]0,1] and Ga

{wECI Iw-(1-1)I <

«},

then

Eas9(w)

1 G, 1-w

holds for the Euler methods Ea. We mention that

Ga C G,3 forall0<(3
U Ga = {wEC I tw<1} aEj0,1j

are satisfied. Therefore, concerning the summability of the geometric series,

we see that the regular Euler methods Ea become more effective with decreasing a and that, when a tends to zero, they reach the effectiveness

tl

of the discrete Borel method (cf. 5.2.11).

Example 5.2.14 (Cesaro methods). On account of 3.4.17(c) for each 9 > 0 the Cesaro matrix C3 satisfies (5.2:10) with a = 1. This implies, by 5.2.12, that 1

Q3s9(w) D, 1-w

Moreover, we remark without proof that for each /3 > 0 we have

{w E C I C8s9(w) E c} = K1 \ {1}

(5.2:12)

and

C s9(w)

1

x,\{1} 1-w

{z E C Iz < 1} . In particular, (5.2:12) proves that Cesaro where K1 methods are not useful in generating analytic continuations. A Let us remark that in the literature all matrix methods, with which we have dealt so far, are examined with reference to the summability of the geometric series. All those methods have the advantage of being workable. The disadvantage, however, is that in general the substar GS[f] is small in comparison to the Mittag-Leffler star S[f]. Summability methods which provide analytic continuation into the whole Mittag-Leffler star have been already studied by Le Roy (1900) and Mittag-Leffler (1908). Also certain Dirichlet methods (cf. [267], p. 145) have this property.

In this context it is also of interest to seek regular matrices (being members of a given class of matrix methods) which sum the geometric series

(or a given power series) on prescribed subsets of C. W. Luh (cf. [154]), W. Luh with R. Trautner (cf. [155]), K. Stadtmuller (Faulstich) with W. Luh and L. Tomm (cf. [79], [80]), and K. Stadtmuller (Faulstich) with W.

Analytic continuation

227

Luh (cf. [78]) along with other mathematicians who have investigated this problem. There are a lot of publications concerning Okada's theorem and other methods, based on ideas in summability to analytically continuable holomorphic functions given by power series. In some closing remarks we emphasize some papers concerning modifications, generalizations and some interesting new ideas.

Remarks 5.2.15. (a) If f is a holomorphic function on a domain D and zo is a singularity of f, then Okada's theorem does not provide an analytic continuation of f to z E {pzo I p E ] 1, oo[} , that is to points `beyond' zo on the half-line from 0 through zo. This problem was begun by Jakimovski in [117] using a modification of the Mittag-Leffier star: he replaces the family of half-lines from 0 through z by certain families of Jordan arcs from 0 through z to oo where two distinct arcs intersect only at 0 and the family satisfies a certain continuity condition. Using the same idea, King (cf. [126]) improves Jakimovski's approach by considering almost convergence instead of ordinary convergence. (b) An essential, as well as restrictive, assumption in Okada's theorem is the condition that G be assumed star-like and simply connected. Gawronski and Trautner (cf. [93]) leave out that condition and prove a generalization of Okada's theorem in the case that G is the maximal open set Q with As9(w) Q 1 rw and GS[f] is replaced by C \ ((C \ Ap) (C \ G)) where

Ap denotes any maximal domain in which the function f is uniquely analytically continuable. (c) Muller, in [177], extends the Hadamard multiplication theorem con-

cerning the analytic continuation of g O f where f (z) = E" a"z", g(z) = F_" b"z" are given power series with positive radii of convergence

and (g o f)(z) := >" a"b"z" is the Hadamard product of f and g. As an application he deduces a general version of Okada's theorem containing, for example, the result of Gawronski and Trautner referred to in (b). (d) In his paper [101] (cf. [103] too) Gro13e-Erdmann gives short, as well as simple, proofs of the Hadamard multiplication theorem and of Okada's theorem in their general forms. He uses essentially new ideas and presents elegant proofs which are shorter than those of MUller and of Gawronski and Trautner. (e) Niethammer, in [180], studies the numerical analytic continuation of power series in connection with Okada's theorem. He discusses those matrix methods which give the domain of continuation greater than the circle of convergence and.which meet certain requirements resulting from algorithmic viewpoints. A measure for the speed of convergence of the transformed series is introduced and an asymptotic estimate of the truncation error is 0 considered.

228

Application of matrix methods

Exercise 5.2.16. Verify that the sequence (gn) of functions defined by gn(z)

:= Vin ``

1

1

F,l.21-yk

(nEN,

k=1

is compactly convergent on {z E C I IzI < 1} and {x E C I Jzj > 11

.

Exercise 5.2.17. Let the assumptions of Okada's theorem 5.2.8 be satisfied, that is, let G be a star-like domain with 1 f G and D1 C G, and let g(w) := 1iw (w # 1). Also let A = (ank) be a matrix. Show that for the matrix A = (ank) the condition (5.2:5) is equivalent to A satisfies (Zsj), {(wk) I w E G} C COA and A(wk) 0 0. G

Exercise 5.2.18. If A is a regular matrix and g(w) :=1w (w then As9(w) 0 11w

1),

Di

Bibliography: [116], [201], [203]; [267], [117], [126], [93], [101], [103], [177], [92], [180]

5.3

Numerical solution of systems of linear equations

In the present section we deal basically with an idea due to W. N. KubIanowskaja (1955/59) and developed by W. Niethammer (1967) and other mathematicians aiming to develop numerical methods for the solution of systems of linear equations. We restrict our interest to aspects of summability and we do not deal with topics like speed of convergence, computing expense, and so on. We introduce RR-methods' and consider generalized Euler methods. We show a way to use certain RR-methods, including generalized Euler methods, to obtain numerical solutions of systems of linear equations. So far we have applied (infinite) matrices to transform sequences into sequences, that is we have considered FF-methods2. By contrast, it is some-

times useful to apply matrices to transform series into series, that is one considers RR-methods. This is illustrated in the present section. Since any matrix may be considered as either an FF-method or an RR-method, we denote a matrix A by A when considering it as an RR-method. First, we introduce some notation.

Notation and Remark 5.3.1. In what follows, we understand a series to be a pair ((au), (xk)) of sequences (a,,) and (xk) E w with xk = Ek=o (k E N°) which is briefly denoted by F, a,,. We put

a

w := { t a I

a,, E K (v E NO) }

(set of all series)

1 R here stands for the German word Reihe which means series in English. 2 F here stands for the German word Folge which means sequence in English.

Numerical solution of systems of linear equations

cs :_ IT, a,,

(set of all convergent series).

E cs}

I

229

If E a, E cs, that is (a,,) E cs, then we use the notation E,, a,, for the sum of the series. If E denotes the summation matrix, then the map

ERF:w -a w, Ea,,

k

(at,) v=0

is bijective, ERF(cs) = c, and ERF(c) = 6s hold obviously.

A

Notation 5.3.2 (series-series method). For each matrix B = (bni,) we use the notations wB

{

aE

I

(a,,) E wB}

(application domain of B)

and csB

{EavEa

(Ebnat)ees} v

1

> a E wB I n

(summation domain of B ). Furthermore, we put

E (E

E a E wB)

(5.3:1)

V

for the B-transform of 57 a,, and

B-1: a := E 1: bn,,a,, n

v

\ E a E csB)

(5.3:2)

for the sum of this `double series'. The members of the domain csB are

said to be summable to the value b- E a by b. In addition, we call the triple (B, csB, B-E) a summation method (or matrix method in RR-form or RR-method for short) where B : wB -i w and f3-E: csB -+ K are defined by (5.3:1) and (5.3:2), respectively. For brevity we denote it also by B. Analogously, in the case of a matrix method in the sense of I.2.12 we speak about a matrix method in FF-form or also of an FF-method. According to whether we consider, for a given matrix, the corresponding

RR- or FF-method we say the matrix has RR-form or FF-form.

Application of matrix methods

230

Analogously to the regularity of summability methods in FF-form we define the notion of regularity of summability methods in RR-form.

Definition 5.3.3 (RR-regularity). A summation matrix b is called RR-regular if cs C & and b- F_ a = F,v av for each F, av E cs. Let B be any matrix, E be the summation matrix, A := EBE-', and let ERF be the mapping in 5.3.1. Then, in the particular case A = I = B the commutative diagram

I

W

w

ERF

ERF I W

W

suggests the examination of whether certain properties of RR-methods (matrices in RR-form) may be reduced to corresponding properties of FFmethods (matrices in FF-form), and conversely. So, for example, it would be easy if RR-regularity could be reduced to FF-regularity, and conversely. That (formal) observation leads to the following definition:

Notation and Remark 5.3.4 (R-F-equivalence). If B is a matrix, then b and A := EBE-1 are called R-F-equivalent if WA = ERF(W ) ,

thus LJ& = E-1 (WA)

(5.3:3)

and at least one of the statements (which are equivalent by (5.3:3))

Ax = ERF I B(ER ' X))

for every x E WA

or

B

(A (E,,, av) = ERF

a E wB

U

is satisfied. If B and A = EBE-1 are R-F-equivalent, we obviously get CA = ERF(CSn)

with

limA x = B- Z (xk - xk_ 1)

for each sequence x = (xk) E CA (x_1 := 0) and CS5 = ERF(CA)

for every series E a E csB.

with B-

a = limA L1

Numerical solution of systems of linear equations

231

Remarks 5.3.5. (a) If A and B are row-finite matrices such that A = EBE-1, then B = E-'AE, and the methods b and A are R-Fequivalent.

(b) If B is a given matrix and B and A:= EBE-' are R-F-equivalent, then b is RR-regular if and only if A is regular (as an FF-method). Proof. (a) Since all matrices under consideration are row-finite, we can use the associative property (cf. 2.2.6) to get

E-'AE = E-'(EBE-')E = {E-'E}B{E-'E) = B. The R-F-equivalence of B and A follows also from the associative property (cf. 2.2.5) because

4-_

x E WA

(EBE-' )x exists E (B (E-'x)) exists

x E ERF {wB)

and

Ax = E (B (E-'x)) = ERF (B(ERFx))

(x E WA).

(b) If B is a matrix such that B and A := EBE-' are R-F-equivalent, then

B is RR-regular

cs C 6s g and B- E a _ 4==.

as,.

for all F a, E cs

ERF(C) C ERF(CA)

and limA(

= v=0

am

for all

E cs

v

C C CA and limA x= lira x for all x E C A is regular. This completes the proofs of the statements in the remarks. The following examples show that in 5.3.5(a) the condition that A and B be row-finite is not necessary for R-F-equivalence.

Example 5.3.6.

If a E C and Ta is the Taylor matrix (cf. 3.5.2(d)), then 7 :_ E-' (TAE) is well-defined and T. and T,, are R-F-equivalent. The extensive proof is left to the reader (cf. Exercise 5.3.23 where the proof is divided into four parts). L1

In Example 5.3.8 below, we determine, for each a > -lam the RRmethod j,,' arising from the Cesaro matrix Ca.. The methods CQ and Ca are R-F-equivalent. First, we give some notation and remarks.

232

Application of matrix methods

Notation and Remarks 5.3.7. For every a E lR let Ea = (snk) be the lower triangular matrix defined by

jn-nk -k-1)

8nk

(O
As we will prove, the statements

E° = I, El = E, E' = .E (r E N)

(5.3:4)

r times and

= EaE'3 = EpEa

Ea+.S

for all a, J3 E R

(5.3:5)

hold. Furthermore, using the definition (cf. 3.1.7 and also 3.1.6) we get

Ca = diag ( fn+a)) E 1n

(a E 1R and -a O Nl).

(5.3:6)

JJJ

We now verify (5.3:4) by induction. The cases r = 0 and r = 1 are obvious.

If r E N and E' = .E is true, then r times

[Er

rn J' k =

31A-L Sno (r) (1)

µ=k

n-k

= E'E = E1+1 since r+1 times

l(n-.u +r-11 E n-JA p=k

(v+r-1 - n-k+rl)

n -k ) (0
V

V=O

= snk i>

[c£ 3.1.5(d)]

Next, we prove that (5.3:5) holds for all a,,8 E R. Because Ea+p = ER+a it is sufficient to show Ea+Q = EaE'. However, that identity follows from n

sna) s(8)

[EaER]nk =

µ

µk

µ=k

(n-p+a-11 jµ-k+J3-1) L n-µ J l µ-k n -k+a+J3-1 n-k (-+O) snk

J

[cf. 3.1.5(e)]

(0
Consequently, (5.3:5) holds. Statement (5.3:6) is clear.

A

Numerical solution of systems of linear equations

233

Example 5.3.8 (Cesaro methods). Let a E R with -a V N be given. Then by 5.3.7 the RR-form Ca of the Cesaro matrix Ca = (c;,k)) (cf. 5.3.5) satisfies

Ca = E-1CaE _ E-1 drag ( (n+a) ) Za+l n

Consequently

[Co]nk = 0

(k > n; k, n E N°) and [Ca]nn

(nEN°).

Moreover, if n > 1 and k < n (k, n E N°), then [C

o]nk =

-

(n-k+a) n-k ( n+a) n

-

(nni-k-ka t n-1-k ) n-1+a n-1 ) (n-1-k+a) l n-1-k (

_

(n-1+a n-1

(n-1-k+a) n-1-k

(n- 1+1 n-1

nnk a n-k

(

n a

-

1

n

n(n - k + a) - (n + a) (n - k) (n + a)(n - k)

(n-n1-k+")

(a)

n ink n n1 As further examples of RR-methods we introduce the generalized Euler methods which induce numerical methods for the solution of systems of (n+a)

linear equations. The motivation for the applications of Euler methods is that the basic idea of those numerical methods are-as in the case of analytic continuation-closely connected with the (compact) summability of the geometric series.

Definition 5.3.9 (generalized Euler methods in RR-form).

Let K1 := {z E C E IzI < 1} be the closed unit disc in C, U be a neighbour-

hood of K1 and assume that p : U -+ C has the following properties: (Vi) p is holomorphic and injective on U.

(V2) p(O) = 0 and p(l) = 1. (V3) p(K1) is star-like (cf. 5.2.5). We consider the matrix P = (pik) defined by the coefficients of the Taylor expansion pk(t) _

Pikts

(kEN° and Iti<1)

i of pk about z = 0. Then P is called the generalized Euler matrix (for p), and the corresponding RR-method P is called the generalized Euler

method (for p).

234

Application of matrix methods

Note the difference from the definition of Sonnenschein matrices. Furthermore, we remark that in the definition of P we made use only of the assumption `p is holomorphic at zero'; we need the other assumptions in the proofs of the main results later in this section.

Properties 5.3.10 (of generalized Euler methods). Let p be given as in 5.3.9 and let P = (pjk) be the generalized Euler matrix for p.

(a) P is a lower triangular matrix because p(O) = 0. Furthermore, P is a triangle if p has a simple zero at zero. It is column-finite if p is a polynomial.

(b) Recursion formula for Pjk (j E N, 1 < k < j) : If the Taylor expansion of p, thus (pjl)j, is known, then we can recursively

determine pjk (j E N, 1 < k < j) by the formula j-k+1

(j,k E N, 1 < k < j).

Pjk = T, Pv1Pj-v,k-1 v=1

More precisely, for given j, k E N with j > k _> 2 we can determine Pjk using the known values P11, P21.... , Pj-k+1,1

Proof. (a) On account of the uniqueness of the representation of power series we get (noting pot = 0 since p(O) = 0 )

pk(t) = (p(t))k =

(PitP)

k

(k E N and t< 1).

(5.3:7)

Consequently, using the Cauchy product of power series, pvk = 0 for 0 < v < k. Therefore, P is a lower triangular matrix. Furthermore, by (5.3:7) it is obviously a triangle if and only if p11 54 0; that is, if and only if p has a simple zero at zero. In addition, we see immediately that P is columnfinite if and only if pk is a polynomial which is equivalent to the property that p is a polynomial by (5.3:7). (b) To prove the stated recursion formula we remark that for k E N and Itl < 1 we have Pk(t)

=

Pk-I(t)P(t)

(P.k_1t) (1a) n

Ej v=0

pj-",k-Itj-,Pvlt

which implies pjk = F_j -0Pv1Pj-v,k-1 (j E N°). Noting that P is a lower 0 triangular matrix, the proof is complete.

With the next example we show that P is rightly called a generalized Euler method.

Numerical solution of systems of linear equations

Example 5.3.11. Let 0 < a < 1 and pa(t)

at

(t

235 1

Then pa satisfies the assumptions (VI), (V2) and (V3) in 5.3.9, and the generalized Euler matrix Ea for p (cf. 5.3.9) obviously satisfies £a = tTa

(0 < a < 1) where Ta is the Taylor matrix of order a (cf. 3.5.2(d)).

Furthermore, as one may easily verify, £a is exactly the RR-form of the Euler-Knopp method E. (cf. 3.4.8), that is

£a = E-IEaE

(0 < a < 1).

Therefore, if 0 < a < 1, then £a and Ea are R-F-equivalent and £a is A

RR-regular (cf. 5.3.5).

As we already mentioned, the (compact) summability of power series by matrices, in particular that of the geometric series, plays an essential role as it does for analytic continuation. First, we make the following definition.

Definition 5.3.12. Let B = (bik) be a matrix and J:k akzk be a power series. If z E C and

akzk E w6, then we put

uB(z)

bjkakzk k

(Therefore we have

B- akzk =

vB(z)

,

i

if E akzk E s6.)

By analogy with 5.2.2 we now define:

Definition 5.3.13 (compact summability of power series). Let B be a matrix, Ek akzk be a power series, vr(z) be defined as in 5.3.12, G C C, and let F : G -> C be a function. Then >k akzk is said to be compactly summable on G to the value F(z) by b if F akzk E 6s F3 for each z E G and Ei vB(z) = F(z) uniformly on each compact subset of G. (Notation: B-F_ akzk F(z).) In the case of the geometric series and the Euler methods we can reduce

the compact summability by (a to the corresponding statement for Ea.

Example 5.3.14 (Euler method). For every a E ]0, 1] the statement Ea

-

zk ca =

-z 1

where Ga := { w f w-C1 111

aA

<

a

holds. This is an immediate consequence of 5.2.13 in connection with the R-F-equivalence of 9,-,, and E. A Next we roughly sketch the starting point leading to iteration methods for the solution of linear systems of equations.

236

Application of matrix methods

Question 5.3.15. Let n E N be fixed and Mn the set of all n x n-matrices with coefficients in K Now, if A E Mn and b E K" are given, then we ask for a solution of the linear system of equations

Ax=b, that is for a y E Kn satisfying Ay = b. Since we would like to use iteration methods for the determination of the solution y, we consider only uniquely solvable systems. We know from linear algebra the equivalence of the following statements: (i) For each b E ' there exists a unique solution of Ax = b. (ii) A is invertible.

(iii) det A ¢ 0. Thus we assume det A # 0, and let further matrices M, N E Mn with

A= M- N and det M# 0 be given. As one may easily verify, for each fixed b E K and for every y E K' the following statements are equivalent: (a) y is a solution of Ax = b. ((3) y is a solution of x = Tx +b, where T := M-1 N and b := M-1b. The system of equalities

x = Tx + b is called a fixed point form of Ax = b. Banach's fixed point theorem leads to the iteration

x(0) E K" (arbitrarily given) and x("+1) := Tx(") + b

(v E N°) (5.3:8)

for the determination of the solution where we assume that the iteration sequence (x(")) converges. Now, if

a(T) :_ {A E C I A is an eigenvalue of T} and

p(T) := max{IAI I A E a(T)}

denote the spectrum of T and the spectral radius of T, respectively, then in numerical analysis it is a well-known fact that the iteration sequence (x(")) in (5.3:8) converges (coordinatewise) for each initial value x{0} E ]K" if and only if p(T) < 1. More precisely, in such a case the iteration sequence

converges to the solution y of Ax = b and we get for any initial value x(0) := b the statement

x("+1) = Tx(") + b

Numerical solution of systems of linear equations

237

_

+1

ET =0

k

b -' (I - T)'1 b = y (v -+ oo),

(5.3:9)

k*

while on the other hand Y+1

Tk

Tk -> k=0

= (I -T)-1

(v -) oo),

(5.3:10)

k

where the convergence in (5.3:9) and (5.3:10) is understood componentwise.

(Note that we obtain an equivalent statement if we replace in (5.3:10) coordinatewise convergence by convergence relative to the so-called spectral

norm `T -+

p(THT)' with TH :=t T. The same is true if we consider

a matrix norm 11 11 with ,ITII < 1 and in (5.3:9) convergence relative to a

norm on K" which corresponds to that matrix norm.) In particular, in the case p(T) < 1 we get an `approximate solution' of Ax = b, if we recursively determine the matrix Ek±o Tk where we start with the identity matrix I and if we put v+1 := (ET1c).

k=0

This observation leads in the case p(T) > 1 to the following problem.

Problem. Let T E M and I - T be invertible. We ask whether it is possible to transform the so-called Neumann series >k Tk by a certain matrix B (in RR-form) into a series

vf(T) with v41 (T) E M such that (I -T)-1 = EvB (T) where the convergence of the `matrix series' is componentwise.

A

Definition 5.3.16 (summation of Neumann series). Let T E M,,, and let B = (b3k) be an infinite matrix. Analogously to the summation of the geometric series (cf. 5.3.12, case ak := 1 (k E N° )), we define: if S E Mn, then Tk is said to be summable by b to S, if the `matrix series'

vB (T) := f bbk Tk for each j E No k

and

:_ E vB(T) converge (componentwise) in M and B-

Tk = S holds.

238

Application of matrix methods

Since the Neumann series in the case n = 1, where T E M1 = K., is the geometric series, the summability of the geometric series by a matrix B may give us information on the summability of the Neumann series by the corresponding method B. This is the content of the following theorem. Theorem 5.3.17 (summation of the Neumann series). Let G be a

simply connected domain (in C) with 0 E G and 1 0 G, and let B be an infinite matrix such that

B-zk =c 1-z* 1

If T E M and a(T) C G, then the Neumann series E Tk is summable by B to the matrix (I -T)-1. (Note that the assumptions contain 10 a(T) which implies that I - T is invertible.) Proof. We start with some preparatory considerations. First we make use of the assumption that B compactly sums the geometric series g(z) = 11 on G. We choose a compact domain A such that zZ

or(T) C A C G. Consequently we have B- 57

zk =

vB {z) _ (5.3:11) E bjkzk = 1 - z j j k uniformly on A. Now, for i E N we derive both sides of the last equality i-times to obtain

d

1

1

dz1-z

(1-z)2

and

d=

_

1

dzil - z

i!

(1-z)i+l

for the right hand side, and bjkzA =

dz i

bjk k(k - 1)

(k - i F 1)zk -i

j k j k=i for the left hand side. On account of the uniform convergence on A, both results coincide. Therefore

(1 - z)i+1 -

bjk t

j j=k

)zA;-i

\

(i E N°)

(5.3:12)

uniformly on A.

Next, let T E Mn and J = S-1TS be the Jordan normal form of T where S E Mn is some non-singular matrix. As is well known the Jordan normal form has the form / Tµ

0

1

0\

J2

with

Js J

Jµ =

E M,,,,

Numerical solution of systems of linear equations

239

where 7 1....

, r, are the eigenvalues of T and nm determined by the eigenvalue rµ, satisfies Eu=1 n,, = n. Later in the proof we apply, for example, the rational function

g:C\{1}- C, z--+(1-z)-1 to T. For that we have to know how to apply it to the Jordan blocks Jµ. So we have for the power Tk the equality J'k

Tk = SJkS-1

0

jk =

with

(5.3:13)

JakJ and

0 Jµk

= (rI + E)k where E =

(5.3:14) 1

I

0

Furthermore, if µ E K, then for the power E' we obviously have 0

Et

0

...

1

0

...

0

1

0

0

=

0

0 1

where 1 sits in the lth diagonal

.

0

0/

(l = 0, ... , nA - 1) and El = 0 (l > nu). For r := nu - 1 we get, by applying the binomial formula to (5.3:14), the identity

J'k =

min(r,k)

()r_1Ei.

(5.3:15)

t

1=0

Moreover, we have

(I-Jl)-1 (I -

T)-'

0

=S

S-1.

(5.3:16)

1

(I -

J,)_' J

By the assumptions of the theorem rµ # 1 (µ = 1, ... , s). Hence, for each A E N. we can convert the blocks in (5.3:16) in the following way:

(I-J,)-'

=

(I-r,2I-E)-1

240

Application of matrix methods

((1- Tr,)I - E)-'

=

E1 ' 1 - Tt,

(1 -TF,)-' ( I +

1

E2

-TV

r

+...+

+ (1 ETµ)2

(1

ET,, )r)

where r = nt, - 1 or, using another style, Ei

r

(5.3:17)

After these preparatory considerations we now prove the theorem, that is we establish

B-ET' = (I -T)-'.

If we use T = SJS-', then we can easily verify that it is sufficient to prove this in the case of the Jordan normal form J of T. In fact, by (5.3:13) and (5.3:16), it is sufficient to prove it for each Jordan block J. That is, it is sufficient to show

(pEN.).

B-EJ'`k=(I-J,.)-'

By (5.3:11) we have JokvB(JJ)

with (cf. (5.3:15))

vj(JJ)

=

E k min(r k) ,

E bjk E I Z 1=0

k

where r =

\ ll

1.

Interchanging the order of summation we get r 00

vj(Ju) _

bjk{,

)

rk-t Et

1=0 k=1

and 00

Evj(Jµ}

r

_

j=0

E

1=0

00

00

Ebjk{t}T-!

E1

j=0 k=1

rr

i

D (1 - T1,)1+1 E1 (I Thus, the theorem is proved.

[by (5.3:12)]

[by (5.3:17)].

0

Numerical solution of systems of linear equations

241

Corollary 5.3.18. Let x = Tx+b be a fixed point form of Ax = b where A E Mn with det A $ 0 and T as well as b are chosen as in 5.3.15. Now, if G, T and B satisfy the conditions in 5.3.17, then

XW

-3 y(v-4oo)

E vB(T) i-o

(componentwise) where y denotes the unique solution of Ax = b.

Proof. This is an immediate corollary of 5.3.17 (cf. 5.3.15 too).

Corollary 5.3.18 gives us the possibility of obtaining methods for the (approximate) solution of systems of equalities by the use of infinite matrices. In this connection the following questions result naturally from the considerations in 5.3.15-5.3.18.

Questions 5.3.19.

i

If a matrix B is given, then, if it exists, one asks for a (largest possible)

simply connected domain G with 0 E G and 1 G such that the geometric series is summable by b to the value 1 . (On the other hand, one may fix G and ask for a B which has that property.) If A E M,,, and therefore T E M,,, are given in accordance with 5.3.15 and 5.3.18, then a matrix B and a domain G with a(T) C G are sought. (In such a case, the sequence (x(')) in 5.3.18 converges to the solution of Ax = b,) Determine the speed of convergence of (x(")). Is it possible to choose B such that (x(")) can be recursively computed, or even such that (x(")) is an iteration sequence? Can B be chosen so as to minimize the computing expense? In 5.3.14 we answered the first question in the case of the Euler method Z (a E ]0, 1] ). In the same way it is proved in 5.3.8 and 5.2.14 (cf. 5.3.5(a) too) that the geometric series is compactly summable by the Cesaro method C,, on G := D1. Now, we answer, with hints for a proof, that question in the case of the generalized Euler methods.

Theorem 5.3.20 (summability by generalized Euler methods). Let p : U -+ C satisfy the conditions (V1)-(V3) in 5.3.9 and P be the generalized Euler matrix for p. If S2

\p(Ki)

then zk

E csp

for each z E Q. We use plane.

with

p(t) := p(t) (t E K1),

and P - r zk =

1

z

C U too} to denote the extended complex

242

Application of matrix methods

Proof. We refer the reader to Niethammer's paper [179]. With the help of Figure 5.3.1 we make do with an illustration of the domain St in the particular case p(t) i }e (a E ]0,1]) and t 34 1-a ), that is the case of regular Euler methods £a. Note that 1 is a bounded domain since

Fig. 5.3.1

p(O) = 0 and that z = 1 sits on the boundary of f because p(l) = 1. That the figure is also typical of the general case is because of the condition (VI). The condition (V3), that is p(KI) is star-like, enters essentially in the proof 0 of 5.3.20.

Theorem 5.3.20 gives us a sufficient condition for the (pointwise) summability of the geometric series by generalized Euler methods for a given function p. If we would like to have a corresponding condition for compact summability we have to strengthen condition (V3).

Theorem 5.3.21 (compact summability). Let r > 1 be given and p : Dr --4 C be a function satisfying (Vj) p is holomorphic and injective on Dr, (V2)

p(0)=0 and p(l) = 1,

(Vs) p(Dr) is star-like, and let P be the generalized Euler matrix for p. Then

P-rk

1 c

1-z'

where 1 = 1 \ t7(Dr) (cf. 5.3.20). Proof. Again we refer the reader to Niethammer's paper [179]. We content ourselves with the comment that for each compact K of f2 there exists an F with 1 < F < r and

K C 1lr :_ C \ P(Kr) with K;. :_ {z E C I IzI < F} . With the help of Figure 5.3 we illustrate this situation in the case of regular

Numerical solution of systems of linear equations

243

Fig. 5.3.2: f2 := U \ p`(Ki )

Euler methods £a where we take r := 1'a We do not deal with the other questions in 5.3.19 since they are of a purely numerical nature. We refer to Niethammer's paper [179] and other papers connected, for example, with the names M. Eiermann, W. Niethammer, W. Schempp and R. S. Varga.

Exercise 5.3.22. Let A be row-finite and FF-regular and B := E-IAE. Prove that b is RR-regular.

Exercise 5.3.23. Let T,, = (tnk) be the Taylor matrix of order a (cf. 3.5.2(d)) and TQ = (tn+l.k+1) be a modified Taylor matrix. Show that Ta satisfies, for each 0 < a < 1, the following statements:

(a) Vx=(xk)Ew : xEwy- 4 VnENo : xk = ID (k__ I0

-7

Hint: Prove first that lime (n)k-" = nt (n E N°) using (3.1:2) in 3.1.9.

(b) V x E w : xEwf-

ExEw

is well-defined and TQ = ca-T.. (c) The matrix Ta (d) T. and T. are R-F-equivalent.

Exercise 5.3.24. Let U be an arbitrarily given neighbourhood of K1 and

p : U --+ C be a function which satisfies (V1)-(V3) in 5.3.9, and let 0 (j E N°) be satisfied by the coefficients of the Taylor series Pjl

p(t) _ Ej pj1tj (Iti < 1). Prove that the generalized Euler method P is RR-regular. P t p(t) for Itl < 1 and show that bk. Hint: First, make use of pxk (k, n E N°) satisfies the inequality bk+t," <_ bk" (k, n E N°). Then prove E,[Pa]. = F,µ a, (a = (a,) E cs) noting 2.1.14. Bibliography: [179]; [181], [75], [182], [74], [183]

244

5.4

Application of matrix methods

Fourier effectiveness of matrix methods

We emphasize that in the present section-in contrast to Section 5.3-we use matrices exclusively in FF-Form. In particular, in the case of a series we apply the matrices to the sequence of its partial sums (and not to the sequence of its terms). Moreover, for the present section we make the

general assumption that which means, in particular, that we deal only with real sequences and series and with matrices with real entries. The starting points and motivation for the present section are two important theorems in calculus. The first one tells us, that the Fourier series of 21r-periodic continuous functions are not necessarily pointwise convergent; the second one is the well-known theorem of Fejer which contains the very strong statement that the C1-transform of the sequence of the partial sums of the Fourier series of any 27r-periodic continuous (real) function f converges uniformly to f. (We will deduce both theorems as corollaries of results which we prove below.)

Proceeding from Fejer's theorem, it is natural to replace C1 by other matrix methods and to call a matrix method A Fourier-effective if for every 27r-periodic continuous (real) function the A-transform of the sequence of partial sums of its Fourier series converges pointwise to the function. To begin, we introduce some notation and prove a simple theorem that shows that a uniformly convergent trigonometric series is the Fourier series of its sum function.

Notation 5.4.1 (Fourier series). In the sequel we denote by

C2, :_ If : R -3 iR

I f is 27r-periodic and continuous}

the set of all continuous, 27r-periodic functions. Furthermore, if a function f : JR -} JR is 27r-periodic and f of Riemann, we put

is integrable in the sense

ak(f) := ak := 7r1 f f(t) cosktdt

(k E N°)

a

and

bk(f) := bk

1f 7r 1

(k E N°).

f (t) sin kt dt R

(Note, if k = 0, then bk = 0.) We call the (formal) series 00

00

+ E (ak cos kt + bk sin kt) k=1

Ak (t)

(5.4:1)

k=0

the Fourier series of f with Fourier coefficients ak and bk (k E Nl° ). For examples of Fourier series see 5.4.17.

Fourier effectiveness of matrix methods 245

The question naturally, arises whether the Fourier series of a function f is convergent or better uniformly convergent, and in such a case whether it converges to the `right' value, namely to the value of f. Concerning this, we show first that a uniformly convergent series of the type (5.4:1) represents a 2r-periodic continuous function f and that this series is exactly the Fourier series of f.

Theorem 5.4.2. Let 2 + E00 1(ck cos kt + dk sin kt) =: f (t) be uniformly convergent for t E lit Then f E C2, , ak (f) = ck and bk (f) = dk (kENO). Proof. It is well known from calculus that a limit of a uniformly convergent series of 27r-periodic continuous functions is also 27r-periodic. Furthermore, for each n E No we get

an(f)

f"

1

=

7r

f (t) cosntdt

n n

CO

2ir

f,r cosntdt 00

+ E1

k_1

=

A

f(ck cos kt cos nt + dk sin kt cos nt) dt n

Cn

on account of the uniform convergence and since 1 7r

f

1

'

fir

sin kt cos nt dt = 0

-7r

if k

(k E N` },

cos kt cos nt dt = to1 if k Onn > 0

(k E IY) .

Since in addition

f- sin ktsinntdt = rc

1

iifkk

54

to

(kEIY)

we obtain analogously bk(f) = dk (k E No ).

The mapping which maps f E C2,r into the n1h partial sum of its Fourier series may be represented as an integral operator as we now verify.

Theorem 5.4.3 (Dirichlet kernel/integral formula). For each f E C2,, and n E N° the equality (cf. 5.4.1)

Ak (t) = 1

sn (f,'t) k=0

holds where

J

n

f (u) D. (t - u) du

246

Application of matrix methods

sin2sin

Dn(v)

1

)v

v

(v E R, 2 # kir (k E Z))

2

is called the Dirichlet kernel. (Note, Dn may be continuously (and differentiably) continued to the points 2 = k7r (k E Z) and is well-defined onIR in that sense.) Proof. Applying the definition of the Fourier series and its coefficients we obtain for all f E C2,r, t E JR and n E N (the case n = 0 is trivial) 7r

sn(f, t)

f (u) du

J

27r

+ k=1 E 7r- fn

f ddf

x

1

f (u)

n

- f f (u)

=

Tr

f (u) (cos ku cos kt + sin ku sin kt) du (,r

`

(

2

sin (2

+ E cos k(t - u))

du

k=1

(t - u)

sin)tu

du,

2

where we make use of the identity 2 sin t

u 2

21

+

E cos k(t - u) k=1

/

= sin n+ 21)(t-U).

Thus the theorem is proved. The following remarks on the Dirichlet kernel will be useful in the sequel.

Remarks 5.4.4. (a) Dn E C2,r for all n E N°. (b) JDn(v)I < n+1 < n + 1 for every n E N° and v E R. Proof. In the proof of 5.4.3 we represented the Dirichlet kernel by

Dn(v) _

+ Ecoskv (nENO, vE

0

[note E := 01. k=1

k=1

Hence, we get Dn E C2, (n E N°) and

IDn(v)I < 2 + T, I cos kvj

2+n

(n E N°, v E J).

k=1

This proves the remark. In a generalization of Fejer's theorem, which we will state in 5.4.15(c), we now apply suitable matrix methods A on the sequence (sn(f, t)) and give a necessary and sufficient condition in terms of the matrix A that the transform of (sn(f, t)) under A converges uniformly to f.

Fourier effectiveness of matrix methods

247

Theorem 5.4.5. Let f E C2,,, s(f,t) := (sn(f,t))fENo, t E R, and let

A = (ank) with E(k + 1)IankI < oo (n E N°)

(5.4:2)

k

be given. Then s(f, t) E WA and

Qn (f, t) := [As(f, t)]n =

! Jf

f (u) Kn (t - u) du a

where

(v E R, n E Nl°)

ankDk(v)

Kn(v)

k

denotes the ,nkh kernel corresponding to A. Remark 5.4.6. Because of 5.4.4(b) and (5.4:2), for each n E N° the kernel

Kn is a 27r-periodic continuous function, that is Kn E C2,, as a limit of a uniformly convergent series of 21r-periodic continuous functions (cf. 5.4.4(a)). 1L Proof of 5.4.5. First we show s(f, t) E WA for all f E C2, and t E R and

that ank Dk (t - u)

converges uniformly for u E R.

(5.4:3)

k

For all n, k° E No we have ko

ko

Ianksk(f,t)I

lankI 1

<

k=0

7r

k=0 IIfIIoo

<-

n

<

A

If(u)Dk(t-u)Idu

[cf.5.4.3]

ir

ko

IankI27r(k+1)

[cf.5.4.4(b)]

k=0

E(k + 1)Iankl < oo

2IIfII

[cf. (5.4:2)],

k

that is s(f, t) E WA. Furthermore, by 5.4.4(b) we obtain

ElankDk(t-u)I < >lankl(k+l) k

(uER, nEN°).

k

Now, the absolute convergence of the series on the right hand side of the last inequality implies for every n E No the desired uniform convergence of the series 1: ankDk (t - u) in u E a k

The proof of the main statement of Theorem 5.4.5 is now easy. For all f E C271, t E JR and n E N° the equalities 6n (f: t)

_

ank sk (f, t) k

248

Application of matrix methods

1 E ank f fr (u) Dk{t - u) du ir

=

[cf. 5.4.3]

k

n

f f(u) E ank Dk(t - u) du n

[cf. (5.4:3)]

k

0

hold.

Next, we consider some examples.

Examples 5.4.7 (for kernels). (a) If A := I, then Kn = Dn (n E N°) is the Dirichlet kernel.

(b) If A := C1, then, for v E R with 2 # kit (k E Z), we obtain

i

Kcl (v) n

n

_ n+1 E Dk k=o

/ (v)

=

2

(sin

n+1

)

2

v12

(n E NO )

2sin 2

by the equalities 1

nn`2sin(k+2)vsin2

n+1a

(2 sin

2)

2

1

1 - cos(n + 1)v

n+1

(2 sin 2) 2

2

sine )2

(sin

n+1

2sin2

if we note

sin a sin )3 = 2 (cos(a - (3) - cos(a + 13))

and 2 sin2 a = 1 - cos 2a.

The kernel Kn 1 is called the nth Fejer kernel (n E NO ).

(c) In the case of the (discrete) Borel matrix Bi = {e -n ,) , for v E R with 2 54 kit (k E Z), the kernels are given by

Ki (y) = B l'

en{cos v-i)

(v2 + n sin v) 2 sin 2

(n E NO),

(5.4:4)

and KBi is called the nth Borel kernel (n E N° ). To prove (5.4:4) we note first that (5.4:2) is satisfied because

e-n E(k + 1)i = 1+n k

Consequently, the nth kernel of Bi can be calculated using 5.4.5 as follows: k

KBl (v)

[BfInk Dk(v) = -n

_ k

k Dk(v) k

Fourier effectiveness of matrix methods 249

n'k

e-n

2 sin Z

=

e -n

k

sinv

1

2i

2

k!

sin {kv +

v 2

)

7t' (ei(kv+$)

E k. k

l

- e-i(kv+z))

_R

2 sin 2 2i en(cos v-1)

1

2 sin 2

2i

e n(cos v-1)

2 sin 2

/ei1+in sin v

ex2-in sin v)

1

v

sin

+ n sin 'v)

We have made repeated use of eia = cos a + i sin a and a-ia = cos a -

isin a (aER).

A

In the following definition we emphasize those matrices A for which (6n (f, t)) converges pointwise to f for each f E C2,,.

Definition 5.4.8 (Fc-effective). If (5.4:2) is satisfied for a given A,

n- f (t) for each f E C2,r and t E Ill;.

then A is called Fourier-effective (for functions in C2,'), or Fceffective for short, if 6n (f, t)

The following preliminary considerations give us a necessary condition for Fc-effectiveness which we will later refine in some sense. Instead of deducing it with classical methods we will make use of the uniform boundedness principle which is well known in functional analysis. In this way we save a technically extensive construction which is inessential in connection with Fc-effectiveness.

Preliminary considerations 5.4.9. As usual,

II II.: C2, -* R, f -+ IIfII. :=

sup

tE[-Tr,7r]

If(t)l

denotes the supremum norm on C2,, and we assume that A is Fc-effective and satisfies (5.4:2). Then, for given n E N° and t E R the map (Czn

An

,

II III) -+ (p , 11) f _* 6n (f, t) ,

is linear (cf. 5.4.5) and continuous because an (f,t) I

=

1A

f(u) EankDk(t-u)du Ir

k

250

Application of matrix methods

<

2 E(k + 1)Iankl IIf1Ioo [cf. 5.4.4(b) and (5.4:2)]. k

Since (C2,,, 11 11.) and (R, 11) are Banach spaces, by the uniform boundedness principle (cf. 6.3.35) the statement

(an(f,t))n E c C m

[A is Fc-effective],

that is the pointwise boundedness of (An' )n, implies the uniform boundedness of (An)n, that is IIAnII

sup {Ion(f,t)I

I

IlfIk <- 1, f E C24 < M < 00

(5.4:5)

is satisfied for some M > 0. On account of the representation of An as an integral operator in 5.4.5, the determination of the operator norm IIAnII is easy, and since the kernel Kn is 27r-periodic by 5.4.6, it is not surprising that IIAnII is independent of t as we will show in the next theorem.

Theorem and Definition 5.4.10 (Lebesgue constants). If A satisfies (5.4:2) and if Kn (n E

and An (n E No , t E R) are defined as in 5.4.5 and 5.4.9, respectively, then the following statements hold: (a) For each t E R and n E N° the identity IIAnII =

f

ir

IKA(u)I du =: Ln

holds. In particular, IIAnII is independent of t. The number LA (n E N° )

is called a Lebesgue constant for A. (b) If A is Fc-effective, then supra LA < 00. (c) If supra LA = oc, then there exist an f E C2, and a t E R such that n (an (f, t))n diverges.

Proof. Statement (b) is an immediaie corollary of (a) and (5.4:5). Furthermore, (c) is the negation of (b). We now prove (a). Obviously, for each f E C2n with Ilfll0 < 1 (cf. 5.4.5 and 5.4.6) we have IT

IAn(f)I = Ion(f,t)I < i f a IKn(t-u)Idu = i fIK(u)Idu. Ir

i

Therefore,

IIAnII <

1 f 1Kn(u)I du

for each n E N0.

To prove '>', we choose e > 0, and construct, for given n E No and t E R, an f E C2,r with II f II00 < 1 and

Fourier effectiveness of matrix methods

I0,nn(f,t)I

IAn(f)I =

251

f IKn(t-u)Idu-E'. ,r

For that we choose a

(k + 1)lankl <

with

K E No

k=K+1

and put _ kA

_ _ and f {u} := sgn Kn {t - u}

K

:=

ankDk

for each u E

k=0

The function f is 27r-periodic since Kn is, but it is not necessarily continuous. However, since kA n, and thus f, changes sign in each interval of length 27r at most finitely often, for f we can choose an with I1 f 11. < 1

f E C2,r

and

II f - f II- < 8(1 + M)'

where M :_ Ek(k + 1)jankl. Hence, we obtain IAn(f)I

=

Ion (f,t)I =

f

f(u)k (t-u)du 7r

-J

f(u)(k,a(t-u) - Kn(t-u))du

- f(J(u) - f(u))Kn(t-u)du jn _ I Kn (t - u) I du - 4 - 4

J

> >

n

J

IKn (t - u) I du p f

>

[note 5.4.4(b)]

n n

, Kn(t-u) - Kn (t-u)Idu I

2

1 7r

Thus, the chosen f has the desired properties.

0

The determination of the Lebesgue constants is especially easy if the row sums of the matrix under consideration are equal to one and if the kernels are non-negative.

252

Application of matrix methods

Corollary 5.4.11. If A satisfies (5.4:2) and E. ank = 1 (n E NO), and if

Kn(v)>0 (nEN°,vER), then L; =1 (nEN°).

Proof. By 5.4.10 and the definition of Kn we obtain for all n E NO, t E R and the constant function f = 1 the statement

=

LA

ff(u)K(t_u)du

IKn (u) I du =

1

X

E ank sk (f, t) _ E ank

= 1,

k

k

since sk (f, t) = 1 for all k E No and t E R. Applying 5.4.10 we now show that there exists a 21r-periodic continuous

function whose Fourier series does not converge to the `right' value-that is, to the value of that function at least at one point. Moreover, we may choose the function such that its Fourier series diverges.

Theorem 5.4.12 (divergence of Fourier series). If A := I, that is K, = Dn (n E N°), then 42

L. := L,l, =

log n + 17(1).

(5.4:6)

In particular, there exists a 27r-periodic continuous function whose Fourier series diverges at least at one point.

Proof. The additional statement is an obvious consequence of Ln oa (n -+ oo) and 5.4.10(c). To prove (5.4:6) we use, for any u E ]0, ir], the identity Dn(u)

=

2 cot 2 sin nu + 2 cos nu

2 cot 2 - 1

1

sin nu + u sin nu + 2 cos nu

and the boundedness of both the first and the last summand in the last term which one may check easily. Thus, we obtain, for all n > 1, the identities n

Ln

=

1it f ,. IDn(u)I du = 2 f" Isin nul

rO L 0

2

n-.

U

2 7r

f

IDn(u)I du

du + 27(1)

(k+i) n I sin nul

u

du + 17(1)

Fourier effectiveness of matrix methods 2

_

n-

sin nu f° u+n du + i)(1) k=O n-1

2 7r

sin nu E

fo

k=1 R

2ir (n

=

253

1

2

it f

k,,

du +

ka

du+i?(1)

2b + n

n

o

sin nu u

du + O(1)

n-1

sin nu

1

o

k=1

u+n

since 0 < sinU nu <

and consequently

n

0<

2 7<

o

!sinnudu<2nIr =2. n U 7r

Further, by Euler's summation formula (cf. [128, p. 540]) the inequalities

7r k+1

C

- ka u+ k,r n n

(1
irk

imply n

n (log n + iD(1)) it

7r

n-1 1

k+ 1

k=1

n-1

1

k-1 u+

kn

n-1

< n E k= n (log n+ f7(1)). k=1

Therefore, for 0 < u < n we have n-1 1

k=1

(log n + 0(1)).

u +

Hence, we obtain for all n > 1, the identity Ln

= i2r

n(log nA

fdu + 0(1)

which is what we had to prove.

0 < oo'

In the proof of the implication `A FC-effective supn LA n of the (cf. 5.4.10) we have made use only of the boundedness, but not

convergence, of (an (f, t)) n. Therefore, we cannot expect the converse implication to be true. Now, we are going to prove that under an additional assumption, which is satisfied in the case of C,,, (ci > 0), the converse

is true even if (an (f, t}) converges uniformly for t E R as n -+ oo. The following lemma prepares us for the proof.

254

Application of matrix methods

Lemma 5.4.13. Let A be a matrix satisfying (5.4:2) and >k ank = 1 (n E NO). Then

17r J

Kn(u) du = 1

(n E N°)

(5.4:7)

R

and, for every f E C2, and t E R, the statement

f

6n (f, t) - f (t) = - + Jn

Sot(u) Kn (u) du =

j J

cpt(u) Kn (u) du

nn'

o (5.4:8)

(f (t + u) + f (t - u) - 2f (t)) (t, u E R).

holds, where cpt (u) :=

z

Proof. The proof of (5.4:7) is contained in that of 5.4.11. To prove (5.4:8) we observe for f E C2,r and t E R: (i) Both Dk and Kn are even and 27r-periodic. Therefore, n

fg7r

n

(ii) 'pt is an even function. So, the first equality in (5.4:8) follows from (i) and the second one holds, 0 since apt, K.n and `u --+ Pt{u}Kn (u) ' are even.

Theorem 5.4.14 (Fc-effectiveness). Let the matrix A satisfy (5.4:2), Yk ank = 1 (n E NO), sup,, Ln =: 1M < co and

[ : p(S) :=

V S E ]0,

L

I K(u) I

0

(n

).

Then

V f E C2,,

:

o'n (f, t) n-* f (t) uniformly for t E R,

(5.4:9)

which implies the FC -effectiveness of A.

Proof. Let f E C2, and s > 0 be given. Since f is uniformly continuous, we may choose a S > 0 so that

Ivt(u)I S

If(t+u)- f(t)I + 2If(t-u)-f(t)I < 2(1+M) (t E [-7r, 7r] and Jul < 5).

For this S we choose an no E No with len(s) < for all t E [-7r, 7r] and n > no we obtain

4

(n _> no). Thus

I6n (f, t) - f (t) I

f

I
[cf. (5.4:8) in 5.4.13]

Fourier effectiveness of matrix methods e

2J

Iwt(u)I IKn(u)I du +

2

J

255

1cot(u)I I K' (u)I du

A

<

2(1

+ M) L" + 211f hoop. (5) (L = ; fo I Kn (u) I du, since K,a is even] n

2+

2

=e

[on account of the choice of 3 and no].

Hence, (5.4:9) is proved for t E [-zr, rr], and thus for every t E R since un (f, -) and f are 27r-periodic. As a corollary of Theorem 5.4.14 we get the Fc-effectiveness of the Cesaro methods of positive order. Thus, statement (c) is the well-known

theorem of Fejer. Theorem 5.4.15 (PC-effectiveness of C.). For every a E R with a > 0 the following statements hold: (a) Ln' = 1 (n E N° , a > 1) and supra Ln' < oo (a > 0). (b) Ca is Fc-effective. f (t) uniformly for t E Il8. (c) b f E C2,r : 6n° (f, t)

n°

Proof. (a) First, let a > 1. Then Kn°(v) > 0 (n E NO and v E R) as we state in Exercise 5.4.18, and thus Ln° = 1 (n E N° ), since Ca clearly satisfies the other assumptions in 5.4.11. Now, let a > 0. In that case we apply the following inequalities which are proved in Zygmund's book [268, p. 94] and hold for n > 1 and v E

-

1a nj :

IKn° M1 < n + 1 < 2n, Man-av-(a+i) IKn°(v)I < for some M. > 0. For n > 1 we obtain, by (5.4:10), that J o

Kc°(v)I dv < 2n

ry

! n

=2

and, by (5.4:11), that J1,'

IKn°(v)I dv

< Man-a =

Man- a

r 7r

(nQ a

v-(a+') dv

- nla) a

Ma < a

Therefore,

sup Ln ° = n

sup' n

7r

I

IKn ° (v) I dv < oo

[c£ 5.4.101.

o

(b) The Fc-effectiveness of Ca (a > 0) is contained in (c).

(5.4:10) (5.4:11)

256

Application of matrix methods

(c) Because Ca (a > 0), as a row-finite matrix, obviously satisfies (5.4:2)

and since >k cnk = 1 (n E N°) and by (a) also sup. Ln° < oo, statement (c) follows from 5.4.14 because, by (5.4:11), V S E ]0, wr]

: 0 < µn(S) _

J

I Kn ° (u) I du

6

< 2(ir - S) ir

M.

n-aS-(a+1)

(n.) }0

o

The Lebesgue constants for Ca (a > 0) were examined in detail by H. Cramer (1919) and he proved the following statements which exceed 5.4.15(a) and which we give without proof.

Remark 5.4.16. If a > 0, then L(a) := limn Lna exists and L(a) = 1 for every a > 1. Also L(a) -> 1 (a -+ 1-) and L(a) -4 oo (a -4 0+) hold.

In the literature the Lebesgue constants of many methods have been determined: for example, those for the Borel method by Larch [146], those

for the Hausdorff methods by Lorch and Newman [147], and those for Sonnenschein matrices by Sledd [219].

Exercise 5.4.17. Determine the Fourier series of the 21r-periodic continuation of the following functions:

(a) f : [-ir, ir[ --+ R, t --* 2. (b) g : [- r, 7r[

R, t -- sgn t.

Exercise 5.4.18. Verify that for each a E IR with a > 1 the Ca-kernels are non-negative, that is KCQ (v) > 0 (n E NO , v E R). Bibliography: [128], [268], [257], [203], [108], [190]

5.5 Notes on Chapter 5 We have already mentioned that there are applications of summability to number theory. The most famous one is the proof of the prime number theorem which can, for example, be deduced from Wiener's Tauberian theorem.

We refer to [29], [202] and [244]. On the other hand, number theoretical properties of countably infinite sets {pk E N I k E No } of natural numbers give rise to the study of the summability behaviour of matrices defined by the sequence (pk), such as the Riesz and Norlund matrices. As a simple example, in the case that p = (pk) is the sequence of all prime numbers (in their natural ordering), V. K. Srinivasan stated in [225], by using results on the distribution of the prime numbers on the real line, that the Riesz matrix Rp is equivalent to the Cesaro matrix C1 and the Norlund method is stronger than C1. As applications of summability methods in function theory we dealt in Section 5.1 with the boundary behaviour of power series and in Section 5.2

Notes on Chapter 5 257

with the analytic continuation of holomorphic functions by means of matrix

methods. We applied suitable matrix methods to the sequence of partial sums of the Taylor series. Now, it is obvious to proceed analogously in the case of other kinds of expansions of functions into series which give rise to applications of summability methods in the branch of special functions. For an overview of the extensive literature use, for example, MathSciNet with `Classification: (40 AND 33)'. Concerning applications in differential equations we refer to [29] and W. Balser's book [11] and recommend the application of MathSciNet with `Classification: (40 AND 34)'.

In the context of the application of summability methods to Fourier series, which we dealt with in Section 5.4, we should mention that special summability methods play an important role in the branches of integral transforms and Fourier analysis. We refer, for example, to [29], [216] (applications of the Borel method), [206], [268], [190] and [108], or to MathSciNet with `Classification: (40 AND 42)' and with `Classification: (40 AND 44)'.

There is a series of applications of summability in probability theory which is connected with key words like strong laws, central limit

theorem, almost-sure convergence, summation of random series, Markov chains and summability methods, and Tauberian theorems in probability theory. We refer the reader to [29] and recommend the use of MathSciNet with `Classification: (40 AND 60)' in order to get an overview of the relevant literature. In these notes we only touched on the main application fields of summability. We should not forget to mention that there are also applications to other fields of mathematics and, for example, in mathematical physics. The last one is related in a natural way to those in the branches of special functions, Fourier analysis and integral transforms.

Part II Functional analytic methods in summability In Chapter 2 and also in Chapters 3 and 4 we exclusively applied classical methods of proof. In contrast to that we now study in Part II the structure of the domain of matrix methods by applying essentially modern methods. For that we make available all functional analytic tools beyond basic knowledge, which we will need in Part II or III (Chapter 6). Essentially, the investigation of the structure of matrix domains is based on the fact that domains of matrix methods are FK-spaces; that is, there exists a unique locally convex topology on matrix domains which is metrizable and complete and such that the coordinate functionals are continuous. In Chapter 7 we develop in outline the FK-space theory which is mainly due to K. Zeller, and also to A. Wilansky. Applying the uniform boundedness principle and the Banach-Steinhaus theorem, which hold for matrix maps between FK-spaces, we get `soft' proofs of some Toeplitz-Silverrnan-type theorems which we proved in Chapter 2 by applying `hard' methods, in particular, by extensive gliding hump arguments. However, the elegance of these proofs should not obscure the fact that they are usually not constructive, which may be a disadvantage in certain situations.

The heart of Part II is Chapter 8 in which we study the structure of matrix domains by applying FK-space theory. For these studies the examination of `distinguished subspaces' which are usually defined by a suitable consideration of the `sections' of the members of the domain plays an outstanding role. These investigations are of independent mathematical interest. Further, they lead, for instance, to general consistency results for matrix methods and results on the existence of bounded divergent and unbounded sequences in the domain of a matrix method. Moreover, they are fundamental for Part III.

6

Functional analytic basis The aim of this chapter is to familiarize the reader with functional analytic tools which we will apply in the remaining parts of the book. We assume

that the reader is already familiar with basic notions about topological spaces, semi-metric spaces and semi-normed spaces. However, to make the book more self-contained, we give, in Sections 6.1-6.3, a compact introduction to these concepts. We concentrate on basic notions, facts, examples and theorems and omit almost all proofs. In view of our main concern, most of the examples are related to sequence spaces. Among other things we recall, in Section 6.2, the Baire category theorem for metric spaces and, in Section 6.3, the Hahn-Banach theorem, the uniform boundedness principle and the Banach-Steinhaus theorem for semi-normed spaces. In Sections 6.4-6.7, we introduce in greater detail the notion of a locally

convex space. This notion and that of continuous linear maps between locally convex spaces are central to the study of the topological structure of domains of matrix methods. Locally convex spaces generalize semi-normed spaces in the sense that their topologies are generated in an obvious way by a family of semi-norms. This means that many notions and results concerning semi-normed spaces (Section 6.4) and linear maps between them (Section 6.5) can be adapted to locally convex spaces. The Hahn-Banach theorem is an important example of this. If (X, ,r) is a locally convex space with dual X' the problem arises of characterizing all locally convex topologies T such that X' is the dual of (X, T). Associated with this problem we examine in Section 6.6 dual pairs and dual topologies. From the point of view of summability it is of interest that we again consider Schur's theorem 2.4.1. In Section 6.7 we deal with the class of semi-metrizable locally convex spaces, and, in particular, with the class of Frechet spaces (F-spaces), that

is the class of complete metrizable locally convex spaces. In Chapter 7 we will see that domains of matrix methods-endowed with their natural topology-are F-spaces. F-spaces generalize Banach spaces and we state in Section 6.7 that important principles like the open mapping theorem, the closed graph theorem, the uniform boundedness principle and the Banach-

262

Functional analytic basis

Steinhaus theorem hold. The notion of uniform boundedness has to be suitably generalized for this purpose. In Sections 10.1 and 11.4 we will apply both a generalization of the

Banach-Steinhaus theorem 6.7.16 and a further closed graph theorem which are based on the notion of `barrelled spaces'. For readers who like to study these sections and are not familiar with barrelled spaces we introduce in Section 6.8 the notion of a barrelled space and present the abovementioned theorems.

6.1 Topological spaces A topological space is a non-empty set provided with a structure which enables among other things the consideration of convergence of sequences (more generally, of nets) and of continuity of maps between topological spaces. One way to obtain such a structure is to define a set of subsets which are to be called open.

Definition 6.1.1 (topological space). Let X be a non-empty set and r be a collection of subsets of X, that isl r C P(X). Then r is called a topology on X if r satisfies the following conditions: (i) OErandXEr. (ii) If I is an index set and Qi E 7, i E I, then U{E j Qi E r. k (iii) For any finite collection Q1, ... , Qk E T, we have ni=1 Qi E r.

The pair (X, r) is called a topological space and the members of r are called open sets of (X, r) or, in short, open (in (X, -r)). We present some simple examples of topological spaces.

Examples 6.1.2. Let X -A 0 be given. Then r :_ {0, X} and r := P(X) are obviously topologies on X. They are called the indiscrete topology and the discrete topology, respectively, on X. 6 The statement of the next theorem follows immediately from the definition of a topological space.

Theorem and Definition 6.1.3 (topological subspace). Let (X, r) be a topological space and 054 Y C X. Then 71y :_ {Q n Y I Q E r} is a topology on Y. Furthermore, rjy is called the topology induced by

(X, n) on Y or the relative topology of r on Y. The pair (Y, ray) is called a (topological) subspace of (X, r). We now introduce the notion of a neighbourhood of a point in a topological space.

Definition 6.1.4 (neighbourhood (system)). Let (X, r) be a topological space, a E X and U C X. Then U is called a neighbourhood of 1 2(X) denotes the set of all subsets of X.

Topological spaces

263

x if there exists an open set Q (that is, Q E r) with x E Q C U. The set U(x) := U, (x) :_ { U C X I U is a neighbourhood of x } is called the

neighbourhood system of x. In the case of the examples in 6.1.2 it is easy to determine the neighbourhood systems.

Examples 6.1.5. Let X # 0 be given. (a) If r is the indiscrete topology, then U(x) = {X } for every x E X. (b) If r is the discrete topology, then U(x) _ {U C X I x E U} since each subset of X is open. L1

The definition of a neighbourhood is based on the notion of an open set. Conversely, we can characterize (non-empty) open sets by the notion of a neighbourhood.

Theorem 6.1.6 (neighbourhoods, open sets). In a topological space a subset is open if and only if it is a neighbourhood of each of its points.

Now, we list essential properties of the neighbourhood system of any point in a topological space.

Properties 6.1.7 (neighbourhood system). Let (X, r) be a topological space and x E X. Then U(x) satisfies the following conditions:

(1) VUEU(x) : xEU. (2) V U E U(x) V V E P(X) : U C V = V E U(x). (3) `d U, V E U(x) : U fl V E U(x).

(4)VUEU(x) 3VEU(x) V y E V : UEU(y). To determine the neighbourhood system of a point in a topological space, knowledge of a neighbourhood basis of that point is sufficient.

Definition and Theorem 6.1.8. Let (X, r) be a topological space, x E X and 13(x) be a system of neighbourhoods of x, that is 13(x) C U(x).

Then 13(x) is called a neighbourhood basis of x (relative to r) if for all U E U(x) there exists a V E 13(x) with V C U; that is, if each neighbourhood of x contains an element of 13(x). Consequently, if 13(x) is a neighbourhood basis of x (x E X), a subset

U of X is a neighbourhood of x if and only if there exists a V E 13(x) with V C U. Moreover, Q E r if and only if for each y E Q there exists a VyE13(y) with V,CQ. Examples 6.1.9. Let (X, r) be a topological space and x E X. (a) Obviously, U(x) is a neighbourhood basis of x. (b) 13(x) :_ {Q E r I x E Q} is a neighbourhood basis of x. (c) If r is the discrete topology, then 13(x) := {{x}} is a neighbourhood basis of x.

IL

264 Functional analytic basis From the properties in 6.1.7 of neighbourhood systems we may deduce corresponding properties of neighbourhood bases.

Properties 6.1.10 (neighbourhood basis). Let (X, r) be a topological space, and let, for each x E X, a neighbourhood basis B(x) be given. Then the following statements hold:

(1) VUEB(x) : xEU. (2) VU,V EB(x) 3W EB(x) : W CUnV.

(3) VUEB(x) 3VEB(x) YyEV 3WEB(y) : WCU. Conversely, we can reconstruct the topology if we know for x E X a system B(x) satisfying (1), (2) and (3).

Theorem 6.1.11 (neighbourhood basis). Let X be a non-empty set, and for each x E X let B(x) be a system of sets such that the properties (1), (2) and (3) in 6.1.10 are satisfied. Then there exists a unique topology

r on X, such that B(x) is a neighbourhood basis of x for each x E X. The topology is determined as follows. By definition, a subset Q of X is open, if for every x E Q there exists a U E B(x) with U C Q. For each x E X the neighbourhood system U(x) of x is the set of all supersets of the elements of B(x). We apply the last theorem to define the product of topological spaces.

Theorem and Definition 6.1.12 (topological product). Let I be a (non-empty) index set, and for each i E I let (Xi, ri) be a topological space

with neighbourhood system Ui(xi), xi E X. Further, let X := 11iEJ X2 be the Cartesian product of Xi (i E I). Then for each x = (xi) E X the system U; I Ui E U1(xi)

B(X)

,

Ui $ Xi for at most finitely many i E I

4H1 satisfies the conditions (1), (2) and (3) in 6.1.10. The unique topology r (cf. 6.1.11), which has the neighborhood basis B(x) for each x E X, is called

the product topology, and (X, r) is called the topological product of (Xi, ri) or, in short, product space. Convergence of sequences in topological spaces will play an essential role subsequently.

Definition and Theorem 6.1.13 (convergence). Let (X, r) be a topological space and f = (xn)neN or (xi, x2,...) for short) be a sequence2 in X. Then f is said to be convergent in (X, r), or rela-

tive to r, if

3xEX VUEU(x) 3noEN Vn>no : xnEU 2 That is, f : N -a X is a mapping with f (n) = x E X.

Topological spaces

265

holds. In this case x is called a limit of f and f is said to converge, or be convergent, to x (relative to r). Notation: If (xn)fEN converges to x in (X, r) we write xn --+ x in (X, r) or xn x or, if there is no doubt about the topology, xn -+ x (n -+ oo). Other notations such as xn

n:4) x, xn -i x or limn-+ xn = x are self-explanatory.

Let x E X and B(x) be a neighbourhood basis of x. Then (xn) converges to x if and only if

VUEB(x) 3n°EN Vn>n° : xnEU. We consider simple examples of convergent sequences.

Examples 6.1.14. Let (X, r) be a topological space. (a) If (xn) is a sequence in X with xn = a for some a E X and n > N for some N E No, then (xn) converges to a. Such sequences are said to

be eventually (or ultimately) constant. (b) If r is the indiscrete topology, then in (X, r) each sequence converges to each x E X since U := X is the only neighbourhood of x. (c) If r is the discrete topology then a sequence in X is convergent if and only if the sequence is eventually constant. This is obvious since B(x) = {{x}} is a neighbourhood basis for each x E X. A

Definition and Theorem 6.1.15 (cluster point). Let (X, r) be a topological space, x E X, B(x) a neighbourhood basis of x and M C X. Then x E X is called an adherent point of 1bt :4=* V U E U(x) U n M# 0 4==> V U E B(x) : U n M$ 0,

a cluster point of M

:4 b U E U(x) 3 y E U n M: y# x

4=* VUEB(x) 3yEUnM :y$x.

We denote by M the set of all adherent points of M and call M the closure of M. Further, M is called closed if X `M is open and this is equivalent to M = M. By definition, M is dense in (X, r) if M = X. Using the De Morgan rules we get from the defining properties of a topology the corresponding properties of the system of all closed subsets in a topological space.

Theorem 6.1.16 (closed subsets). Let (X, r) be a topological space and C, be the collection of all closed subsets in (X, r). Then C, has the following properties:

(i) 0,XEC,. (ii) If I is an index set and Ai E C,, i E I, then f1=E1 A2 E C,. (iii) I f A1, ... , Ak E C, is a f i n i t e collection, then U 1 A2 E C,. In particular M = fAEC,., MCA A and M is closed for each M C X. Next we consider some standard examples.

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Functional analytic basis

Examples 6.1.17. Let (X, T) be a topological space.

(a) If r is the indiscrete topology, then 0 and X are the only closed subsets of X. Further, each non-empty subset of X is dense in X. (c) If T is the discrete topology, then all subsets of X are open and closed. Moreover, if M is any subset, then it does not have any cluster points. (d) If R is given its natural topology, then each real number is an adherent point of Q and of R\ Q. Thus both Q and R\ Q are dense in IR endowed with its natural topology. A The notion of continuity will play an important role.

Definition and Theorem 6.1.18 (continuity). Let (X, TX) and (Y,Ty) be topological spaces, x E X and f : X -> Y be a map. Let Bx (x) and By (f (x)) be neighbourhood bases of x in (X, TX) and of f (x) in (Y, Ty), respectively. Then (i) f is continuous at x E X -1

VUEUT,,(f(x)) : f (U)EUT,,(x) V U E LIT, (f (x))

3 V E UTX (x)

: f (V) C U

-1 V U E By (f (x)) : f (U) E LIT,, (x)

bUEBy(f(x)) 3V EBX(x) (ii) f is continuous

:= V x E X

f is continuous at x -1

V

f(V)CU. -1

(Q)

V

(C)

where CX and Cy denote the set of all closed sets in (X, TX) and (Y, Ty ), respectively. We now list some simple facts about continuous maps.

Remarks 6.1.19 (continuous maps). Let (X,TX) and (Y,Ty)

be

topological spaces and let f : X ---f Y be a map. (a) If f is constant, then f is continuous. (b) If Ty is the indiscrete topology or Tx is the discrete topology, then f is continuous.

(c) If X C Y and f : X --* Y, x --> x is the inclusion map, then f is continuous if and only if TyIX C TX.

(d) Let 0 0 M C X and x E M. Then the continuity of f at x implies that of f I M at x (relative to TX I M ). The converse implication fails in general.

(e) Let (Z, TZ) be a topological space and let g : Y ---+ Z be a map. If f is continuous at x and g is continuous at f (x), then the composition go f is continuous at x. In particular, the composition of continuous maps is continuous. 0

Topological spaces

267

Theorem and Definition 6.1.20 (sequential continuity). Let (X, TX) and (Y, ry) be topological spaces, x E X and let f : X -* Y be a map. If f is continuous at x, then f is sequentially continuous in x, that is

b (x,i) in X : xn. -* x in (X, rX) = f (x,,) --+ f (x) in (Y, 7y). In general, sequential continuity does not imply continuity.

The map f is called sequentially continuous if it is sequentially continuous at each x E X.

Theorem 6.1.21 (continuous maps into product spaces). Let (X, r) be the product space of topological spaces (X2, rt), i E I, according to 6.1.12.

(a) Then for each j E I the projection -7r? : X -* Xj, x = (x2) ---+ xz is continuous, and the product topology r is the weakest (cf. 6.1.22) topology on X such that every 7rj (j E I) is continuous.

(b) If (Z, rZ) is a topological space and3 f E Abb(Z, X), then f is continuous if and only if for each j E I the map 1rj o f is continuous. We compare topologies in a natural way.

Definition 6.1.22 (comparison of topologies). Let X be a nonempty set and let r1 and r2 be topologies on X. Then r2 is called stronger (or finer) than r1 and r1 is called weaker (or coarser) than r2i if r1 C r2 holds.

Remarks 6.1.23. (a) If X is a non-empty set, then the discrete topology is the strongest topology on X and the indiscrete topology is the coarsest topology on X.

-3

(b) If r1 and 12 are topologies on X, then the identity map id : (X, 71) (X, r2) is continuous if and only if r1 is stronger than r2. On account of 6.1.20, an immediate consequence is the following statement which is important subsequently. If r1 and r2 are topologies on X and Ti is stronger than r2, then each sequence which converges to x E X /s relative to r1 also converges to x relative to T2.

Theorem 6.1.24 (comparison of topologies). Let X be a nonempty set and let r1 and 72 be topologies on X. For each x E X let 51(x) and B2 (x) be neighbourhood bases of x in (X, r1) and (X, 72), respectively. Then the following statements are equivalent: (a) r1 is stronger than T2.

(b) IV xEX VUES2(x) 3VEI31(x) : VCU. The following theorem deals with the continuity of inclusion maps which will play an important role subsequently. 3 Abb (Z, X) denotes the set of all maps f : Z -+ X.

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Functional analytic basis

Theorem 6.1.25 (inclusion map). Let (X, r) be a topological space and (Y, rly) a subspace of (X, r) in the sense of 6.1.3. Then the inclusion

map f : Y -+ X, y -+ y is continuous, and rl y is the coarsest topology on Y such that f is continuous. In closing this section we introduce the notion of Hausdorff spaces and of compact spaces.

Definition 6.1.26 (Hausdorff space). A topological space (X, r) is called Hausdorff if for all x, y E X with x # y there exist U E U(x) and V E U(y) such that U fl V = 0. We now formulate some important properties of Hausdorff spaces.

Properties 6.1.27. Let (X, r) be a Hausdorff space. (a) Each (topological) subspace of (X, r) is also Hausdorff. (b) One-element subsets, thus finite subsets, of X are closed in (X, r).

Definition 6.1.28 (compact space/subset). (a) A topological space (X, r) is called compact if it is Hausdorff and if for each open covering of X there exists a finite subcovering of X; that is, if for each collection

/r` C r (of open sets) with X C UQc,. Q there exist finitely many sets Qn E r' with X C Uk-1 Qk. QI, (b) A subset Y of a topological space (X, r) is called compact if Y = 0 or (Y, rI Y) is compact.

Remarks 6.1.29. (a) If X is finite and (X, r) is Hausdorff, then (X, r) is compact. In particular, each finite subset of a Hausdorff space is compact. (b) Each compact subset of a Hausdorff space is closed. (c) Each closed subset of a compact space is compact.

For simplification we introduce the following notation.

Definition and Remark 6.1.30 (relatively compact). A subset A of a topological space (X, r) is called relatively compact if its closure A is compact. Using 6.1.29(c) we see that each compact subset of a Hausdorff !S space and each subset of a compact space is relatively compact.

Theorem 6.1.31 (continuous image of compact sets). Let (X, TX) and (Y, ry) be Hausdorff spaces and f : X --* Y be a continuous map. If M C X is compact, then f (M) is compact.

6.2

Semi-metric spaces

Semi-metric spaces form an important class of topological spaces.

Definition and Remark 6.2.1 (semi-metric/metric space). Let X be a non-empty set and d : X x X --* R be a map. Then d is called a semi-metric on X if the following conditions hold for all x, y, z E X :

Semi-metric spaces

(semi-definiteness),

(1) d(x, y) > 0 and d(x, x) = 0 (2)

269

(symmetry),

d(x, y) = d(y, x)

(triangle inequality).

(3) d(x, y) < d(x, z) + d(z, y)

If in addition

(definiteness),

(1') d(x, y) > 0 when x # y

then d is called a metric on X. If d is a semi-metric (respectively metric)

on X, then (X, d) is called a semi-metric space (respectively a metric space). Further, d(x, y) is called the distance between x and y. In A

particular, each metric space is a semi-metric space.

Remarks 6.2.2. Let X be a non-empty set and let d and h be semimetrics on X. Then the following statements hold: (a) If a > 0, then ad is a semi-metric on X, and it is a metric if and only

if d is a metric and a > 0. (b) d + h is a semi-metric, and it is a metric if and only if d(x, y) # 0 or h(x, y) 54 0 for all x, y E X with x 54 y.

(c) d := sup{d, h} : X x X -4 R, (x, y) -> sup{d(x, y), h(x, y)} is a semi-metric, and it is a metric if and only if the corresponding condition in (b) is satisfied. (d) K x,y,z E X : ld(x,y) - d(x,z)(< d(y,z)

(second triangle inequality).

L1

Most of the following examples will play a role subsequently.

Examples 6.2.3. (a) If X := R, then a metric d is defined in each case by d(x, y) := Ix - yI and d(x, y) := I arctan x - arctan yl. (b) Let X := R" where n E N is fixed. For x = (x1, ... , x") E R" and y = (yi,...,y") E R" we put n

CE

Jxk-yk('

k=1

dp(x,y) :_

p

//

m ax

kEN

if1
Then, for every 1 < p < oo, dp is a metric on R. In particular, the metric

d2 is called the Euclidean metric or Euclidean distance. (c) C together with the Euclidean distance d2 defined by d2(a + ib, c + id) :_

(a - c)2 + (b - d)2

(a, b, c, d E R)

is a metric space.

(d) Let K := R or K :=

C,

let D be an arbitrary non-empty set and let

M(D):={f:DKI

supf(x)<xED

I

270

Functional analytic basis

the set of all bounded maps on D. Then d.: M(D) x M(D), (f, g) --4 sup If (x) - g(x) I xED

is a metric and it is called the supremum metric. (e) If we put D := N° in (d), then we have

m:=r°):=M(N°)= jx=(xk)Ew I t

SUP lxkl <00

kEM°

J

and

dd(x, y) = sup Ixk - ykI (x = (xk), Y= (Yk) E m) kEN°

which defines the supremum metric d,, on m. We now introduce the notion of e-neighbourhoods which enables us to consider semi-metric spaces as topological spaces.

Definitions and Remarks 6.2.4 (topology of (X, d)). Let (X, d) be a semi-metric space.

(a) If x E X and e > 0 are given, then U, (X)

{z E X I d(x,z) < e}

is called an e-neighbourhood of x or an open ball with radius e and centre x. The collection of all e-neighbourhoods of x is denoted by Uep, (x). Since the notion of an e-neighbourhood depends on d we use also

the notations Ud(x) and L1 ,8(x) instead of U,(x) and Uep,(x), respectively.

(b) For each x E X the system Uep,(x) satisfies the conditions (1), (2) and (3) of 6.1.10. Thus, by Theorem 6.1.11, there exists a unique topology rd such that Uep, (x) is a neighbourhood basis of x (x E X). The topology

Td is called the topology generated by the semi-metric d. (c) The open sets in (X, Td) are the subsets Q of X which contain, for each x E Q, an e-neighbourhood of x. Furthermore, the neighbourhoods

of any x E X are the supersets of the e-neighbourhoods of x, that is

U(x)={UCX f 3e>0: U,(x)CU}.

(d) The closed balls Ke(x) := {y E X I d(x,y) < e} with centre x and radius e > 0 are neighbourhoods of x since they contain U, (x). Further, K5 (x) is closed in (X, d), that is in (X, rd).

(e) For every x E X each of the following systems is a neighbourhood basis of x : {U£(x) I E>01, {KK(x) I e>01, {U, (x) 1 n E N} , {K1(x) I n E N}

,

{ UQ (x) I n E N} and { Kan (x) I n E N} where an >

0 and an -+ 0. (f) Let x E X and M C X. Then x is an adherent point of M if and only if there exists a sequence (xn) in M which converges to x in (X, d). Further, x is a cluster point of M if and only if there exists a sequence

Semi-metric spaces

271

(x,,) in M with x,, x which converges to x in (X, d). (g) (X, rd) is Hausdorff if and only if d is a metric. In particular, if d is a metric then the limit of any convergent sequence in (X, d) is uniquely A determined. We continue with two pathological examples.

Examples 6.2.5 (indiscrete semi-metric, discrete metric). Let X be a given non-empty set.

(a) A semi-metric d on X is defined by d(x, y) := 0 for all x, y E X and is called the indiscrete serpi-metric. Obviously, UE (x) = X for all e > 0 and x E X, and thus d generates the indiscrete topology. (b) The metric d on X defined by

' y) =

ifx#y 0,

if x = y

(x,yEX)

is called the discrete metric. We obviously have Ufx_

X

if1<e
{x}

if 0 < e < 1

(x E X and e > 0),

therefore, d generates the discrete topology. (Note that UI (x) = {x} 54 KI (x) = X if X has more than one point.) A

Theorem and Definition 6.2.6 (subspace). Let (X, d) be a semimetric space and 0 # Y C X. Then the restriction dy := dl y,,y : Y x Y -+ R, (x, y) --* d(x, y) is a semi-metric on Y which is a metric if d is. Furthermore, dy is said to be induced by d on Y and (Y, dy) is called a subspace of (X, d). For simplification of notation, we use d instead of dy when no confusion can arise. Obviously, UdY (x) = Y fl Ud (x) for all e > 0 and x E Y; thus, dy generates on Y the subspace topology rdIy induced by rd. Given a countable collection of semi-metric spaces we can define a 'natural' semi-metric on the product. An important special case is w = W40 . For that we consider the Frechet combination of countably many semi-metrics.

Theorem and Definition 6.2.7 (Frechet combination). Let X be a given non-empty set.

(a) If d is a semi-metric on X, then

d:XxX---+IR, (x,y)

d(x'y) 1 + d(x,y)

is a semi-metric on X which is a metric if and only if d is.

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Functional analytic basis

(b) If (dk) is a sequence of semi-metrics on X, then the Frechet combination d of (dk) defined by 2-k

d(x,y):= k=1

dk(x,y) 1 + dk(x,y)

(x,yEX)

(6.2:1)

is a semi-metric on X. It is a metric if and only if for all x, y E X with x # y there exists a j = j (x, y) E N such that di (x, y) > 0.

Example 6.2.8. Let C(R) :_ If : R -1 R I f is continuous} be the set of all continuous functions on 1R. Since C(R) 0 M(R) there is no natural

metric on C(R) like the supremum metric on M(R). That is why we consider the Frechet combination d of the semi-metrics dk : C(R) x C(Ilt) -+ ]R, (f, 9)

doo(f I(-k,k], 9I(-k,kj) =

sup I1(x) - 9(x)I xE-k,k]

where k runs through N. Obviously, dk is a semi-metric since dkl[-k,k] =

d,,. and do is a metric on M([-k, k]). Moreover, d is a metric on C(R)

since for all f,9 E C(R) with f # g there exists an x E R with f (x) # g(x) and therefore a k E N with x E [-k, k] which implies doo(fl[-k,k],9l(-k,k]) > 0. Thus d(f,g) > 0.

L1

We now use the Frechet combination of semi-metrics to define a natural semi-metric on the product of semi-metric spaces.

Theorem and Definition 6.2.9 (product space). Let I := N {kEN11
d{x, y}

k=1 a0

k.1

dk(xk,yk)

ifl=lam

2_k dk xk>?Ik

if I = N

l+dk zk,U )

(x = (xk),y = (yk) E X).

Moreover, d is a metric if and only if dk (k E I) is a metric. The space (X, d) is called a product space and d is called a product semi-metric. If d is a metric, it is called a product metric. (Note, the topology generated by the product metric is equal to the product topology in the sense of 6.1.12.) 4 In the case I = N. we mean by (xk) the tuple (x1,. .. , xn) and in the case I = N the sequence (XI,x2,...).

Semi-metric spaces

273

Proof. The essential idea to prove the main statement of the theorem is to consider, for each k E I on X, the semi-metric dk defined by dk(x,y) := dk(xk,yk) (x = (xk), Y = (Yk) E X).

Then we consider in the case that I is finite the sum d := EkEl dk, and in the case that I := N the Frechet combination of dk (k E N). We now give some examples of product spaces.

Examples 6.2.10. (a) If X:= W', then the metric dl in 6.2.3(b) is the product metric if we consider on R the Euclidean distance. (b) Let X = w = KN° = 11kEN° Xk with Xk = K (k E N°). Then

d,, (x, y) :_ E 2-k k-o

Ixk - ykI I + Ixk - yk I

(x = (xk) , y = (yk) E w)

defines the product metric on w. We denote by r-, the topology generated by d,,. IL Next we consider convergence of sequences in different types of semimetric spaces. We start with a general convergence criterion which is based on the fact that Ueps (x) is a neighbourhood basis.

Theorem 6.2.11 (c-no-criterion). Let (X, d) be a semi-metric space, x E X and (xn) be a sequence in X. Then

xn --axin (X,d) e=,,, V e > 0 3noEN Vn>no :

be>0 3noEN Vn>no : d(xn,x)<e (d(xn, x)) E co.

Theorem 6.2.12 (convergence in certain semi-metric spaces). Let X be a non-empty set, (d2) be a sequence of semi-metrics on X, x E X and (xn) be a sequence in X.

(a) Convergence relative to the sum of semi-metrics: If d := En I d2 (cf. 6.2.2(b)), then (xn) converges to x in (X, d) if and only if for each i E Nn the sequence (xn) converges to x in (X, d2). (b) Let the semi-metric dk : X x X -4 JR be defined by

dk(x,y) :=

dk(x,y)

(x,y E X)

1 + dk(x,y)

(cf. 6.2.7(a)). Then (xn) converges to x in (X,dk) if and only if (xn) converges to x in (X,dk).

(c) Convergence relative to the Frechet combination: Let d be the Frechet combination of the semi-metrics d2 (cf. 6.2.7(b)).

Then (xn) converges to x in (X, d) if and only if for each i E N the sequence (xn) converges to x in (X, d2).

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Functional analytic basis

Now, it is easy to characterize the convergence of sequences in finite or countable products of semi-metric spaces.

Theorem 6.2.13 (convergence in the product space). Let (X, d) be the product space of at most countably many semi-metric spaces (Xi, di) (i = 1,2,...). Further, let {x{n)) with x(n) = (xsn))i=1,2,... be a sequence

in X, and x = (xi)i-1,2,... E X. Then (X(n)) converges to x in (X,d) if and only if (x(n)) converges componentwise to x; that is, if for each i = 1, 2, ... the sequence (x;'))n converges to xi in (Xi, di). We now apply the last theorem to w.

Example 6.2.14 (coordinatewise convergence). Let (w, d,) be given as in 6.2.10(b), x E w and let (X(n)) with x(n) = (x=n))1 be a sequence in w. Then (X(n)) converges to x in (w, d,,) if and only if it converges coordinatewise to x; that is, (x=n)) n converges to xi for each i E NO. In the next part of this section we deal with the continuity of maps on semi-metric spaces.

Theorem 6.2.15 (sequential continuity). Let (X, d) be a semi-metric space, (Y, ry) be a topological space, x E X, and let f : X -* Y be a map. Then f is continuous at x if and only if f is sequentially continuous

at x. To characterize the continuity of maps between semi-metric spaces we now use the fact that in semi-metric spaces the set of all e-neighbourhoods of any point x is a neighbourhood basis of x.

Theorem 6.2.16 (e-8-criterion). Let (X, dX) and (Y, dy) be semimetric spaces, x E X, and let f : X --i Y be a map. Then f is continuous at x if and only if

Ve>0 3b>0 byEX : (dX(y,x)
Definition 6.2.17 (comparison of semi-metrics). Let X be a nonempty set, and let d1 and d2 be semi-metrics on X. Then d2 is called stronger than d1 and d1 is called weaker than d2 if the topology rd, generated by d2 is stronger than the topology rd, generated by dl. By definition, d1 and d2 are equivalent if rd1 = rd,. We continue with the consideration of some simple examples.

Examples 6.2.18. (a) Let X := JR. and let d be the discrete metric and d2 be the Euclidean metric on X. Then d is stronger than and is not equivalent to d2. (b) Let X := R, and let d2 be the Euclidean metric and let d : R x 1[t - +

Semi-metric spaces

275

R, (x, y) -a arctan x - arctan yI be the metric in 6.2.3(a). Then d2 and d are equivalent. (c) The metrics dF (p E [1, oo]) on R' (n E N) introduced in 6.2.3(b) are (pairwise) equivalent. L Completeness of semi-metric spaces will be an essential tool in subsequent investigations. So we are going to introduce this notion, to present some important facts on completeness and to give some examples. We start with the definition of a Cauchy sequence.

Definition and Theorem 6.2.19 (Cauchy sequence). Let (X, d) be a semi-metric space and (xn) be a sequence in X. (xn) is a Cauchy sequence (in (X, d) )

Ve>0 3noEN b`v,n>no : d(x,,,xn)<e Ve>0

3no E N V n > no : d(xno,xn) <E.

Properties 6.2.20 (Cauchy sequences). In any semi-metric space (X, d) the following statements hold: (a) Each convergent sequence is a Cauchy sequence.

(b) If a Cauchy sequence (xn) has a subsequence which converges to x E X, then (xn) itself converges to x. Now, we emphasize those semi-metric spaces in which each Cauchy sequence converges.

Definition and Properties 6.2.21 (complete semi-metric space). A semi-metric space (X, d) is called complete if every Cauchy sequence converges in (X, d).

(a) If Y is a non-empty closed subset of a complete semi-metric space (X, d), then the subspace (Y, dy) is also complete. (b) If (X, d) is a metric space and (Y, dy) is a complete subspace of (X, d), then Y is closed in (X, d). However, if d is a semi-metric, but not a metric, then this statement fails. (c) If (X1, dl ), (X2, d2),... are at most countably many semi-metric spaces, then the product space (X, d) (cf. 6.2.9) is complete if and only if f o r each i = 1, 2, ... the space (X2, di) is complete. (d) In contrast to the sequential continuity of continuous maps between semi-metric spaces, the continuous image of a Cauchy sequence is not neces-

sarily a Cauchy sequence. (Consider the identity map i where d and d2 are as in Example 6.2.22(b).)

d)

d2)

(e) (X, d) is complete if and only if each sequence (xn) in X, which satisfies Fn d(xn+1 , x,) < oo, converges.

A

Examples 6.2.22 (completeness). (a) If X is a non-empty set and d the indiscrete semi-metric or the discrete metric, then (X, d) is complete.

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Functional analytic basis

(b) The metric space (1K, d2) is complete where d2 is the Euclidean distance. However, (R, d) is not complete when d is the metric

d : R x R --> R, (x, y) ---* I arctan x - arctan yI

(cf. 6.2.3(a)). Note that d and d2 are equivalent and that (n) is a nonconvergent Cauchy sequence in (X, d).

(c) The product Kn of K (where K carries the Euclidean distance) endowed with the product metric, that is n

d : K x10 --r R, d(x,y) _ E Ixk - ykI (x = (xk),y = (yk) E K), k=1

is a complete metric space (cf. (b) and 6.2.21(c)). (d) (w, d,,) is a complete metric space (cf. 6.2.10(b) and 6.2.21(c)). (e) (M(D), is a complete metric space (see the following proof). (f) (m, d,,,) is a complete metric space. (To prove it, put D := NO in (e).) (g) (c, are complete metric spaces. (They are closed and (CO, subspaces of (m, as we will sketch below.) (h) (21, d1) is a complete metric space. (We will give the proof below.) (1) The metric space (cp,

not converge (to (k+1) E co \

is not complete since

k=° kk+1 ek} n

does

).

Proof. We will give the proofs of (e) and (h) and sketch that of (g). (e) Let (fn)nEN be a Cauchy sequence in (M(D), that is V E > 0 3 no E N d v,n > no : sup l xED

fn(x)I <e-

(6.2:2)

Fix x E D. Because I fv(x) - f (x)I < SUP Ifv(y) - fn(y)I yED

we get by (6.2:2) that the sequence (fn(x))nEN is a Cauchy sequence in (1K, d2) where d2 is the Euclidean metric on K. Consequently, the sequence (fn(x))fEN converges in (K, d2). We denote the unique limit by f (x). This defines a function

f:D --+ K, x--3 lim fn(x) n-aoo We now prove:

(i) f is bounded, thus f E M(D), and

(ii) fn --) f in (M(D), which proves the completeness of (M(D),

d).

Semi-metric spaces

277

(i) On account of (6.2:2) there exists an no E N with

Vn>no

fno(x)I <1.

In particular, we get for all x E D and n > no

Ifn(x)I < 1+Ifno(x)I < 1+dd(fno,0), where 0 denotes the zero map on D. By letting `n -; oo' in (K, d2) we get

I f (x) I < 1 + doo (fno, 0)

far every x E D,

and thus f E M(D). (ii) If e > 0 is arbitrarily given, then we may choose an no E N according to (6.2:2). Then we get, for v, n > no and each x E D, the inequalities

If(x) - fn(x)I

<-

<

I1(x) - ff(x)I + Ifv(x) - fn(x)I If(x) - ff(x)1 +c.

Letting `v -> oo' in (K, d2) we obtain If (x) - fn(x)I :5,F

for all

x E D

and consequently

d,o(f, fn) = sup If (x) - fn(x)I < E. xED

This proves fn --* f in (M(D), (g) To prove that (c, d,,) is complete, by 6.2.21(a) it is sufficient to show For this let (fn) n with fn = (x(n})k be a that c is closed in (C°°, sequence in c which converges in (Q°O, to f = (xk) E P°°. In particular,

we have xfn) ---* xk (n -* oo) for every k E No. Since f is a bounded

sequence, by the Bolzano-Weierstrass theorem we may choose an E lg xk, --+ (v -* oo). If n is the and a subsequence (xk) limit of fn, that is en = limk.,o xkni, then limv-- x(n) = Cn. The whole situation may be illustrated by the following diagram: xkv

is close to

(i v is large)

en

I

I

is close to (if n is large)

is it close to ? (if n is large)

I

xk

is close to (if v is large)

I

The diagram suggests that the sequence (Cn) of the limits converges to .. Since the limit of the sequence (Cn) is uniquely determined, the sequence

278

Functional analytic basis

(xk) has exactly one accumulation point, namely , which implies (xk) E c. Moreover, we get limk.oo xk = limn. oo limk.+ xkn). Therefore, (c, dam) is complete. To prove that (co, dam) is complete, by 6.2.21(a) it suffices to show that

co is closed in (c, .). For this we note that lim : c --3 K, (xk) --+ limk xk is continuous on (c, do.). Thus, co = {(xk) E c I limk xk = 0} = lim-1({O}) is closed since {0} is closed (cf. 6.1.18(ii)). (h) To prove that is complete, we assume that (x(n)) with x(n) = (xkn)) is a Cauchy sequence in d1). Since PI C t' and because for all x = (xk), y = (yk) E el the inequality dc°(x, y) =SUP Ixk - ykl <- E Ixk - ykl = di (x, y) kEN°

k=0

holds, (x(n)) is also a Cauchy sequence in However, is complete by (f), that is (x(n)) converges to some x = (xk) E e°° relative to

.. We now show that x E el and limn dl (x(n), x) = 0 which proves that (X(n)) converges (to x) in For an arbitrarily given e > 0 we choose an no E N such that dl (x(n), x(")) <

(n, v > no)

(6.2:3)

and, since limp i. d,,.(x("), x) = 0, for each N E N we can choose a vN E N with vN > no such that

d (x("), x) <

lE

(v > UN).

N2

(6.2:4)

Thus we get for each given N E N and all n > no and every v > vN the inequalities N

D

N

xk(n) - xk I

N

k k k - xkI E Ix(n) - x(")I + EIx(") k=0 k=0

k=O

<

dl (x(n), x(")) + N d. (x("), x)

<

2+2=E

[on account of (6.2:3) and (6.2:4)].

Hence, since 1V E N is arbitrarily given, we have 00

VE>0

V k=0

which implies x(n) - x E e' for every n > no, thus x E e1, and 0 dl (x(n), x) = 0, that is (x(n)) converges to x.

Semi-metric spaces

279

The next topic in this section is compactness in the case of metric spaces. For this we introduce the notion of bounded sets in semi-metric spaces.

Definition and Remark 6.2.23 (bounded set). Let (X, d) be a semimetric space and A C X. Then A is called bounded if A is contained in an e-neighbourhood of at least one point of X. Further, a sequence

in X is called bounded if A :_ {x I n E N} is bounded. As one may easily check, A is bounded if and only if for each x E A there exists an e-neighbourhood of x containing A. 4L

Examples 6.2.24. Let (X, d) be a semi-metric space. (a) Each finite subset of X is bounded. (b) Each subset of a bounded set is bounded. (c) A finite union of bounded subsets is bounded. (d) Every Cauchy sequence in (X, d) is bounded. (e) If supx,yEx d(x, y) < oo, then X is bounded. In particular, this holds in the cases of the discrete metric and the indiscrete semi-metric. IL

Theorem and Definition 6.2.25 (compact metric space). Let (X, d) be a metric space. (a) Each compact subset is bounded and closed. However, the converse fails in general. (Consider X := R and the metric d in 6.2.22(b); then X is bounded and closed, but it is not compact.) (b) (X, d) is compact if and only if (X, d) is sequentially compact, that is each sequence in X has a subsequence which converges (in (X, d) ). (c) If (X, d) is compact, then it is complete. In closing this section we deal with the Baire category theorem and the notions connected with it.

Definition and Examples 6.2.26 (nowhere dense subset). Let (X, r) be a topological space and Y C X. Then Y is called nowhere dense (in (X, -r)) if Y contains no non-empty open set, otherwise Y is called somewhere dense (in (X, r) ). In (R, d2) the set N is nowhere dense and both Q and R \ Q are somewhere dense. 0

Definition and Remarks 6.2.27 (first and second category).

Let

(X, r) be a topological space and Y C X. Then Y is said to be of first category or meagre if Y is an at most countable union of nowhere dense subsets; otherwise, Y is said to be of second category. A space of second category is also called a Baire space. (a) Nowhere dense subsets are meagre. (b) An at most countable union of meagre subsets is a meagre subset. (c) Q is meagre and dense in (R, d2).

280

Functional analytic basis

(d) X is of second category in (X, T) if and only if X 36 U:EN Y; holds for each sequence (Y1)iEN of closed nowhere dense subsets of X. A The Baire category theorem, which follows, is a basic tool in functional analysis.

Theorem 6.2.28 (Baire category theorem, [252, 1-6-6]). If (X, d) is a complete metric space, then X is of second category in (X, d).

6.3 Semi-normed spaces, Banach spaces In this section we deal with semi-metric spaces (X, d) in the special case that X is a linear space and the semi-metric d is generated by a seminorm. This guarantees that the topological structure is consistent with the algebraic structure of X in the sense that addition of vectors and multiplication of vectors by scalars are continuous.

Definition and Remarks 6.3.1 (semi-norm, norm). Let X be a linear space over K (K:= R or K:= Q. Then a map p : X -a Ill is called semi-norm on X if for all x, y E X and a E K the following conditions are satisfied:

(1) p(x) > 0 (semi-definiteness), (2) p(ax) = jajp(x) (positive homogeneity), (triangle inequality). (3) Ax + y) < p(x) + p(y) A semi-norm p is called norm, if-in addition-the condition (1*) p(x) > 0 for each x 0 0 (definiteness) is satisfied. If p is a semi-norm (respectively a norm) on X, then (X, p) is

called a semi-normed space (respectively a normed space). Each norm is a semi-norm and, in the case of a norm, we often use the notation 11 11 instead of p. For any semi-norm we have p(O) = 0.

Remarks and Definition 6.3.2 (generated semi-metric). (a)

L1

Let

(X, p) be a semi-normed space. Then a semi-metric d on X is defined by

d(x, y) := p(x - y) for all x, y E X.

Moreover, d is a metric if and only if p is a norm. We call d the semi-

metric (metric) generated by the semi-norm (norm) p. The topology generated by d is also called the topology generated by p. If Y is a linear subspace of X, then ply is a semi-norm. We denote it also by p and call (Y, p) a subspace of (X, p). By definition, (X, p) is complete if (X, d) is complete. A complete normed space is called a Banach space. If Y is a closed linear subspace of a Banach space, then (Y, p) is a Banach space (cf. 6.2.21(a)).

(b) If X is any linear space over K and d is a semi-metric on X, then d is generated by a semi-norm p if and only if

Semi-normed spaces, Banach spaces

(i) b x, y, z E X : d(x + z, y + z) = d(x, y)

281

(translation invariance)

(ii) `d x, y E X V a E K : d(ax, ay) = )al d(x, y). In this case, we have p(x) = d(x, 0) for all x E X, and we say that (X, d)

is semi-normable. If d is a metric, then p is a norm and (X, d) is said & to be normable. In the following theorem we state that each normed space can be embedded in a Banach space as a dense subspace.

Definition and Theorem 6.3.3. Let (X, p) be a normed space and (Y, q) be a Banach space with X C Y. Then (Y, q) is called a completion

of (X, p) if qIx = p and X is dense in (Y, q). For every normed space there exists a completion (which is unique up to isomorphism). We now characterize bounded sets in semi-normed spaces.

Theorem 6.3.4. Let (X, p) be a semi-normed space and A C X. Then the following statements are equivalent:

(a) A is bounded in (X, p) (with respect to the generated semi-metric). (b) sup {p(x) I x E A} < oo. (c) xn --Y 0 (n -+ oo) for each sequence (xn) in A. Based on the notion of the product of semi-metric spaces we now define the product of finitely many semi-normed spaces. However, we should mention that the product of countably many semi-normable semi-metric spaces is not semi-normable in general (cf. 6.3.8(d)).

Definition and Theorem 6.3.5 (product semi-norm). Let n E N, let (X1 , pi ), ... , (X, pn) be semi-normed spaces, let di be the semi-metric generated by pi and d the product semi-metric on X :_ rl = 1 Xi according to 6.2.9. Then (X, d) is semi-normable and, in particular, d is obviously

generated by the product semi-norm n

p : X ---> R, x = (xi) -

p(xj). :=1

We call (X, p) the product of (Xi, pi) . The product semi-norm p is a norm if and only if each pi (i E Nn) is a norm.

In a natural way each semi-normed space is related to a normed space as we will now state.

Definition and Theorem 6.3.6 (quotient space). Let (X, p) be a semi-normed space, let Y:= Kern p :_ {x E X I p(x) = 0} and let5

X/Y:={[x]:=x+YIxEX} 5 For M, N C X, a E K and x E X we use the notations N + M (x+y I xEN, yEM}, x+M:={x}+M and aM={ax I xEM}.

282

Functional analytic basis

be the algebraic quotient space. Then the map 1111 : X/Y -4 R

,

[x]--*p(x)

is a norm and (X/Y, II II) is called a quotient space (relative to (X, p) ). We now work out the specific feature of the neighbourhood systems of semi-normed spaces.

Notation and Remarks 6.3.7 (neighbourhood system). Let (X, p) be a semi-normed space and d be the semi-metric generated by p. Obvi-

ously,Ue(x)={y EX I d(x,y) =p(x-y) <e} for allxEXandr>0. By 6.3.2(i) we get

Ve>0 VxEX : Ue(x) = x+Ue(0) and therefore for all x E X the identities

Ueps(x) _ {x+U I U EUeps(0)} _: x+Ueps(0) and U(x) = x+u(0). For that reason we put, in the case of semi-normed spaces, U

Up := U (O)

0 and Ue:=UE :=Ue(0) fore>0. Examples 6.3.8. (a) The zero map p : X -* R, x -a 0 is a semi-norm on each linear space X over K and it generates the indiscrete semi-metric.

(b) The map I : K -> R, x -+ IxI is a norm on K and generates the I

Euclidean distance. (c) As is well known, K" is a linear space over K. Since for each p E [1, oo] the metric dp in 6.2.3(b) satisfies the conditions (i) and (ii) in 6.3.2(b), the metric dp is generated by a norm which we denote by II llp We have

IIxIIP =

if1
(x = (xl, ... , xn) E l[C" ).

if P = oo

Further, for each j E N" a semi-norm qj on K" is defined by qj (x) := I xj I

(x = (x1,...,x") E K"). (d) Similarly to the last example, for each j E N°, a semi-norm qj on w is defined by qj(x) := Ixj I (x = (xk )k E w). Moreover, one may show that the product metric d, on w (cf. 6.2.10(b)) cannot be generated by a (semi-)norm (since each neighbourhood of zero in (w, d,,) contains a nontrivial subspace of w). In 6.4.23 we will give another proof of the last statement.

(e) If D is a non-empty set and M(D) the linear space of all bounded functions f : D ---> K, then

IIfll. := sup If(x)l xED

(f E M(D))

defines a norm II lloo on M(D). Obviously, II III generates the supremum

metric d,,. and is called the supremum norm.

Semi-normed spaces, Banach spaces

283

(f) If we put in particular D := Nl° in (e) then we get that (m, 11 11.) is generates the metric d". introduced in a normed space. Obviously, 6.2.3(e). (g) (e,11111) is a normed space and 1111, generates dt .

(h) Generalizing e = Qi we define for 0 < p < oo the linear space of all

absolutely p-summable sequences by 00

ep

{x = (xk)kEN° E W

I

L..: lxklP < 001 k=O

In the case 1 < p < oo the map UP 11

lip

eP ---> R, x = (xk) ---+

Ilx11p

C37 Ixklp k

is a norm which is called the £P-norm. We will come back to the case

0
(M(D), II

1l.), (co,11 11.), (e, 11111) and, as one may prove,

(PP, 11

(m, li III}, lip) for p E

[1, oo] are Banach spaces.

In the next part of this section we will encounter further examples of Banach spaces which will play an essential role in our work.

Definition and Remarks 6.3.9 ((absolutely) convex, absorbing). Let X be a linear space over K and M C X. Then M is called

convex :e--* baE[0,1] : aM+(1-a)MCM, balanced :4--* b a E K : (lal < 1 = aM C M), absolutely convex :4=* tea, f3 E K : (lal+l,31 < I

aM+f3M C M),

absorbing :4=#- VxEX 3A>0KaEK: (Jai >a==* xEaM). (a) The set M is absolutely convex if and only if it is convex and balanced. (b) Any intersection of convex subsets of X is convex.

(c) If aEF, xEX and MCX, then: If M is convex, so are x + M and a M. If M is balanced, so is a M.

Definition and Remarks 6.3.10 (Minkowski functional). Let X be a linear space over 1K.

(a) If p is a semi-norm on X, then UE is absorbing and absolutely convex for each e > 0. (b) If M is a non-empty absorbing absolutely convex subset of X, then

pM:X-+R, x-+pM(x):=inf{AERI A>0 and xEAM} is called the Minkowski functional of M, and pM is a semi-norm satisfying {x E X I pM(x) < 1} C M C {x E X I pM(x) < 1} .

284

Functional analytic basis

The topological structure of semi-normed spaces is consistent with the algebraic structure of linear spaces, as we will now state.

Theorem 6.3.11 (continuity of `+' and

If (X, p) is a semi-

normed space, then

+ :X xX --3X, (x,y)-*x+y and

(a,x) --+ ax are continuous where X x X and K x X are endowed with the product semi-norm (generating the product semi-metric, cf. 6.2.9).

In the following theorem we characterize the continuity of (further) semi-norms on semi-normed spaces.

Theorem 6.3.12 (continuous semi-norms). Let (X, p) be a seminormed space and q be a semi-norm on X. Then the following statements are equivalent: (a) q is continuous. (b) q is continuous at zero.

(c) 3M>0 VxEX :

q(x) <Mp(x). Proof. We only sketch the proof roughly. The implication

is

trivially true. To prove that `(b)=*(c)' and `(c)=(a)' hold one may use the c-6-criterion for continuity (cf. 6.2.16) and the characterization of sequen0 tial continuity (cf. 6.2.15), respectively.

Definitions and Remarks 6.3.13 (linear map). Let (X, p) and (Y, q) be semi-normed spaces (over K). A map T : X ---+ Y is called linear if

b'a,/EK Vx,yEX : T(ax+Qy)=aT(x)+/3T(y) holds. It is also called a linear operator and, when Y

a linear

functional (or linear form). We use following notation:

Hom(X, Y) := {T : X -* Y I T is linear} B(X, Y) := {T : X -+ Y I T is continuous and linear} B(X) := B(X, X) , X' := Hom(X,IK) and X' := B(X,K). Here, K has the norm q := I We call, X* the algebraic dual of X and X' is called the dual space (or dual) of (X, p). We will also use the notation (X, p)' instead of X' when different semi-norms are under consideration. B(X, Y) is obviously a linear subspace of Hom (X, Y) and 16 X' is a linear subspace of X*. Using the sequential continuity (cf. 6.2.15) and the characterization of the continuity of semi-norms (cf. 6.3.12) we get a very practical characterization of the continuity of linear operators between semi-normed spaces.

Semi-normed spaces, Banach spaces

285

Theorem 6.3.14 (continuous linear maps). If (X, p) and (Y, q) are semi-normed spaces and T : X --* Y is a linear map, then the following statements are equivalent: (a) T E B(X, Y), that is T is continuous. (b) T is continuous at zero.

(c) 3 M > 0 Vx E X : q(T(x)) < Mp(x). Proof. The implication `(a)'(b)' is trivially true whereas `(b)'(c)' follows from `(b)#. (c)' of 6.3.12 since q c T : X -+ IR is a continuous seminorm on X. We prove '(c)#, (a)' by showing the sequential continuity of T (cf. 6.2.15). Choose M > 0 according to (c) such that

q(T(x)) < Mp(x) for all x E X,

(6.3:1)

and consider any x E X and any sequence (xn) in (X, p) converging to x; in particular, p(xn - x) -i 0. Hence, by (6.3:1), (T(xn))aEN converges 0 to T (x) in (Y, q). Thus, T is continuous at x.

Remark 6.3.15. Let (X, p) and (Y, q) be semi-normed spaces and let T : X --} Y be linear. Then the following statements are equivalent: (i) T E B(X, Y). (ii) T(Kl(0)) is bounded, that is sup {q(T(x)) I p(x) < 1} < oo. (iii) T(Ui(0)) is bounded (in (Y,q)). (iv) T (A) is bounded for all bounded subsets A of X. Statement (iv) shows why continuous linear operators between seminormed spaces are also called bounded linear operators. A Before we go into the study of continuous linear maps in greater detail we apply Theorem 6.3.14 for characterizing equivalent semi-norms.

Definition and Theorem 6.3.16. Let X be a linear space and p, q be semi-norms on X. Then p and q are called equivalent if the generated semi-metrics are equivalent. The following statements are equivalent:

(a) p and q are equivalent.

(b) 3M,N>0 VxEX : Np(x)
Proof. Applying 6.3.16(b) we obviously get that relative to equivalent semi-norms the same sequences are, respectively, convergent and Cauchy sequences.

We now give an example of equivalent norms.

0

286

Functional analytic basis

Example 6.3.18.

11

Ilbv (cf. 1.2.3) and

I Ibv:bv- R, x=(xk) --+IliMxkI+EIxk-xk-1l k

are equivalent norms on by. The proof is left to the reader.

0

On account of 6.3.14(c) we can obviously endow B(X, Y) with a natural semi-norm.

Theorem and Definition 6.3.19 (operator semi-norm). Let (X, p) and (Y, q) be semi-normed spaces, T : X --+ Y be linear and'

11T11:=inf{M>0 1 V x E X : q(T(x)) <Mp(x)}. Then the following statements hold: (a) T E B(X, Y) if and only if 11TI1 < oo. (b) (B(X, Y), 1111) is a semi-normed space, and in the case that X # {0}, the semi-norm 1111 is a norm if and only if q is a norm. 1111 is called the

operator semi-norm or operator norm, respectively. (c) V T E B(X,Y) `d x E X q(T(x)) < IlT1Ip(x). (d) VT E B(X,Y) :7 IITII = sup{q(T(x)) I p(x) = 1} = sup {q(T(x)) I p(x) < 1}

_-

sup

q(T (x))

PI P(x)

# 0

Theorem 6.3.20. Let (X, p) be a semi-normed space and (Y, q) be a Banach space. Then (B(X, Y), 1111) is a Banach space. In particular, (X`,11 11) is a Banach space. Here 11 11 denotes the operator norm.

Remark 6.3.21 (norm and pointwise convergence). Let (X, p) and (Y, q) be semi-normed spaces, and let (Tk) be a sequence in B(X, Y) which converges with respect to the operator semi-norm 11 to T E B(X, Y). 11

Then (Tk) converges pointwise to T; that is, Tk(x) -+ T(x) for each x E X in (Y, q) since (cf. 6.3.19(c)) we have

q(Tk(x) - T(x)) = q((Tk - T) (x)) < IITk

- T11 p(x) -> 0 (k -3 oo),

that is q(Tk(x) - T(x)) --+ 0 (k -+ oc).

Theorem 6.3.22. Each linear map T between a finite-dimensional normed space (X, p) and any semi-normed space (Y, q) is continuous.

Let X be a linear space (overy) and f E X*. Then the codimension of the kernel Kernf :_ {x E X I f (x) = 0} of f is less than or equal to 6 Note that, by convention, inf 0 := oo. 7In the case of the first and third equality we assume p 0 0.

Semi-normed spaces, Banach spaces

287

one, that is codimx (Kern f) < 1. Obviously, in the case of a continuous linear functional f on a semi-normed space, the kernel of f is closed as it is the inverse image of the closed set {0}. The following theorem tells us that continuous linear functionals can be characterized by that property.

Theorem 6.3.23 (continuous linear functionals). Let (X, p) be a semi-normed space and let f E X. Then the following statements are equivalent:

(a) f is continuous, that is f E X'. (b) Kern f is closed in (X, p). (c) 3M>0 VxEX : I.f(x)I <Mp(x). An essential tool in functional analysis is the Hahn-Banach theorem which tells us-roughly speaking-that each continuous linear functional on a subspace of a semi-normed space can be continued to a continuous linear functional on the whole space. We refer to the Hahn-Banach theorem

also as the Hahn-Banach extension principle. Theorem 6.3.24 (Hahn-Banach theorem). Let X be a linear space over K, p : X -p R a semi-norm and8 Y < X. If M > 0, and f E Y' satisfies

If (x) I < Mp(x) then there exists a g E X* such that

gIY = f

and

for all x E Y,

(6.3:2)

lg(x)I < Mp(x) for all x E X;

that is, g continues (continuously) f from Y to X retaining the equality (6.3:2) with the same M. We now draw some useful corollaries from the Hahn-Banach theorem.

Corollary 6.3.25. Let (X, p) be a semi-normed space, Y < X, and let f E (Y,pIY)`. Then there exists a g E X' such that gIY = f and III = lifll. Proof. If f = 0, then g := 0 has the desired properties. If f # 0, then we may apply the Hahn-Banach theorem for M := Ilf II The next corollary of the Hahn-Banach theorem guarantees a certain richness of the dual of (non-trivial) semi-normed spaces. Corollary 6.3.26. Let (X, p) be a semi-normed space and Y be a closed

proper subspace of (X, p). Then, for every xo E X \ Y, there exists a g E X' with g(xo) = 1 and Y C Kerng, that is gIY = 0. Proof. For any xo E X \ Y we consider Z:= {axo + y I a E K, y E Y J and the linear functional f : Z -4 K, axo + y --3 a. Note, Y is closed in (Z, pl z) , f (xo) = 1 and Y = Kern f C Kerng. Thus f is continuous by 6.3.23. Applying 6.3.25 (to f and Z) we get the desired result. 8 Recall that Y < X means that Y is a subspace of X.

288

Functional analytic basis

As a corollary of 6.3.26 we get an important tool for the investigation of the topological structure of the domains of matrix methods.

Corollary 6.3.27. Let (X, p) be a semi-normed space and Y be a linear subspace of X. Then the following statements are equivalent:

(a) Y is dense in (X, p), that is V = X.

(b) `d f E X' : (Y C Kern f = f = 0). Proof. If Y = X, and if f E X' with Y C Kern f is given, then X = Y C Kern f = Kern f C X; thus f = 0. Conversely, if Y g X then 6.3.26 gives us an f E X' which does not satisfy statement (b).

As a further corollary of 6.3.26 we get a statement about the sum of closed subspaces.

Corollary 6.3.28 (sum of closed subspaces). Let (X, p) be a semi-

normed space, V be a closed linear subspace of it and W be a finite-dimensional linear subspace of X. Then the sum V + W = {x + y E X I x E V, y E W} is also a closed subspace of (X, p). Proof. It is obviously sufficient to consider one-dimensional subspaces W. Then the statement is essentially an application of 6.3.26. We leave the details to the reader. A further corollary of 6.3.26 gives us a sufficient condition for a seminormed space to be separable. We omit the proof.

Definition and Theorem 6.3.29. (a) By definition, a topological space is separable if it contains an at most countable dense set. (b) Let (X, p) be a semi-normed space and (X',11 11) be its dual endowed with the operator norm. Then (X, p) is separable if (X', II II) is.

In the next application of the Hahn-Banach theorem we consider the dual of the dual space of a normed space (endowed with the corresponding operator norm).

Definition and Theorem 6.3.30 (second dual). Let (X, II IIx) be a normed space with dual X' and operator norm II lix, The dual X" :_ (X')' of (X', II Iix') is called second dual (of (X, II lix)). We denote the corresponding operator norm by II ilx". By Theorem 6.3.19(d) we have

IlFilx" = sup {IF(f)I I f E X' with ilf lix' < 1 }

(F E X").

Furthermore, the following statements hold:

(a) For each x E X the linear functional Fx : X' -+ K., f -+ f (x) is continuous and IIFzIlx,, = Ilxli x.

(b) If Y := {F= I x E X}, then (X, II lix) is isometrically isomorphic to

(Y, II IIx«) by virtue of the map T : X -+ Y, x --+ F.. In particular, (Y, 11 IIx«)=(X,11 lix); that is, X is-up to the isometric embedding-a linear subspace of its second dual (X", 11 lix,,).

Semi-normed spaces, Banach spaces 289

Definition and Remark 6.3.31 (pointwise/uniform bounded). Let (X, p) and (Y, q) be semi-normed spaces, and let I be a non-empty set of linear operators from X into Y, that is 41 C Hom (X, Y). Then 41 is called pointwise bounded if {T(x) I T E tF} is bounded in (Y,q) for each x E X.

We say that 4' is uniformly bounded (or norm bounded) if sup I

I T E It} < oo

is satisfied. In particular, in the case of uniform boundedness we have (I C B(X, Y).

A sequence (Tn) in Hom (X, Y) is called pointwise bounded (respectively uniformly bounded) if IT. 1 n E N} is pointwise bounded (respectively uniformly bounded). As we just remarked, the members of a uniformly bounded family of operators are continuous. The next theorem tells us that equicontinuity and uniform boundedness are equivalent. For this we recall the definition of equicontinuity.

Definition 6.3.32 (equicontinuity). A family $ of maps between semi-metric spaces (X, dx) and (Y, dy) is defined to be pointwise equicontinuous at x E X if be > 0 3S > 0 b f E 4' by E X : dx (x, y) < S

dy (f (x), f (y)) < e ;

it is called uniformly equicontinuous or, in short, equicontinuous if

be>0 3S>0Vf E4' Vx,yEX : dx(x,y)<S

dy(f (x), f (y)) < E.

Theorem 6.3.33 (equicontinuity). Let (X, p) and (Y, q) be seminormed spaces and let 0 54 4' C Hom (X, Y). Then the following statements are equivalent: (a) 4' is equicontinuous.

(b) tie>0 9S>0 VTE4' VxEX : p(x - 0) = p(x) < S

q(T(x) - T(0)) = q(T(x)) <,E.

(c) 3M>0 VTE4' YxEX : q(T(x))<Mp(x). (d) 4' is uniformly bounded. Uniform boundedness obviously implies pointwise boundedness whereas the converse statement fails in general as the following example proves. Example 6.3.34. Let (X, p) := (cp, II IIco) and (Y, q) Then the sequence (Tn) where n

Tn : tp --+ K, x = (Xk)kENO --i T0(x) := E xk k=o

is obviously pointwise bounded, but it is not uniformly bounded since IITnII = n (n E N°).

A

290

Functional analytic basis

The uniform boundedness principle, Theorem 6.3.35, tells us that completeness of the domain space is sufficient in order that pointwise boundedness implies uniform boundedness. (Note, (X, p) in 6.3.34 is not complete.)

Theorem 6.3.35 (uniform boundedness principle). Let (X, p) and (Y, q) be semi-normed spaces, let (X, p) be complete and 0 0 P C B(X, Y). If

is pointwise bounded, then

is uniformly bounded.

We now formulate two versions of the Banach-Steinhaus theorem. For this we need some definitions.

Definition 6.3.36 (pointwise convergence/Cauchy sequence). Let X be a non-empty set, (Y, r) be a topological space and let (f") be a sequence in Abb (X, Y). Then (fn) is called pointwise convergent (relative to r) if there exists an f E Abb (X, Y) such that (fn(x)) converges to f (x) for every x E X (in (Y, r) ). In this case, f is called a pointwise

limit of (f,,), and (fn) is said to be pointwise convergent to f (in Abb (X, Y) relative to r); if, in addition, fn, f E S C Abb (X, Y) then we

speak of pointwise convergence in S (relative to r). If (Y, d) is a semi-metric space, then (f,a) is called a pointwise Cauchy sequence (relative to d) if (fn(x)) is a Cauchy sequence in (Y, d) for each x E X.

Remarks 6.3.37. (a) Under the assumptions of 6.3.36 and if (Y, d) is a semi-metric space, a pointwise convergent sequence is obviously a pointwise Cauchy sequence. Conversely, a pointwise Cauchy sequence is pointwise convergent if (Y, d) is complete. (b) Let (X, p) be a semi-normed space, (Y, q) be a normed space, and (Tn)

be a sequence in B(X, Y) which converges pointwise to T E Abb (X, Y). Then T is obviously linear, but it is not necessarily continuous, that is T E B(X, Y) is not necessary. As an example of this we can consider the A example in 6.3.34. As corollaries of the uniform boundedness principle we now give two versions of the Banach-Steinhaus theorem which gives a criterion for the continuity of the pointwise limit of a sequence of continuous linear operators. We start with the usual one.

Theorem 6.3.38 (Banach-Steinhaus). Let (X, p) be a complete seminormed space, (Y, q) be a semi-normed space, and let (Tn) be a sequence in B(X, Y). If (Tn) is point wise convergent to

T : X --> Y, x -> lim Tn(x) (in (Y,q)), n-*00 then T E B(X, Y) and JIT11 < lim

f flTnfl < sup flTn{1 < oo. nEN

(6.3:3)

Semi-normed spaces, Banach spaces

291

Proof. (Tn) is assumed to be pointwise convergent, and thus pointwise bounded and consequently uniformly bounded by 6.3.35. This gives

q(T(x)) = limo q(TT(x)) < sup IITnII p(x) nEN

(x E X)

and therefore the continuity of T by 6.3.14. Refining the above estimation by considering a subsequence Tn,, with lim infn,,,,, IITnII = limk, JITn, 11 we get the desired inequalities in the theorem. Applying 6.3.38 and the uniform boundedness principle one may prove the following general version of the Banach-Steinhaus theorem.

Theorem 6.3.39 (Banach-Steinhaus, general version). Let (X, p) be a complete semi-normed space, (Y, q) be a Banach space, and let H be a dense subset in (X, p). Furthermore, let (Tn) be a sequence in B(X, Y). If (T,,) is pointwise bounded and (T,,IH) a pointwise Cauchy sequence, then (Tn) is pointwise convergent, the pointwise limit T of (Tn) is linear and continuous, and (6.3:3) holds. In closing this section we consider isometric isomorphisms and adduce further examples of Banach spaces from known Banach spaces.

Theorem 6.3.40. Let (X, p) and (Y, q) be semi-normed spaces and let T: X -* Y be an isomorphism with T E B(X,Y) and T-1 E B(Y,X). Then (X, p) is complete if and only if (Y, q) is complete. In particular, (X, p) is a Banach space if and only if (Y, q) is a Banach space. Furthermore, V is a closed subspace of (X,p) if and only if T(V) is a closed subspace of (Y, q).

Proof. We give here only the main idea of the proof. Applying 6.3.14(c) to T and T-1 we get that (xn) is a Cauchy sequence in (X, p) if and only if (T(xn)) is a Cauchy sequence in (Y, q). We draw two simple corollaries from the last theorem.

Corollary 6.3.41. (a) Let (Y, q) be a semi-normed space, X be a linear space and T : X -+ Y be an isomorphism. Then

p : X --f

,

x-+p(x):=q(T(x))

is obviously a semi-norm on X. Further, the space (X, p) is complete if and only if (Y, q) is complete. In particular, (X, p) is a Banach space if and only if (Y, q) is a Banach space. (b) Let (X, p) and (Y, q) be semi-normed spaces and T : (X, p) --3 (Y, q) be an isometric isomorphism (that is, T is an isomorphism and satisfies q(T(x)) = p(x) for each x E X). Then (X, p) is complete if and only if (Y, q) is complete. In particular, (X, p) is a Banach space if and only if (Y, q) is a Banach space.

292

Functional analytic basis

On the base of 6.3.41 we get more examples of sequence spaces which

are-together with their natural norm-Banach spaces. Examples 6.3.42. (a) (bs, 11 jib,) and (cs,11 16) are Banach spaces. (b) by and bvo endowed with II Jib,, and I Ibo are Banach spaces. Proof. (a) Since E : (bs,11 Jib,) -+ (m,11 II,o) , E : (CS, 11 11b.) --+ (c, 1111,0) , (xk) -> (Ey=o

X.') and are isometric isomor-

(Xk)

phism,, (bs, II Jib.) and (CS, 11 I1b8) are Banach spaces on account of 6.3.41(b)

and 6.3.8(i). (b) Obviously, E-1 : (bv,11 !!b,) -+ (1,11111), (xk) -+ (xk -xk_1), where x_1 := 0, is an isometric isomorphism. Thus (bv, II is a Banach space because of 6.3.8(i) and 6.3.41(b). Also (bvo,11 11b,,) is a Banach space: bvo = Kern (lim Ibv) and because lim Ib E bv', on account of j-1

I lim xI <_ IIxl1 = sup 1 xj I = sup X0 - E (X" - xy+1) <- IIXIIbv 9

1

V=0

for each x = (xk) E by, the set bvo is closed in (bv, II I1b,,), see 6.3.23. The normed spaces (bv, I Ibv) and (byo, I Ib,) are Banach spaces since 11 llbv and I

lb, are equivalent (cf. 6.3.18 and 6.3.17).

As we stated in 6.3.20 the dual X' of a semi-normed space (X, p) II is a Banach space. We now state that for each T E B(X, Y) there corresponds, in a natural way, a map T' E B(Y', X'). endowed with the operator norm 11

Definition and Theorem 6.3.43 (dual map). Let (X, p) and (Y, q) be semi-normed spaces and let T E B(X, Y) be given. Then the map

T':Y'-a X', f ->T'f := f o T is well-defined and continuous, that is T' E B(Y', X'). Furthermore, IITII =

IIT'Il (Here, Y' and X' are endowed with the operator norm.) The map

T' is called the dual or adjoint map of T. 6.4

Locally convex spaces

The aim of this section is to introduce the notion of a locally convex space generalizing the notion of a semi-normed space. We refer to the properties of neighbourhood bases in topological spaces which we have listed in 6.1.10 and which allow the reconstruction of the topology as we have stated in Theorem 6.1.11. The following example motivates the notion of a locally convex space.

293

Locally convex spaces

Example and Definition 6.4.1 (locally convex space).. Let X be a linear space over K and P be a non-empty family9 of semi-norms on X. For x E X we put

B(x) := x+8:= f x+ B I B E B} with

-

B:=

ei>0andpEP(i=1...n)}. J

(Recall that UP :_ {x E X 'J p(x) < e} is the e-neighbourhood of zero relative to p E P.) Then B(x) (x E X) enjoys the properties (1), (2) and (3) in 6.1.10, as we will prove below. The topology Tp, which is uniquely

defined by 6.1.11, is called the topology generated by P, and (X,Tp) is called a locally convex space and lcs for short. We also denote (X, Tp) by (X, P). A topological space (X, T) and its topology r is called locally convex if X is a linear space over K and if there exists a family P of semi-

norms on X such that r = Tp. We call 8 the canonical neighbourhood basis of zero relative to P, which we also denote by Bp. We now verify that B(x), x E X, has the properties (1), (2) and (3) in 6.1.10. Obviously, (1) is satisfied since 0 E UP for each p E P and e > 0, that is 0 E B holds for every B E B. It is sufficient to show (2) in the case of x = 0; that is, for B. However, we have U fl V E B for all U, V E B; therefore, we may put W := U fl V. Obviously, in (3) it is also sufficient to consider x = 0. Thus, let n

u = nUE'

(nEN, ei>0 and piEP(i=1,...,n))

i=1

be given. We put V := U and choose a y E V, which means particularly pi(y) < ei (i = 1, ... , n). If we consider n

8i:=ei-pi(y) (i=i,...,n) and W := y+nUa,, i=1

then W C U and W E y + B = B(y) as one may easily check. 0 Remark 6.4.2. By the definition of a locally convex space (X, P) and the neighbourhood basis B(x) of x E X we get immediately that for every

a E X the translation by a, that is T. : X -t X, x -- a + x, is

a

continuous.

9 By definition, a family P of semi-norms on X is a map from an index set I into

the set P of all semi-norms on X; thus P : I -+ P, which we denote also by P = (pi

I

i E 1), P = (pi)iet or, for short, if the index set is clear, by P = (pi). However,

sometimes we also use the notation P for the set given by the family P. Conversely, if a set P = {pi i E I} of semi-norms is given, then P also denotes the family (pi i E I) I

and (p I p E P), respectively.

I

294

Functional analytic basis

Examples 6.4.3. (a) Each semi-normed space (X, p) is a locally convex space (X, P) with P :_ (p), and the topologies generated by p and P are equal, that is rp = rp. In particular, (K, 11) is a locally convex space. (b) On w we consider the family P:= (q? I j E N°) of the semi-norms

qj:w->R, (xk)-->IxjI (jEN°). Then (w, P) is a locally convex space and we denote its topology by r". The locally convex space (w, rr,) will play an essential role in the subsequent chapters. As we will see, the topology r47 introduced here is equal to that introduced in 6.2.10(b).

(c) The space C(R) of all continuous functions f : IR -+ I[8 together with the topology generated by the family P :_ (p; j E N) of the semi-norms pi defined by

pi(f) := sup If(x)I zE[-j,jj

(j E N, f E C(R))

is a locally convex space and-as one may show-the topology rp is equal to that generated by the metric in 6.2.8. A

Definition and Example 6.4.4 (subspace). If (X, r) is a locally convex space, Y is a linear subspace of X and if r is generated by the family P of semi-norms, then (Y, rly) is a locally convex space and rly is obviously generated by Ply :_ (ply I p E P) . In this sense we call (Y, Ply) a subspace of the locally convex space (X, P). IL In the next point we examine relative to the properties (1), (2) and (3) in 6.1.10 the neighbourhood bases of locally convex spaces, in particular the neighbourhood basis of zero.

Properties and Notation 6.4.5 (neighbourhood basis). Let (X, P) be a locally convex space with topology rp. (a) Analogously to the case of semi-normed spaces we have U(x) = x +U

for the neighbourhood system U(x) of x E X where we put U := U(0) for the neighbourhood system of zero. The elements of U are called neighbourhoods of zero. Thus, we obtain the neighbourhood system of any point x by translation of the neighbourhood system of zero by x. (b) If B is any system of subsets of X, then it is a neighbourhood basis

of zero in (X, P), if and only if for each x E X the system x + B is a neighbourhood basis of x in (X, P). (c) There exists a neighbourhood basis B of zero in (X, P) with the following properties:

(1) VU,VEB 3WEB: WCUf1V. (2) VUEB VaEK\{0} : allEB. (3)

V U E B : U is absolutely convex and absorbing.

295

Locally convex spaces

(For example, the canonical basis Bp enjoys the properties (1)-(3).) Moreover, we may choose 5 such that (4) b U E B : U is closed in (X, P) is satisfied (cf. Exercise 6.4.31). (d) Conversely, for each linear space X over K and for every non-empty system B of subsets of X which satisfies (1) and (3) of part (c) there exists

a unique topology r such that (X, r) is a locally convex space and B is neighbourhood basis of zero in (X, r). Remark: For example,

P:=(pu I UEB) withpu(x):=inf{aERIa>0 and xEaU} is a family of semi-norms which generates r, where pu is the Minkowski functional of U (cf. 6.3.10). Proof. The proofs of the statements. in (a), (b) and (c) are straightforward and are left to the reader. Now, we sketch a proof of (d). Let B be a system of subsets of X which satisfies the properties (1) and (3) in part (c), and let P be the family of all Minkowski functionals pu, U E B (cf. above). Then P generates a locally convex topology rp. We have to verify that B is a neighbourhood basis of zero in (X, P). (Since B, as one may show, enjoys the properties (1) and (3) in 6.1.10 and because, by 6.1.11, it is

a neighbourhood basis relative to a unique topology, Tp is the unique topology with the desired properties.) That this is the case follows if each U E B is a neighbourhood of zero in (X, rp) and for every neighbourhood W of zero there exists a V E B with V C W. For this, one verifies for any

U E B and p := pu the inclusion Ui C U C 2U1 and notes that Ul is a neighbourhood of zero in (X, P).

Remark 6.4.6. The property (3) in 6.4.5(c) justifies the terminology for a locally convex space: Since convexity is translation invariant we get by this property that every point in a locally convex space has a neighbourhood basis consisting of convex sets, that is the space carries locally a convex structure. A In the case of locally convex spaces the property of being a Hausdorff space can be easily characterized by the notion of a total family of seminorms which we now introduce.

Definition and Theorem 6.4.7 (total family of semi-norms). Let X be a linear space over K and P be a non-empty set of semi-norms on X. Then P is called total or separating if it enjoys the property

VsEX\{0} 3 p E P : p(x)54 0.

(6.4:1)

A locally convex space (X, P) is Hausdorff if and only if P is total.

296

Functional analytic basis

Proof. If P is not total, then there exists an x E X \ {0} with p(x) = 0 for each p E P. That implies x E U for each U E Bp, and thus for every neighbourhood of zero. Consequently, there do not exist neighbourhoods U and V of zero and x, respectively, which satisfy u n v = 0. Therefore, (X, P) is not Hausdorff. Conversely, let x, y E X with x # y be given. Then, because of (6.4:1) there exists a p E P such that p(x - y) 0. Setting a := zp - y), then obviously (x + UP) ft (y + UP) = 0, and, therefore, (X, P) is Hausdorff because x + UE E U(x) and y + U£ E U (y). Examples 6.4.8. The locally convex spaces (w, ri,) and (C(R), P) (cf. 6.4.3(b),(c)) are Hausdorff spaces as one may easily verify.

A

Our next aim is to verify that-similarly to the case of semi-normed spaces-the topology of locally convex spaces is consistent with the structure of linear spaces; that is, addition and multiplication by a scalar are continuous operations. For that we need the notion of the `product of locally convex spaces'.

Definition and Example 6.4.9 (product of lcs). Let 136 0, and let (Xi, Pi), i E I, be locally convex spaces. Furthermore, let X := rLEI Xi be the Cartesian product of the linear spaces Xi and iri : X ---+ Xi be the projection of X onto Xi (i E I), and let

P:=U{p°Iril PE Pi} iEI

Then X endowed with a topology r is called the topological product of the locally convex spaces (Xi, Pi), if r is generated by the family P of semi-norms. In such a case, r is called the product topology, and (X, r) is a locally convex space. Note that r is equal to the product topology in the sense of 6.1.12. Obviously, (w, r,s,) (cf. 6.4.3(b)) is the (countable) product of (K, p

with itself: we have I := No, Xi := K and Pi In 6.4.27 we remark that the notion of the product of locally convex spaces is consistent, respectively, with that of the product of finitely many semi-normed and of countably many locally convex spaces whose topology is generated by a semi-metric. We now state that addition and multiplication (by a scalar) on locally convex spaces are continuous.

Theorem 6.4.10 (continuity of + and ). Let (X, P) be a locally convex space. Then addition

+ :X xX -iX, (x,y) --ix+y and multiplication by a scalar

Locally convex spaces

297

:KxX -->X, (a,x)-ticax are continuous. Here, X x X and K x X carry the product topology in accordance with 6.4.9.

Proof. We prove the continuity of' +' and leave the proof of the continuity to the reader. of We denote by B and B the canonical neighbourhood basis of (X, P) and (X x X, P), respectively, where (cf. 6.4.9)

P:={pozrl IpEP}U{polr2 I pEP}. The continuity of `+' is verified, if we have shown

V (x, y) E XxX V U E B 3 V E 13 b (v, w) E (x,y)+V : v+w E x+y+U. Therefore, let (x, y) E X x X and U E B be given. Then, by the definition

of 8 we have U = l 1 Up' for some ei > 0 and pi E P. We now put n

n Pioni

V :_

Uej/2

i=1

Pton2 . UE;l2

n i=1

For each (v, w) E (x, y) + V we may choose (s, t) E V with (v, w) = (x, y) + (s, t). Therefore we have v + w = x + y + s + t and 3+t E U; thus v + w E x + y + U, on account of the choice of V. This completes the

0

proof.

The continuity of semi-norms on locally convex spaces can be characterized similarly to that on semi-normed spaces.

Theorem 6.4.11 (continuous semi-norms). Let (X, P) be a locally convex space and let q : X ---+ R be a semi-norm on X. Then the following statements are equivalent: (a) q is continuous. (b) q is continuous at zero.

(c) 3 n E N 3p1i...,pnEP 3M>0VzEX :

q(x)

n

M >pj(x). j=1

In particular, each p E P is continuous. Proof. We only sketch the proof. The implication `(a) (b)' is trivially true. (b) . (c) : Let q be continuous at zero. Because q(0) = 0 this is equivalent to n

b' e> 0 3U := n Ups ESP b x E U: q(x)<E.

(6.4:2)

j=1

Now, if we choose, f o r c := 1, numbers n E N and E1, ... , En > 0 and semi-norms p1, ... , pn according to (6.4:2), then, as one may prove, (c) is

298

Functional analytic basis

satisfied if we put M:= !with 6 := min {e3 I j = 1, ... , nI . (c) (a) : The proof is straightforward if we note that for an arbitrarily given z E X the inequality in (c) holds particularly for x := y - z, y E X, and that fq(y)-q(z)I < q(y-z) is satisfied by the second triangle inequality for semi-metrics (cf. 6.2.2(d)).

Next, we characterize the comparison of locally convex topologies by continuity statements on corresponding families of semi-norms which generate the topologies.

Theorem 6.4.12 (comparison of lc topologies). Let X be a linear space over K and let Tp and rq be locally convex topologies on X, which are generated by families of semi-norms P and Q, respectively. Let Up and UQ be neighbourhood bases of (X, P) and (X, Q), respectively. Then the following statements are equivalent: (a) Tq is weaker than rp.

(b)VUEUq 3VEUp: VCU. (c) V q E Q : q is continuous on (X, Tp).

Proof. (a) q (b) : Applying 6.1.24 and in view of 6.4.5(b) we get that rq C Tp holds if and only if

VXEX VUEUQ 3VEUp: x+VCx+U which is obviously equivalent to (b).

(b)

(c) : Let q E Q and e > 0 be given. Since U? is a neighbourhood

of zero in (X, Q), there exists a U E Uq with U C US and, by (b), a V E 13p such that V C U. Then we have Iq(x) -q(0)I = q(x) < e for every x E V. Thus q is continuous on (X, P) (at zero, see 6.4.11). (c) . (b) : Let (c) be true and U E Uq be arbitrarily given. Then there

exists a U' := nj 1 U,' E B with U' C U for some n E N, qj E Q and Ej > 0 (j E Na). On account of the continuity of qj (at zero) there exists Vj E Up such that qj(x) < ej for each x E Vj (j = 1, ... , n), which implies V:= nn Vj C U* C U. Thus, we have statement (b). 1

Theorem 6.4.12 gives us the possibility to `thin out' the family P which generates a given locally convex topology by leaving out `superfluous' seminorms. In general that will facilitate the handling of the topology.

Corollary 6.4.13. Let X, Q, P, rp and Tq be given as in Theorem 6.4.12. If, in addition, Q C P holds and if each p E P \ Q is continuous on (X,TQ), then Tq = Tp. Corollary 6.4.13 implies, in particular, that a locally convex topology generated by a family Q of semi-norms is not changed if we add continuous semi-norms to Q.

Locally convex spaces

299

Proof of 6.4.13. From the additional remark in 6.4.11 and from 6.4.12 `(c)

0

(a)' we get -rp C rQ as well as TQ C rp.

Corollary 6.4.14. If (X, P) is a locally convex space, then the topology -rp generated by P is the weakest locally convex topology r on X such that every p E P is continuous on (X, r). Proof. Obviously Tp has the property that each p E P is continuous on (X, Tp). If r is a topology such that every p E P is continuous on (X, r), then Tp is weaker than T by 6.4.12 `(c) = (a)'. 0 Following the main interest of this book we give examples of locally convex sequence spaces based on the notion of the 8-dual introduced in 2.3.1. These examples will play an essential role in subsequent chapters.

Example 6.4.15. Let X and Y be sequence spaces with cp C Y C X16. Then for each y = (yk) E Y a semi-norm py on X is defined by (x = (Xk) E X),

py(x)

and Py := (py I y E Y) generates a locally convex topology Tp,, on X. Since cp C Y the topology rp. is Hausdorff and rp,. D r,, I x . In particular, Tp, = T,,, in the case X := w and Y :_ W. In 6.6.8 we state that in the case p C X the topology Tp,, is exactly the `weak topology' of the `dual pair' (X, Y) which is in general denoted

by o(X,Y). Proof. By the definition of the 6-dual py is well-defined for each y E Y and it is obviously a semi-norm. Further, Py is total; thus rp,, is Hausdorff by 6.4.7, since if x = (xk) E X \ {0}, then there exists an n E NO with

xn54 0 and we get py(x)=IxnI#0 for y:= e' 2 C Y. According to 6.4.3(b) and 6.4.4 the topology T,Ix is generated by (qj l x I j E N°) where qj(x) := (x11 (x = (Xk) E w). Because PI x W C Y we have e? E Y (j E N), therefore PIx C Py since qj I x = pe, (j E N°). Thus the additional statement in 6.4.11 tells us that each qj Ix is continuous on (X, Tp,,) , from which we get T,,, Ix C Tp,, by 6.4.12. Now, we verify Tp, = T,,,, in the case X := w and Y := ip = w13.

We have P := {qj I j E N°} C P,,. If p E Pg,, then there exists a y = Ek=° ykek E cp with p = py. Hence, for every x = (Xk) E w we obtain p(x) = py(x) =

: plylloo

qk(x)

IxkI :5 (1 + IIyIfco) k=0

k=0

Thus p is continuous on (w, -r,,) (cf. 6.4.11), and, consequently, we have T4, J Tp,,. Therefore T,,, = Tpw, by 6.4.13.

0

300

Functional analytic basis

As we will verify next, the convergence of sequences in locally convex

spaces (X, P) can be reduced to the convergence of sequences in seminormed spaces. However, we should remark that in general-in contrast with the case of semi-normed spaces-we cannot characterize notions like cluster point, adherent point, continuity and so on by the consideration of the convergence of sequences; for that one has to consider the more general notion of nets1o

Theorem 6.4.16 (convergent sequences in lcs). Let (X, P) be a locally convex space, x E X and let (xn) be a sequence in X. Then the following statements are equivalent:

(a) (xn) converges in (X,Tp) to x. (b) (x,, - x) converges in (X, 7-p) to zero. (c) (xn) converges for each p E P in the semi-normed space (X, p) to x, that is p(xn - x) --> 0 (n -* oo). Proof. (a) * (b) : This is an immediate consequence of the (sequential) continuity of the translation Ta by taking a := -x and a := x in turn. (c) : Note that each p E P is (sequentially) continuous (cf. 6.4.11). (b) (c) . (b) : Let 5 be the canonical neighbourhood basis of zero and V = nk1 U i E B be given where k E N, si > 0 and pi E P (i E Nk). If (c) holds, then for each i E Nk there exists an ni E N with pi (xn - x) < Ei for all n > ni, that is with x, - x E U£ i for all n > ni. Putting no := max{nl,. . , nk} we get xn - x E nk1 U£ = V for all n > no, that is

0

statement (b) holds.

Remark 6.4.17. In accordance with 6.4.9 let (X, P) be the product of the locally convex spaces (Xi, Pi), i E I. Let (x(n)) with X(n) = (x;n))iE1 E X be a sequence in X and let x = (xi) E X. Then (x(n)) converges to x in n converges to xi (X, P) if and only if for every i E I the sequence in (Xi, Pi). To prove this we consider the following equivalent statements:

(x(n)) converges to x in (X, P)

b'pEP : V i E I b p E Pi

=ViEI

p(x{n}-x)

:

0

[by6.4.16]

p(7ri (xinl - x))

n-; 0

n° V p E Pi : p(xin) - xi)

0

[definition of P]

[definition of iri ]

b i E I : (xin) )n converges to xi in (Xi, Pi)

[by 6.4.16].

Example and Definition 6.4.18. Let x E w, and let (x(n)) with x(n) (xkn)) k E w be a sequence in w. Then, by 6.4.17 and the example in 6.4.9,

I

to Generalizing the notion of a sequence, a net in a non-empty set X is a map f X where I is a directed set.

Locally convex spaces

301

(x(")) converges to x = (xk) in (w, r,,,) if and only if (x(n)) converges coordinatewise to x; that is, if and only if lim".... xk"1 = xk for each k E NO. For that reason r, is also called the topology of coordinatewise (or pointwise) convergence on w. A We now relate the notion of a bounded set in a locally convex space to that in a semi-normed space.

Definition and Theorem 6.4.19 (bounded set). Let (X, P) be a locally convex space and let A C X. Then A is called bounded in (X, P), if supxEA p(x) < oo holds for each p E P; that is, if A is bounded in the semi-normed space (X, p) for every p E P. The following statements are equivalent:

(a) A is bounded. (b) For every semi-norm p which is continuous on (X, P) the set A is bounded in the semi-normed space (X, p).

(c) For every U E U there exists an a > 0 such that A C a U. (d) -Lx,, -> 0 for each sequence (x") in A. The statements (b) and (c) show that the notion of boundedness in locally convex spaces depends only on the topology and does not depend on a special family of semi-norms which generates the topology.

Proof of 6.4.19. (a)

(b) : That the implication holds is an immediate consequence of the definition of boundedness and the characterization of continuous semi-norms p in (X, P) (cf. 6.4.11). (b) =::* (c) : Let (b) be satisfied, and let U be a neighbourhood of zero in (X, P). Then by Exercise 6.4.30(a) there exist a continuous semi-norm p and an e > 0 with UP C U. Putting a (1 + supxEAp(x)), then obviously a > 0 and A C aUE C aU, that is (c) holds. (a) : If (c) is satisfied and p E P is given, then for Up there exists (c)

an a > 0 with A C aUl, that is sup.EA p(x) < a < cc. (d) .. (a) : This is an immediate consequence of `(a) 4-* (c)' in 6.3.4. 0 Remark 6.4.20. From the definition of a bounded set in a locally convex space we get immediately that each finite set, every subset of a bounded set, the finite union of bounded sets and the set of all terms of a convergent sequence are bounded, since the same is true in semi-normed spaces. 0 Further examples of bounded sets are treated in Exercise 6.4.32. Example 6.4.21. If we consider a subset A of w, then it is bounded in (w, r,,,) if and only if it is contained in an (infinite-dimensional) `rectangular block' with finite edge length; that is, if and only if there exists a sequence

(caj), aj > 0, with

-aj
(x=(xk)EA, jEN°).

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Functional analytic basis

If in addition A C m, then A is bounded in (m, 11 supZEA IlxiI- < oo; that is, there exists an E > 0 with

-e<xj < E

if and only if

(x=(xk)EA,?ENO),

which means that A is a subset of an (infinite-dimensional) `cube' (with finite and constant edge length). A We have introduced locally convex spaces as a generalization of seminormed spaces. The next theorem gives us a characterization of those locally convex spaces which are semi-normable.

Definition and Theorem 6.4.22 (of Kolmogoroff). A locally convex space (X, r) is semi-normable, that is there exists a semi-norm on X which generates r, if and only if it has a bounded neighbourhood of zero.

Proof. If (X, r) is semi-normable by a semi-norm p, then the unit ball is a bounded neighbourhood of zero. Conversely, if U is a bounded neighbourhood of zero, then we may assume that it is closed and absolutely convex (cf. 6.4.32 and 6.4.5). Now, as one may prove, the Minkowski functional pu for U generates the (locally convex) topology r.

Examples 6.4.23. The locally convex spaces (w, r;,) and (C(R), P) in 6.4.3 are not semi-normable. Proof. We prove here that (w, r,,,) is not semi-normable whereas the (similar) proof for (C(R), P) is left to the reader. Obviously, (w, r,,,) is not semi-normable by the Kolmogoroff theorem if and only if each V E 6 is unbounded in where B is the canonical

neighbourhood basis of zero. Now, let v = n,1 Ufki (E B) with some n E N, E, > 0 and qki E P (i = 1,... , n) where P := (qj i j E N°) with qj (x) := (xj I (x = (xk) E w). We choose an N E l" \ {k1,. .. , k,, } and consider x(') := reN. Because qki (X( r)) = 0 (i = 1, ..., n), that is X(r) E V,

and gN(x(r)) = r we get that V is unbounded in (w,Thus, (w, T,) is not semi-normable.

Remarks 6.4.24. We have just verified that (w, -r,,) is not semi-normable and does not have a bounded neighbourhood of zero. However-as we will state in the next theorem-the topology r,,, can be generated by the metric d: w x w -4 R defined by 00

d(x, y)

E 2_? j=o

1xj - yj' l+Jxj -yjl

(x = (xk), y = (yk) E w) .

Thus the set U :_ {x E w f d(x, 0) < I } is a neighbourhood of zero in (w, r,,,) and bounded as a subset of the metric space (w, d). On the other hand, U is unbounded as a subset of the locally convex space (w, ru,). Therefore, we have to distinguish carefully between boundedness in semi11 metric spaces and boundedness in locally convex spaces.

Locally convex spaces

303

In addition to the semi-normability of locally convex topologies the question of their semi-metrizability is also of interest.

Theorem 6.4.25 (semi-metrizable lcs). If (X, r) is a locally convex space, then the following statements are equivalent: (a) (X, r) is semi-metrizable, that is there exists a semi-metric on X which generates r. (b) There exists an at most countable neighbourhood basis of zero. (b*) There exists an at most countable neighbourhood basis of zero which exclusively consists of closed and absolutely convex sets. (c) There exists an at most countable family of semi-norms which gener-

ates r. (d) There exists a translation invariant semi-metric which generates r. Additional statements: I f r is generated b y an at most countable f a m ilyP= , p , . ..) . . ) of semi-norms, then the Frechet combination d defined by

(X - y) d(x, y) := E 2-" 1 pn + p" (x - y)

(X, y

E X)

n-1,2,...

is a translation invariant semi-metric which generates r. Moreover, (X, r) is metrizable; that is, there exists a metric which generates r, if r is generated by an at most countable total family of semi-norms or, equivalently, it is generated by a translation invariant metric.

Proof. (a) . (b) : This implication holds since a neighbourhood basis of zero is given by the family of all balls with centre 0 and radius n .

(b) =*- (b*):LetC={C1I iEI} with I=N° or I=N NO, nEN°, be an at most countable neighbourhood basis of zero. Furthermore, let 13 be a neighbourhood basis of zero in (X, r) consisting of closed and absolutely convex sets (cf. 6.4.5(c)). Now, we may choose for each i E I a Bi E 13 with Bi C Ci and denote the family of all Bi (i E I) by 13`. Then 13` is obviously an at most countable neighbourhood basis of zero which includes exclusively closed and absolutely convex neighbourhoods of zero. (b*) (c) : If U = {Ul, U2, ...} is an at most countable neighbourhood basis of zero consisting of closed and absolutely convex sets, then, as hinted in the proof of 6.4.5, r is generated by the family of all Minkowski func-

tionals pu, (n = 1, 2, ...). (c) . (d) : Let P = (pl, p2, ...) be an (at most) countable family of semi-norms which generates r. To prove the implication `(c) (d)' it is sufficient to verify that the semi-metric d, which is defined in the additional statements and is obviously translation invariant, generates r. The proof is technical and is left to the reader. (d) . (a) : This implication holds trivially. In particular, the part `(c) => (d)' contains a proof of the first part of the

304

Functional analytic basis

additional statements. The second follows immediately from `(a) q (c)', `(a) = (d)', 6.4.7 and the fact that metric spaces are Hausdorff.

Examples 6.4.26. By 6.4.3 and 6.4.25(c) we get that (C(R), P) and (w, r,) are semi-metrizable; they are even metrizable since in each case the family of semi-norms generating the topology is total (cf. 6.4.8).

a

Remark 6.4.27. On the basis of 6.4.12 and 6.4.25, respectively, we may verify that the definition of the product of locally convex spaces in 6.4.9 is consistent with the finite product of semi-normed spaces and with the product of at most countably many semi-metrizable locally convex spaces.

We do not deal further with it. However, let us note that it is true in particular for (w, r,J) in 6.4.9.

L1

Completing the considerations on the metrizability of locally convex spaces, we now give an example of a sequence space endowed with a translation invariant metric which does not generate a locally convex topology.

Example 6.4.28. For 0 < p < oo we defined in 6.3.8(h) the linear space P of all absolutely p-summable sequences and considered the PP-norm in the case 1 < p < oo. Now, for any 0 < p < 1 the map dp defined by dp(x, y) := 1: 1xk - ykjP

(x = (xk), y = (yk) E P)

k

is, as one may show, a translation invariant metric and the topology r generated by dp is not locally convex, thus not normable, as we now verify. If r were be locally convex, then by 6.4.5(c), there would exist an absolutely convex neighbourhood U with U c {x E PP I dp(x, 0) < 11 =: U1 and for U an e > 0 with U, := {x E Pp I dp(x, 0) < e} C U. This would imply for 9:= 2 and e' the statement 51"Pe' E UE C U (r E N°) and therefore y(3)

61!p

J+1 r_0 er E U

where we make use of the fact that U is absolutely convex and that for an absolutely convex set U, with x1, ... , x,, E U and al, ... , an E K, the implication

n

n

i=1

i.1

aixi EU holds. On account of 0 < p < 1, this contradicts

dp(y(J), 0) = b (j + 1)-P(j + 1) = 5 (j + 1)1-p > 1, if j E No is chosen sufficiently large.

Locally convex spaces

305

In the additional remarks in 6.4.25 we stated that in the case of locally

convex spaces (X, P), where P = (pl, p2, ...) is an at most countable family of semi-norms, the Frechet combination d defined by

d(x, y) :_ E 2-n n=1,2,...

Pn(x - y)

(x, y E X)

1 + Pn (x - y)

is a translation invariant semi-metric which generates the topology r of (X, P). Now, we proceed to see whether d is generated by a semi-norm, that is we consider the map

t s: X -+ R: x ---+ d(x, 0) = E 2-n n=1,2,...

Pn(x) 1 +Pn(x) .

In general, ! ! is not a semi-norm except when pn = 0 (n = 1, 2, ... However, it is called a para-norm which we now define.

Definition and Remarks 6.4.29 (para-norm). Let X be a linear space. Then a map ! : X -> R is called a para-norm (on X) if it !

satisfies the following conditions for all x, y E X :

(a) !x! > 0 and !0! = 0. (b) ! - x! =!x!. (c) !x + y! < !x! +!y!. (d) If (tn) is a sequence in K with to -* t E K and (xn) is a sequence

in X with !xn - x! --3 0, then !tnxn - tx! -+ 0. A para-norm is called total, if !x! = 0 implies x = 0. The following remarks are obviously true: (1) Each semi-norm (norm) is a para-norm (total para-norm). (2) If ! ! is a para-norm (total para-norm) then

d!!:X xX-3R, (x,y)--*Ix-y! is a translation invariant semi-metric (metric). On that score, each paranorm ! ! generates a topology (which is generated by a translation invariant semi-metric, namely by d! ! ).

(3) If (X, P) with an at most countable family P = (pl, p2i ...) is a locally convex space, then ! t

2-n P. (x)

: X -+ !R : x --+ d(x, 0) n=1,2,...

1 +Pn(x)

is a para-norm and it is total if P is total. (4) A locally convex space (X, r) is semi-metrizable if and only if r is generated by a para-norm (in the sense of (2)). A

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Functional analytic basis

Exercise 6.4.30. Let (X, P) be a locally convex-space, M C X, x E X and P, := (p: X ---> R I p is a continuous semi-norm on (X, P)) . Prove the following statements:

(a) The set B

UP

e > 0 and p E P.) is a neighbourhood basis of

zero in (X, P). (b) x is an adherent point of M in (X, P) if and only if x is an adherent point of M in (X, p) for each p E P,.

(c) For the closure M of M in (X, P) the identity v = ()PE p M(X'p) denotes the closure of M in (X , p). (d) The closure of a linear subspace of X is again a linear subspace. M(X'P)

holds where

Exercise 6.4.31. Let (X, P) be a locally convex space, and let IC :=

16Kfi InE N, ei>0 and piEP(i=1,...,n) i=1

,

where KE := Kf (0) and KE (x) := {y E X I p(x - y) < e} denotes the closed e-neighbourhood of x E X relative to p E P.

Verify that the elements of k are closed in (X, P) and that K is a neighbourhood basis of zero in (X, P) which satisfies the properties (1), (2) and (3) in 6.4.5(c).

Exercise 6.4.32. Prove, in the case of a locally convex space (X, P), the following statements:

(a) If .A E K and if A and B are bounded subsets of (X, P), then A+ B, A and AA are bounded too. (b) Each compact subset of X is bounded. Bibliography: [209], [250]

6.5

Continuous linear maps and the dual space of a locally convex space

In Section 6.4 we learnt that, as in the case of semi-normed spaces, the topology of locally convex spaces is consistent with the structure of linear spaces. As a consequence of that we have that the continuity of semi-norms can be characterized similarly as in the case of semi-normed spaces. Thus, we expect that we may also characterize continuity of linear maps between locally convex spaces in an similar way to that in the case of semi-normed spaces. We now confirm that expectation with the following theorem.

Theorem 6.5.1 (continuous linear maps). Let (X, P) and (Y, Q) be locally convex spaces, and let T : X --> Y be a linear map. Then the following statements are equivalent:

Continuous linear maps and the dual space of a locally convex space 307

(a) T is continuous. (b) T is continuous at zero. (c) For each q E Q the semi-norm q o T is continuous.

(d) VgEQ 3nEN 3p1i...,pnEP 3M>O VxEX:

n q(Tx) < M E pj(x). j=1

Proof. The equivalence of (a) and (b) follows immediately from the linearity of T and from 6.4.5(a) and (b), respectively. (a) (c) : Each q E Q is continuous on (Y, Q) by 6.4.11. Thus q o T is continuous as the composition of continuous maps. (c) .. (d) : For every semi-norm q on Y the composition map q o T is a semi-norm on X. Therefore, we get the equivalence of (d) and (c) if we (c)' in 6.4.11 to q o T instead of q. apply `(a) (c) (b) : Here we adapt the proof of `(c) (b)' in 6.4.12. There we replace qj by qj o T and statement (b) by

VUE5Q 3VE13p: T(V)CU where 13p and 5Q denote the canonical basis of (X, P) and (Y, Q), re-

0

spectively.

Analogously to the case of semi-normed spaces we define the (topological) dual space.

Definition 6.5.2 (dual space). If (X, r) (respectively, (X, P)) is a locally convex space, then we call the set of all continuous linear functionals

on (X, r) the dual space or the dual of (X, r) (respectively, (XP)) and denote it by (X, r)` (respectively, (X, P)`) or by X' in short, if there is no confusion.

As is well known, linear functionals are practically determined by their kernels and the image of one point outside of the kernel. Moreover, as we state in the following theorem, the continuity of linear functionals is also characterized by their kernels.

Theorem 6.5.3 (continuous linear functionals). If (X, P) is a locally convex space and f : X -+ K is a linear functional, then the following statements are equivalent:

(a) f is continuous (that is, f E X'). (b) Kern f is closed in (X, P).

(c) 3nEN

n

3M>OVxEX : If(x)I <MEpj(x). j=1

Proof. The implication `(a) = (b)' holds since Kern f is closed as a preimage of the closed set {O} under the continuous map f. The statement `(a) q (c)' corresponds exactly to the special case '(Y, Q) :_ (K, 11 (})' of

308

Functional analytic basis

(d)' in 6.5.1. Therefore it remains to verify '(b) (a)'. For that let f : X ---p K be linear and Kern f be closed in (X, P). We may suppose Kern f # X because the zero map is trivially continuous. We choose a z E X with f (z) # 0, that is z % Kern f. Since Kern f is assumed to be closed in (X, P), we may choose a U E Bp with 6 = (z + U) f1 Kern f. Now, if f is discontinuous, then f (U) = K follows from Exercise 6.5.17. In particular there would exist a y E U with f (y) = -f (z), which would `(a) '

imply f(z+y) = f(z)+f(y) = 0, that is y+z E Kernf, and z+y E z+U contradicting the choice of U. Thus f is continuous.

O

An essential tool in functional analysis is the Hahn-Banach theorem concerning extensions of continuous linear functionals. We now generalize it to the case of locally convex spaces.

Theorem 6.5.4 (Hahn-Banach). Let (X, P) be a locally convex space, Y be a linear subspace of X and f : Y --* K be a linear functional on Y. If f is continuous on (Y, Ply), that is (cf. 6.5.1 and 6.5.3) if n

If(x)l<MEpj(x) for eachxEY

(6.5:1)

.i=1

where pi, ... , pn E P and M > 0 are suitably chosen, then there is a continuous linear functional g : X -4 K such that n

gly = f and lg(x)I < M E pi(x) for every x E X .

(6.5:2)

Proof. The extension of a linear functional f which enjoys (6.5:1) to a linear functional g which satisfies (6.5:2) is an immediate application of the Hahn-Banach theorem 6.3.24 in the case of semi-normed spaces because a finite sum of semi-norms is also a semi-norm. Here, the continuity of g 0 follows from `(c) . (a)' in 6.5.3.

Now, we draw some corollaries from the Hahn-Banach theorem.

Corollary 6.5.5 (of 6.5.4 and 6.5.3). If (X, P) is a locally convex space and Y is a closed (strict) subspace of (X, P), then for each xo E X \Y

there exists a g E X' such that g(xo) = 1 and gay = 0. Proof. For any xo E X \Y we consider Z:= {axo+y I a E K and y E Y} and the linear functional f : Z ---* K, axo + y -+ a. Since Y = Kern f is closed in (X, P) and thus in (Z, PIz), the functional f is continuous by 6.5.3. Obviously, each continuous extension of f has the desired properties, 0 and Theorem 6.5.4 gives an continuous extension.

The next corollary gives us an answer to the question of whether there exist non-trivial continuous linear functionals on a given locally convex space.

Continuous linear maps and the dual space of a locally convex space 309

Corollary 6.5.6. If (X, P) is a locally convex space and P contains a semi-norm p # 0, then X' ,-f {0}.

Proof. Let p E P and xo E X with p(xo) # 0 be arbitrarily given. Then Y := Kemp is closed since p is continuous and we may apply 6.5.5 to get

an f E X' with f (xo) = 1, that is with f # 0. Corollary 6.5.7 (decomposition of linear functionals). Let (X, P) be a locally convex space, and let f E X', that is n If(x)j

< MEp1(x)

(x EX)

for some M > 0, n E N and pi, ... , pn E P. Then there exist (depending on M, n and pl, ... , pn) functionals fl, ... , fn E X' with n

f = E fa

and l fj(x) I < M pi (x)

(x E X , j E Nn ).

j=1

Proof. In the case n = 1 there is nothing to prove. We may suppose M = 1 since otherwise we replace f by f. If we have proved the statement in

the case n = 2, then the case n > 2 follows immediately by induction. Thus we consider the situation

If(x)I < p(x) + q(x)

(x E X),

where p, q E P are suitably chosen. Obviously,

h : X x X --* R, (x, y) - p(x) + q(y) is a semi-norm. Moreover, by setting

g((x,x)) := f(x)

(x E X) (x, x) I x E X } of X x X, a linear

we define, on the linear subspace Y functional g which satisfies

Ig((x,x))j < h((x,x))

((x, x) E Y).

Applying the Hahn-Banach theorem we may extend g to a linear functional

G on X x X such that IG((x, y)) 1 < h((x, y)) = p(x) + q(y)

((x, y) E X x X)

holds; in particular, G E (X x X, h)'. As one may easily check, the continuous linear functionals fl and f2 on X defined by

fj(x) := G((x, 0))

and f2(x) := G((0, x))

satisfy the conditions

f = ft + f2, jfi(x)j < p(x) and

I f2(x)I < q(x)

Thus, fl and f2 have the desired properties.

(x E X).

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Functional analytic basis

Remark 6.5.8. If (X, P) is a locally convex space, then

U (X,p)') = (X, P)' pEP

follows immediately from 6.5.7.

Ll

Next we discuss the closure and the closedness of the sum of two linear subspaces of a locally convex space, where one of them is finite-dimensional.

Theorem 6.5.9. Let (X, P) be a locally convex space, and let V and W be linear subspaces of X. Then the following statements hold: (a) If V is closed and W is finite-dimensional, then V + W is closed.

(b) V+(u) = IVED(U) if u O V

(u E X).

Proof. (a) Since we will apply (a) only in the case of a metrizable space (X, P) we give a proof only for that case. Since W is finite-dimensional, W is obviously the finite sum of onedimensional linear subspaces of X. Furthermore, because (V+WI)+W2 = V + (WI + W2) holds for all subspaces WI and W2 of X, it is sufficient to prove (a) in the case that dim W = 1 and W ¢ V. (Note, if W C V, then the statement holds trivially.) Now, let W = (w) with w 0 V, and let (xn) be a sequence in V + W which converges in (X, P), say to x E X. Since (X, P) is metrizable, the closedness of V + W is proved if we verify that x E V + W. By our assumptions we may choose for every n E N a vn E V and an an E K with xn = vn + an w. If we knew that (an) converged in (K, 1 1), say to a E K, then we could conclude that (an w) converges in (X, P) to aw E W. Consequently, (vn) = (xn - an w) would converge to x - a w and, since V is closed, we would get x - a w E V, so x E V + W. However, since V is closed, by 6.5.5 there exists a g E X' with g(w) = 1 and giv = 0. Because g is linear and (sequentially) continuous we get by the convergence of (xn) to x the statement an = g(vn) + an g(w) = g(xn) --+ g(x)

(n -+ oo).

(b) For an arbitrary linear space V and any u E X we prove V + (u) _ V + (u). By (a), applied to the closed linear subspace V (cf. 6.4.30(d)), V + {u) is closed, that is V + (u) C V + (u) = V + (u). To prove V + (u) c V + (u) we consider an f E X' with V + (u) C Kern f. We have V C Kern f = Kern f (cf. 6.5.3) and u E Kern f. Thus we obtain V+(u) C Kern f ; moreover, 6.5.21, applied to M := V + (u) and S := V + (u), gives us V + (u) C V + (u). Hence we have V + (u) = V + (u). The next theorem contains a full description of all locally convex topologies on finite-dimensional linear spaces. For that we need the notion of a topological isomorphism.

Continuous linear maps and the dual space of a locally convex space

311

Definition 6.5.10 (topological isomorphism). Let (X, TX) and (Y,Ty) be locally convex Hausdorff spaces. Then (X, TX) and (Y,ry) are called topologically isomorphic, if there exists an isomorphism

T : (X,rx) -* (Y,ry) such that T and

T_1

are continuous. In such

a case, T is called a topological isomorphism. Theorem 6.5.11. Let (X, T) be a locally convex Hausdorff space, and let dim X = n for a certain n E N. Then the following statements hold: (a) (X,r) is topologically isomorphic to (Kn, 11 111). (b) If q is a locally convex Hausdorff topology on X, then T = 17. (c) (X, r) is a Banach space.

Proof. The details of the proof are left to the reader (cf. 6.5.18 and [252, Section 6-3)).

(a) If n = 1, then take an xo E X \ {Q} and consider the natural isomor-

phism defined by axo -+ a (a E P. Applying 6.5.9(a), the statement follows for n E N by the method of induction. (b) The identity map i : (X, r) ---+ (X, 77) is a topological isomorphism as a composition of topological isomorphisms (cf. (a)). (c) If T : (X,r) -> (K", II 111) is a given topological isomorphism, then 11 := 11 II1 oT is a norm (which generates r). The completeness of (X, II II) is obvious, since (T(xn)) is a (convergent) Cauchy sequence in (K, 11 111), if (xn) is a Cauchy sequence in (X,1111), and T-1 is continuous. 0 11

We consider some examples involving the dual space of the locally convex space in 6.4.15.

Example 6.5.12. Let X, Y be sequence spaces with cp C X and cp C Y C Xa, and let rp,, be the locally convex topology generated by Py :_ (py I y E Y) where the semi-norm py is defined as in 6.4.15. Then the following statements hold:

(a) x = >k xke'k for each x = (xk) E X (relative to rp,,, cf. 6.4.16). In particular, W is dense in (X, rp,, ).

(b) Every f E X' has the representation

f(x) _ I: xkf(ek)

(x = (Xk) E X),

k

and Y is (up to isomorphism) a linear subspace of the dual space of (X, rp,, ), where the isomorphism is defined by

T:Y-+X', y=(yk)--+fy

with

fy(x)Exkyk(xEX).

(6.5:3)

k

(In 6.6.7 we will state that even Y = (X, rp,, )' up to isomorphism holds.) The proofs of the statements are left to the reader (cf. 6.5.19).

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Functional analytic basis

A further important application of the Hahn-Banach theorem is the separation theorem which is of a geometric nature: the idea is to separate two disjoint convex subsets by a hyperplane. We start with some definitions.

Definition and Remark 6.5.13. Let X be a linear space over K. (a) A subset H of X is called a hyperplane of X if there exist a real linear functional f $ 0 and an a E R such that H = {x E X I f (x) = a}. In this case, we have H = xo + Kern f and f (xo) = a. In particular, if X carries a locally convex topology r, then H is closed in (X, r) if and only if f is continuous. (b) In addition, let H be a hyperplane of X which is determined by the

linear functional f : X -4 R so that H = {x E X I f (x) = a), and let A and B be subsets of X. Then we say that H separates A and B

:4=* VxEA VyEB: f(x)a or

VxEA VyEB: f(x)>aand f(y)
H strictly separates A and B

VxEA VvEB: f(x)a

or

V X E A V y E B: 1(z)> a and f (y) < a.

Theorem 6.5.14 (separation theorem). Let (X, r) be a locally convex space, and let A and B be non-empty disjoint convex subsets of X. Then the following statements hold: (a) If A is open, then there exists a closed hyperplane which separates A

and B. (b) If A and B are open, then there exists a closed hyperplane which strictly separates A and B. (c) If A is compact and B is closed, then there exists a closed hyperplane which strictly separates A and B. We omit the proof since it is not important for an understanding of the subsequent considerations. In the last part of this section we investigate the continuity of families of linear maps, which results in the notion of equicontinuity. We refer to the corresponding notion in the case of semi-metric and semi-normed spaces (cf. 6.3.33) and remark that the notion, which we now introduce in the case of locally convex spaces, is consistent with it.

Definition and Remark 6.5.15 (equicontinuity). Let (X, P) and (Y, Q) be locally convex spaces, and let P be a non-empty family (or set)

of linear maps of X into Y. Then fi is said to be equicontinuous if for each neighbourhood U of zero in (Y, Q) there exists a neighbourhood V of zero in (X, P) such that

Continuous linear maps and the dual space of a locally convex space 313

VTE4' : T(V)CU.

(6.5:4)

In such a case, every T E 4' is continuous at zero, and hence at each x E X.

Note that in (6.5:4) T(V) C U holds simultaneously for all T E 4'. That is, a single neighbourhood V works for all T E 4'. A Generalizing 6.5.1 we get the following characterization of equicontinuity of families of linear maps between locally convex spaces.

Theorem 6.5.16 (equicontinuity). Let (X, P) and (Y, Q) be locally convex spaces with neighbourhood bases Bx and By of zero, and let 4' be a non-empty family of linear maps from X into Y. Then the following statements are equivalent: (a) t is equicontinuous.

(b) bUEBy 3VEBx VTE41 : T(V)CU.

(c) `dgEQ3nEN3p1i...,p,tEP3M>OVTE4VxEX: q(T(x))

<M

n

> pj(x). j=1

Proof. The proof that `(a) t* (b)' follows immediately from the definition of a neighbourhood basis. The proof that `(a) q (c)' is similar to that of

0

6.4.11. We omit it.

Exercise 6.5.17. Let (X, P) be a locally convex space and f : X -+ K be a linear functional. Prove the equivalence of the following statements:

(a) f is discontinuous. (b) b U E Bp : supXEU If (X) I = oo.

(c) b'UEBp : f(U)=K. Exercise 6.5.18. Give a detailed proof of 6.5.11. Exercise 6.5.19. Verify the statements in 6.5.12.

Exercise 6.5.20. Prove that a locally convex space (X, P) is Hausdorff if and only if for all x E X \ {0} there exists an f E X' with f (x) $ 0. Exercise 6.5.21. Let (X, P) be a locally convex space, and let S and M be non-empty subsets of X. Show the following equivalence.

M C (S) t

b f E X' : (S C Kern f

M C Kern f ).

Remark: A non-empty subset S of a locally convex space (X, P) is called fundamental if the closure of the linear span of S equals X, that is if {S) = X. Therefore, considering M := X in the preceding equivalence we get that S is fundamental if and only if S C Kern f implies f = 0 for each f E X'.

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Functional analytic basis

Exercise 6.5.22. Let (w, r(,) be defined as in 6.4.3(b), and let P be a non-empty subset of w'. Prove that 1 is equicontinuous if and only if

3noEN Vn>no `d f Et): f(en)=0 and sup { l f (et)I If E 4*

and n E N° } < oo.

Bibliography: (209], [250]

6.6 Dual pairs and compatible topologies An important method in functional analysis consists of examining X by investigating its dual space X'. In this connection it has proved to be useful to consider a variety of locally convex topologies on X and X'. Similar to considerations in linear algebra, the bilinear form

(, ) : X x X' -4 K, (x, f) -+ f (x)

(6.6:1)

plays a distinguished role. We make the following definition of a dual pair.

Definition 6.6.1 (dual pair). Let X and Y be linear spaces over K, and let (,) : X x Y -> K be a bilinear map satisfying

VxEX\{0} 3yEY: (x, y) #0

(6.6:2)

b y E Y \ {0} 3 x E X : (x, y) 54 0.

(6.6:3)

and

In this situation we call (X, Y) a dual pair. We have already encountered some examples of dual pairs:

Examples 6.6.2. (a) Let (X, P) be a locally convex Hausdorff space with dual space X. Then (X, X') with the bilinear map defined by

(x, f) := f (X)

(x E X, f E X')

(6.6:4)

is a dual pair. We see that (6.6:2) and (6.6:3) are satisfied by 6.5.20 and by the definition of 'f 54 0', respectively. Subsequently, by the dual pair (X, X') we always mean the dual pair (X, X') together with the bilinearmap defined in (6.6:4).

(b) If X is a vector space over K and X* is its algebraic dual (that is, the set of all linear functionals on X ), then (X, X *) is a dual pair, where the bilinear map is defined analogously to (6.6:4). Thus, we get (6.6:2) as follows. If x E X \ {0} is given, then we complete {x} to an (algebraic)

basis B of X and define f E X' on B by f (x) = 1 and f (z) = 0 if

Dual pairs and compatible topologies

315

zEB\{x}. (c) The pair (w, gyp) together with its natural bilinear map defined by (x, y)

1: xkyk

(X = (xk) E U j, y = (yk) E W)

k

is obviously a dual pair.

(d) If (X, Y) together with a bilinear map (, ) : X x Y -+ K is a dual pair, then (Y, X) together with

(, )* : Y x X -+ K, (y,x) -+ (y,x)* := (x,y) is a dual pair too. In particular, in the case of any locally convex Hausdorff

space (X, P) with dual space X', the pair (X', X) with the bilinear map defined by

(f E X', X E X)

(f, X) := f(x)

is a dual pair. Subsequently, we always understand the dual pair (X', X) to be the pair (X', X) together with this bilinear map. p

As in the examples (a) and (b), in the case of dual pairs (X, Y) there results a natural connection between Y and a linear subspace of X* and analogously between X and a linear subspace of Y*.

Remark 6.6.3. Let (X, Y) be a dual pair. Then

and

T:Y--_X*, y-a fy

with

fy:X -4K, x-+(x,y)

T:X_*Y*,x-3fx

with

ff:Y-+K, y--+ (x, y)

are injective linear maps from Y into X* and from X into Y*, respectively, as one may check. (Note, the injectivity of T and T comes from (6.6:3) and (6.6:2).) Thus Y and X are up to isomorphism a subspace of X * and Y*, respectively.

a

For duality theory, that is the investigation of dual pairs, the notion of the weak topology, which we now introduce, is fundamental.

Definition and Remarks 6.6.4 (weak topology). Let (X, Y) be a dual pair. For each y E Y the map

py:X -4 R, x->py(x):=I(x,y)I is obviously a semi-norm on X. The locally convex topology generated on X by the family (py I y E Y) of semi-norms is denoted by u(X, Y) and is

called the weak topology on X (of the dual pair (X, Y) ). The weak topology (X, o, (X, Y)) is Hausdorff since (py I y E Y) is total by (6.6:2).

316

Functional analytic basis

Because (Y, X) is according to 6.6.2(d) a dual pair if (X, Y) is, v(Y, X)

is well-defined and is called the weak topology on Y (of the dual pair (Y,X)). In particular, in the case of a locally convex Hausdorff space (X, P) with

dual space X', the weak topologies o(X, X') and o(X', X) are defined on X and X', respectively (cf. Example 6.6.2(a)). A

Remarks 6.6.5. Let (X, Y) be a dual pair. (a) By 6.4.14 we get that the weak topology o(X, Y) of the dual pair (X, Y) is the weakest locally convex topology such that each p E P := (p, I y E Y) (py as in 6.6.4) is continuous. (b) For a given y E Y and a given locally convex topology r the linear functional

fv:X -+ K, x-+ (x,y) is continuous on (X, T) if and only if p,, is continuous at zero. This is because pu(x - z) = Iff(x - z)I = Iff(x) - fy(z)I

(x,z E X).

Thus, we get from part (a) that o(X, Y) is the weakest topology on X such that each functional in

T(Y) = {f EX' 1 3yEY tlxEX : f(z)=(x,y)} (cf. 6.6.3) is continuous, that is T (Y) C X'. In particular, the space Y is (up to isomorphism) a linear subspace of (X,o(X,Y))'. (c) Further to the statements in (a) and (b) we may even have that o(X, Y) is the weakest topology on X, for which all f E T (Y) are continuous. (The proof of this statement is left to the reader in Exercise 6.6.25.) A To prove Y = (X, o(X, Y))' up to isomorphism we need a result from linear algebra which is proved for example in [209, p. 32, Lemma 5]. We state this result in the following lemma.

Lemma 6.6.6. Let X be a linear space and fo, ... , fn E X*. Then either fo

is a linear combination of f,,. .. , fn

or

3 a E X : fo(a) = 1 and f? (a) = 0 (j = 1, ... , n). Theorem and Remark 6.6.7. If (X, Y) is a dual pair, then we have Y = (X, o(X, Y))' up to isomorphism where an isomorphism is given by the map T defined in 6.6.311. By 6.6.5(c) the weak topology o(X,Y) is the weakest topology with that property (with respect to T). 11 As usual in analysis we do not distinguish between the

T:Y -+Z with T(Y)C ZC X'.

map T : Y -+ X` and

Dual pairs and compatible topologies

317

Proof. Let T : Y ---+ X* be defined as in 6.6.3. By 6.6.3 the map T is linear and injective. Furthermore, we have already verified T (Y) C (X, a(X, Y) )' in 6.6.5. Thus it remains to show T (Y) D (X, a (X, Y) )' . For that let f E (X,a(X,Y))' be given. On account of the continuity of f at zero (cf. 6.4.1), for each a with 0'< e < 1 there exist 5k,.. . , 5n > 0 and y1, ... , yn E Y such that n

n

If

Ub, '

< e for all x E V := ;_1

up ='

(6.6:5)

;-1

where pa is defined as in 6.6.4 and zj := -1 yj (j E Nn). We now put

fo := f and ff := Tz, (j E Nn). By Lemma 6.6.6 either f is a linear combination of fl, . . . , fn or there exists an a E X such that f (a) = 1 and f? (a) = 0 (j E Nn). The last statement contradicts (6.6:5) because 0 = I fj (a) I = pz, (a), which means a E V. Thus the first statement holds and we have for some Al, ... , An E K the statement n

n

f = E Aifi = E AJTz? = T > AJzi j=1

j=1

that is, f is the image under T of y := E

j=1

alzj E Y, which in turn

implies that (X, a(X, Y))' C T (Y). This is what we had to prove.

O

Example 6.6.8. Let X and Y be sequence spaces with cp C X and ,p C Y C XO. Then (X, Y) together with the bilinear map defined by (2, y)

1: xkyk

(x = (xk) E X, y = (ilk) E Y)

k

is a dual pair. (Note, (, ) is defined since Y C XQ, and since (6.6:2) and (6.6:3) are satisfied because cp c X fl Y.) The topology a(X, Y) is exactly the topology rp,, considered in 6.4.15, and T in 6.6.3 equals T in 6.5.12 whose inverse T-1 : X' ---> Y is given by f - * (f (ek)), f E X', as one may easily check with 6.5.12(b). In particular, a(w,
0

Remark 6.6.9. If (X, r) is a locally convex Hausdorff space with dual space X', then Theorem 6.6.7 applied to the dual pair (X, X') tells us that the weak topology a(X, X') is weaker than T. In particular, we get that convergence of a sequence in X with respect to r implies weak convergence, that is convergence with respect to a(X, X'). The converse implication fails in general as we now verify. We consider the space (co, 11 II,,) and the sequence (ek). Since Ilek - eT = 1 if k 54 r, (ek) is not convergent in

318

Functional analytic basis

(co, 11 11.). We now prove that (ek) is weakly convergent to zero. For that we use the fact that n xEnj

Exkek = (xo, ... , xn, 0, ...)

for each x E co

x (II IIoO)

k=1

n->qo

0 when x = (xk) E co. Thus we - xIj = supk>n+1 Ixkl have x = E' 1 xkek for x E co. Hence, for each f E c' we get since

11X[n]

n

for each x E co.

f (x) = l nm f (xjni) = l nm E xk f (ek) = E xk f (ek) k=1

k

In particular, (f (ek)) E co = t C co (cf. 2.3.3(a)); thus f (e k)

k- 0

for every f E co'.

The weak topology a(co, co`) is generated by the semi-norms

pf :co-*R, x---+I(x,f)I =If(x)I

(f Eco')

(cf. 6.6.4 and 6.6.2(a)). Consequently pf(ek - 0) = If(ek)I _+ 0

(k -* oo)

for every f E co',

that is (cf. 6.4.16) ek --+ 0 in (co, a(co, co')).

A

However, as we will show in Theorem 6.6.24, there exist non-trivial examples of locally convex spaces (X, r) for which weak convergence of sequences implies convergence in the topology r. We now return to the question of whether, for a given dual pair (X, Y),

there exist in addition to a(X,Y) other locally convex topologies r on X with Y = (X, r)` (up to isomorphism). As we have already stated in 6.6.5(b) such topologies are necessarily stronger than a(X,Y) if the isomorphism between Y and (X, r)` is the map T defined in 6.6.3.

Definition 6.6.10 (dual topology). Let (X, Y) be a dual pair and r be a locally convex topology on X. Then r is called a topology of the

dual pair, or compatible with the dual pair (X, Y), if Y = (X, r)' holds (up to isomorphism according to the map T defined in 6.6.3).

Remark 6.6.11. Since each compatible topology of a dual pair (X, Y) is stronger than the weak topology and the weak topology is Hausdorff, each IL compatible topology of (X, Y) is also Hausdorff.

Example 6.6.12. Let (X, P) be a locally convex Hausdorff space with dual X'. Then the topology rp generated by P is a compatible topology of the dual pair (X, X'), because in this case the map T in 6.6.3 is the

embedding T : X' -i X*, f --+ f.

A.

Dual pairs and compatible topologies

319

Theorem 6.6.13. Let (X, Y) be a dual pair, and let A be a convex subset (for example, a linear subspace) of X. Then the closure q of A is the same in all topologies compatible with the dual pair (X, Y).

Proof. For an arbitrary topology r compatible with the dual pair (X, Y), we(X'Y) prove that the closure T of a convex set A is equal to its weak closure

We have T C A(X'Y} since r D o(X, Y). Now, let xo A . Applying Theorem 6.5.14(c) we get an f E X' such that f (xo) 0 f (A). Thus, there exists a S > 0 with If (xo - x) I > 5 for all x E A. Then U {x E X I If (x) I < 5} is a neighbourhood of zero relative to o(X, X') _ o, (X, Y) and we have An (xo + U) = 0. Consequently, xo

we have proved A D A (X'Y). Hence, A =

A(x,Y}

Thus

0

Before we continue the study of compatible topologies we give a further method of generating locally convex topologies.

Definition and Remarks 6.6.14 (bounded sets, polar topology). Let (X, Y) be a dual pair.

(a) If 0 0 M C Y, then M is weakly bounded, that is M is bounded in the (locally convex) space (Y, o(Y, X)), if and only if V X E X : sup ((x, y)I < oo

(cf. 6.6.2(d) and 6.4.19).

yEM

(b) If 0 # M C Y is weakly bounded, then a semi-norm qM on X is obviously defined (cf. (a)) by qM(x) := Sup,EM I(x,y)I (x E X). (c) Let M be a collection of non-empty weakly bounded subsets of Y with Y = UMEM M. Then the topology rQ on X generated by the family

Q:= (qj I ME M)

(qr defined as in (b))

of semi-norms is called the topology of uniform convergence on the elements of M. Each topology generated in this way is also called a polar topology. The weak topology o(X, Y) is a polar topology on X, namely the topology of uniform convergence on one-element subsets (or, equivalently, on finite subsets) of Y. Because Y = UMEM M, the topology of uniform convergence on the elements of M is stronger than a(X, Y) and therefore

tl

Hausdorff since o ,(X, Y) is.

We do not explain here the motivation of the notion polar topology; however, it is easy to explain the notion of the topology of uniform con-

vergence on the elements of M. Namely, a sequence (x,,,) converges in (X, TQ) if and only if there exists an x E X such that qM (x,, - x) = sup I (xn - x, y) I --i 0 (n -> oo), yEM

320

Functional analytic basis

that is I (xn - x, y) I -> 0 (n -+ oo , uniformly for y E M) holds for each M E M. Because of their importance we emphasize, besides the weak topologies, two further polar topologies.

Definition and Remark 6.6.15 (strong and Mackey topology). Let (X, Y) be a dual pair. (a) The strongest polar topology on X, namely the topology of uniform convergence on all non-empty weakly bounded subsets of Y, is called the strong topology and is denoted by #(X, Y). (b) Let M be the set of all non-empty, weakly compact (that is, a(Y, X)compact), absolutely convex subsets of Y. Then the topology of uniform convergence on the elements of M is called the Mackey topology on X and is denoted by r(X,Y). It is well-defined by 6.6.14(c). We have Y = UMEM M since for each y E Y the set M.:_ {Ay ! JAI < 1} is absolutely convex and a(Y, X)-compact because T : (IS, 1) -+ (Y, o(Y, X)), A -+ Ay is continuous and Mb = T ({A E K I JAI < 1)) is a(Y, X)-compact as the image of the compact set {A E K I CAI < 1). Moreover, each o(Y,X)compact set is a(Y, X)-bounded by 6.4.32(b). A locally convex space is called a Mackey space if it carries the Mackey topology. (c) o(X, Y) C r(X, Y) C ;6(X, Y). (d) If dim X < oo, then all polar topologies on X coincide, in particular o(X,Y) = r(X,Y) = 9(X, Y), cf. Theorem 6.5.11.

(e) Obviously, r(X,Y) = 1(X,Y) if and only if each o(Y,X)-bounded A

and closed subset of Y is a(Y, X)-compact.

A large class of spaces carrying a polar topology is the class of all locally convex spaces as we will state in the following theorem.

Theorem and Notation 6.6.16 (lcs and polar topologies).

Let (X, P) be a locally convex Hausdorff space with topology rp and dual X'. Then rp is a polar topology of the dual pair (X, X'). More precisely, rp is the topology of uniform convergence on the elements of M where M is the set of all non-empty equicontinuous families 4; of linear functionals on (X, P). That is (cf. 6.5.16), 41 # 0 and

3nEN 3p1,...,pnEP 3M>O Vf E4> b'xEX if

n

(x)I < M E pi (x). (6.6:6) j=1

For that reason, rp is also called the topology of uniform convergence

on equicontinuous subsets of X.

Dual pairs and compatible topologies

321

Proof. By the definition of the elements 4+ E M we get two statements: (i) '1 is weakly bounded since (cf. 6.6.14(a) and 6.6:6) n

pj(x) < oo

sup If (x)I < M

(x E X).

?=1

(ii) For each qt (defined as in 6.6.14(b)) n

qt (x)

<MEpj (x)

(x E X),

i=1

that is q4, is continuous on (X, Tp). 41 (note, If} E M holds for each f E X') Now, by (i) and X' = the topology TM of uniform convergence on the equicontinuous subsets of X' is well-defined. Moreover, (ii) and 6.4.12 imply TM C T. To prove the converse inclusion r C TM it is sufficient (by 6.4.12) to

verify that for each p E P there exists a -t E M with

p(x) < %(x)

(x E X).

(6.6:7)

Let any p E P be given. Applying the Hahn-Banach theorem 6.5.4 we get

V a E X 3f. EX': (fa(a) = p(a) and b x E X: Ifa(x)I C p(x)). The set 4§ of all such functionals fa (a E X) satisfies the condition in 0 (6.6:6), that is ' E M, and the desired inequality (6.6:7) holds. In the case of a normed space (X, p) the dual space together with the operator norm 1111 is a normed, thus a locally convex space, and by 6.6.16

generated by the operator norm is a polar topology of the dual pair (X', X"), where we put X" := (X',11 11)'. We now prove that it is even a polar topology of the dual pair (X', X). the topology T11

11

Theorem 6.6.17. If (X, p) is a normed space, then the operator norm 1111 generates the strong topology /3(X', X) on the dual space X'.

Proof . We consider the dual pair (X', X) together with the bilinear map

defined by (f, x) := f (x), f E X', x E X (cf. 6.6.2(a) and (d)). The strong topology ,6(X', X) is the topology of uniform convergence on all non-empty a(X, X')-bounded subsets of X. For any given e > 0 we have V f E X' V X E Up : If(x)I <_ Ilf11p(x) <_ Ilf11e,

(6.6:8)

that is Uf is a(X, X')-bounded. Moreover, X = UE>o UP obviously holds. Now, we put M := {U' I e > 0} and verify 711 11 = TM = Q(X',X) where and TM denote the topologies generated by the operator norm and T11 the topology of uniform convergence on the elements of M, respectively. 11

322

Functional analytic basis

First we verify r11 II = TM By definition, the topology TM is generated by the family Q = (qq I e > 0) of semi-norms where

qe(f) := sup lf(x)l zEUF

(f E X')

Since on the one hand

Ilfll = sup If(x)! = sup ff(x)l = q1(f) p(z)<1

zEUr

(f E X')

and on the other hand, because of (6.6:8), the inequality qE(f) <_ II fli e

(e > 0, f E X')

holds, the norm I( II on (X',TM) and each qE (e > 0) on (X', I( 1() are continuous. Consequently, we get rN II = TM by 6.4.12. Moreover, TM C (3(X', X) holds since /3(X', X) is the strongest polar topology of (X',X). If, in addition, we know that each weakly bounded subset is also bounded in (X, p), we may conclude that f(X', X) C Tjy and thus that TM = /6(X', X). To that end, if M is a non-empty weakly

bounded subset of (X, p); then M is bounded in (X, p); that is, there exists an e > 0 with M C UP, which implies qM (f) = sup I f (x) 1 < qq (f) zEM

for each f E X',

that is qM is continuous on (X',TM). This implies, however, that ,d (X', X) C TM because, by definition, the topology )3(X, X) is generated

by the family Q := (qM I M C X is non-empty and weakly bounded) .

0 Theorem 6.6.18 (of Mackey). Let (X, Y) be a dual pair. Then in evTherefore, the proof of 6.6.17 is complete by the following theorem.

ery topology compatible with (X, Y) the same sets are bounded.

This theorem is not only an essential tool for the proof of 6.6.17, but also in itself of general mathematical interest. Since, however, its proof is not important for the understanding of the subsequent chapters, we omit it. We refer the interested reader to [252, Theorem 8-4.1]. With the aim of characterizing the compatible topologies of a dual pair (X, Y), we have the following result. The weak topology v(X, Y) is the weakest compatible topology (cf. 6.6.7): The following theorem of Mackey and Arens, which is fundamental in functional analysis, tells us that the

Mackey topology is the strongest compatible topology of the dual pair (X, Y).

Theorem 6.6.19 (Mackey-Arens). Let (X, Y) be a dual pair, and let T be a locally convex topology on X. Then T is a topology compatible with (X, Y) if and only if o (X, Y) C r C T(X, Y) holds. In particular, the Mackey topology is the strongest topology compatible with the dual pair.

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323

The proof of this theorem is deep and we omit it (cf. [252, Theorem 9-2.3]).

If (X, r) is a locally convex space, then the Mackey-Arens theorem says that r(X, X') is the strongest topology compatible with the dual pair (X, X') and that o(X, X') C T C T(X, X'). In the following theorem we deal with the continuity of linear maps between locally convex spaces and give a partial answer to the question whether continuity remains true if we replace the topologies with the weak topologies and Mackey topologies, respectively.

Theorem 6.6.20. Let (X, rx) and (Y, ry) be locally convex spaces and T : X -a Y be a linear map. Then the following statements hold: (a) If T is TX -ry -continuous, then T is weakly continuous, that is o(X, X')-c(Y, Y')-continuous. (b) If T is weakly continuous, then T is T(X, X')-r(Y, Y')-continuous. We also omit this proof (cf. [252, 11-1.4 and 11-2.6]).

Now, we make use of the Mackey-Arens theorem to prove the fact that metrizable locally convex spaces carry the Mackey topology. This is essential for the subsequent chapters. Theorem 6.6.21. Each metrizable locally convex space (X, r) with dual X' carries the Mackey topology, that is r = T(X, X').

Proof. Let (X, r) be a metrizable locally convex space. To prove T = r(X, X') we have to verify r(X, X') C T (cf. 6.6.19). Since (X, T) is metrizable we may choose a neighbourhood basis

U := {U. I n E N}

with Un+1 C Un (n E N)

of zero (for example, Un := Uld1n, where d is a translation invariant metric

generating -r). Let V be an absolutely convex neighbourhood of zero in (X, r(X, X')) . It is sufficient to prove the existence of an n E N with Un C V. For this we assume that there does not exist such an n. Then we may assume Un ¢ nV for each n E N since otherwise Uk C n Un C V would hold for some k E N since n Un is a neighbourhood of zero in (X, r). Thus we may choose for each n c- N an xn E Un with xn V nV. Since xn -- 0 is satisfied because Unfl C Un (n E N), the set {xn I n E N} is bounded in (X, r) and therefore also bounded in (X, r(X, X')) by Mackey's theorem 6.6.18 and the Mackey-Arens theorem 6.6.19. Consequently, by 6.4.19 there

exists an a > 0 such that

{xkI kENJ CaVCnV for all nEN, n>a, (V is a balanced set!) contradicting the choice of xn.

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Functional analytic basis

In connection with compatible topologies of a dual pair (X, Y), in particular with the statements in 6.6.16 and 6.6.19, the following questions suggest themselves. If r is a topology compatible with (X, Y) then Y = (X, r)' = X' holds by virtue of the isomorphism

T:Y--+X', y-t fy

with

fy(x):=(x,y) (xEX).

The question arises whether the compatible and polar topologies of the dual pairs (X, Y) and (X, X') are the same. If so, how does a given polar topology of one of the dual pairs correspond to a polar topology of the other? In particular, are the corresponding weak topologies (respectively, Mackey topologies) coincident? The answer is in the affirmative and-in accordance

with expectation-the isomorphism T is the key to the solution. Remarks 6.6.22. Let (X, Y) be a dual pair, r be a compatible topology on X, X' = (X, 'r)' and A C Y. Then the following statements hold: (a) A is a (Y, X )-bounded if and only if T (A) is a (X', X )-bounded. This implies that the polar topologies on X relative to the pair (X, Y) and (X, X'), respectively, are identical in the following sense: T is a topology of uniform convergence on the elements of a collection M of o(Y, X)-bounded subsets of Y if and only if T is a topology of uniform convergence on the

elements of the collection MT := {T(M) I M E M}. In particular, we have o(X, Y) = o(X, X'), r(X, Y) = r(X, X') and ,#(X, Y) = 3(X, X'). (b) The isomorphism T : (Y, o(Y, X)) ---* (X', o(X', X)) and its inverse T-1 are continuous. In particular, A is o(Y, X)-compact if and only if T (A)

is o(X', X)-compact. This implies first of all that r(X, Y) = r(X, X') and then by 6.6.19 that the compatible topologies on X relative to (X, Y) and L1 (X, X') are the same.

Proof. First we remark that Y = (X, r)' = X' holds by virtue of the isomorphism T : Y -f X', y -+ fy with fy(x) := (x, y) (x E X). (a) Let A C Y. We prove that A is a(Y, X)-bounded if and only if T (A) is o(X', X)-bounded. We recall (cf. 6.6.4 and 6.6.2(d)) that the topology o (Y, X) is generated by the family (px I x E X) of semi-norms, where

: Y -+ R, y -+ pa (y) := I (x, y) I , whereas o(X', X) is generated by (pZ x E X) where px

px

: X' ---+ R, f -- T. (f) := If (x) I .

Therefore, for all x E X and y E Y we have (6.6:9) p. (y) = I(x,y)I = Ify(x)I = p= (f,) = T.(Ty), and, because T is bijective, we obtain, for each x E X, the identities

suppx(y) =suppx(Ty) = sup px(z). yEA

yEA

zET(A)

The assertion now follows from 6.4.19. Furthermore-because T is bijective-we also get immediately from (6.6:9) that o(X, Y) = o(X, X').

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325

If M is a collection of subsets of Y we put MT :_ {T (M) I M E M) . Since T is bijective, M is a collection of non-empty subsets of Y with Y = UMEM M if and only if MT is a collection of non-empty subsets of X' with X' = UNEMT N. It follows that a topology r` on X is a topology of uniform convergence on the elements'of a collection M of subsets of Y if and only if T is the topology of uniform convergence on the elements of

,MT (of subsets of X') since-because T is bijective-on the one hand

T(M)EMT,

MEM and on the other hand

qM(x) = sup I(x,y)I = sup Ify(x)I = sup I (Ty)(x)I yEM

=

yEM

sup If(x)I = gT(M)(x)

yEM

(MEM,xEX)

f ET(M)

holds for the generating semi-norms. If we choose in particular

M := {M C Y I M# 0 and o(Y,X)-bounded} , then the preceding considerations prove ,6(X, Y) = ,3(X, X'). (b) By 6.5.1 the isomorphism T and its inverse T-1 are continuous relative

to the topologies o(Y, X) on Y and o(X', X) on X' because by (6.6:9) we get for each x E X the identity px(Ty) = p2(y) for all y E Y and px(T-1(f )) = gx(f) for all f E X'. Since the image of a compact subset under a continuous map between Hausdorff spaces is compact, a subset A of Y is a(Y, X)-compact if and only if T (A) is o(X', X)-compact. Applying (a), and noting that the image of an absolutely convex set under a linear

map is absolutely convex, we get r(X,Y) = r(X,X') in the case of the Mackey topologies. Since o(X,Y) = a(X,X') and r(X,Y) = r(X,X') it now follows from the Mackey-Arens theorem that the compatible topologies

of (X, Y) and (X, X') are the same.

0

We close the section with examples of compatible and polar topologies of dual pairs which will play a role in the subsequent chapters.

Examples 6.6.23. (a) We consider the dual pair (w, W) together with its natural bilinear form defined in 6.6.8. Then all compatible topologies on w coincide with r,,,; in particular, we have r,, = o(w, cp) = r(w, gyp) = rl(w, cp).

(b) Noting m13 = e we consider the dual pair (m, 2) with its natural bilinear form defined in 6.6.8. In this case we have o(m, e) Cg r(m, e) C Q(m, £) = r11 II_ where r11 II denotes the topology generated by the supremum norm II III. The proofs of (a) and (b) are left to the reader (cf. Exercise 6.6.26).

These examples will be continued in 7.2.5.

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Functional analytic basis

In closing this section we will show that there exist non-trivial cases of locally convex spaces for which weak convergence of sequences implies convergence: Namely, if we consider Schur's theorem 2.4.1, we will see that in a norm convergence of sequences is equivalent to weak convergence.

Theorem 6.6.24 (reformulation of Schur's theorem). For any a = (ak) E e and given sequence (a(n)) with a(n) = (ank)k E 2 the following statements are equivalent:

(a) a(n) -4 a

in (2, 11

111).

(b) a(n) -i a in o(Q, 2')) (c) a(n) -- a in (t, m)). Note, we consider in (b) the weak topology of the dual pair in 6.6.2(a) and in (c) that of the dual pair (e, m) together with the bilinear map in 6.6.8, where we use PQ = m.

Proof. The implication '(a) (b)' is obviously true by 6.6.9. (b) q (c): As we will prove in 7.5.4(c) we have t' = m up to the isomorphism

T

y = (yk) -- f y with fy(x) = 57 xkyk (x = (xk) E Q). k

Therefore u(t, fl =

m) (with respect to the named dual pairs) as we

verified in 6.6.22.

(c)

(a): If (c) holds and we consider the matrix A := (ank), then the m)-convergence of (a(n)) to a means exactly

1:(ank - ak)xk

n-

0

for all x = (Xk) E m,

k

that is m C CA

with

limA x =

akxk

(x = (xk) E m).

k

By `(a)

(c)' in Schur's theorem 2.4.1 we now get

h(A) := lim sup 37 lank T

- akd = 0, that is lim Ila(n) -

all = 0.

k

Thus we have the 11 I11-convergence of (a(n)) to a.

0

Exercise 6.6.25. Verify the statement in 6.6.5(c).

Exercise 6.6.26. Prove the statements in 6.6.23(a) and (b).

Exercise 6.6.27. Let (X, r) be a metrizable locally convex space with dual X', let 13 be a neighbourhood basis of zero in (X, -r), and let

Frechet spaces 327

4 C X'. Verify the equivalence of the following statements: (a) (P is bounded in (X', #(X', X) ).

(b) 3 U E B : supfEt,ZEVIf(x)I < 00. Bibliography: [122], [209], [250], [252]

6.7

Frechet spaces

In subsequent chapters we will mainly consider K-spaces, that is sequence spaces with a locally convex topology stronger than the topology of coordinatewise convergence. For the examination of the topological structure of such spaces we consider, among other things, linear maps which may be represented by infinite matrices. For that we need-similar to the case of linear maps between semi-normed spaces-tools like the Banach-Steinhaus

theorem, certain closed graph theorems and so on. In the case of seminormed spaces completeness is an essential condition for the validity of those theorems and we may expect that the situation is similar in the case of semi-metrizable locally convex spaces.

In Section 6.2 we studied the notion of a complete semi-metrizable space. It is not a topological notion since it depends on the semi-metric which generates the topology as the example in 6.2.22(b) shows. Using, as in 6.4 and 6.5, the following simple idea, we get, in the case of semimetrizable locally convex spaces (X, P), a topological notion of completeness. Roughly speaking, (X, P) is said to be complete if for each p E P the semi-normed space (X, p) is complete. We adapt the notion of Cauchy sequence to the case of locally convex spaces.

Definition and Remarks 6.7.1 (Cauchy sequence). Let (X, P) be a locally convex space and (xn) be a sequence in X. Then (xn) is called a Cauchy sequence (in (X, P)) if it is a Cauchy sequence in the seminormed space (X, p) for each p E P, that is

VpEP YE>0 3noEN Vk,n>no : p(xk-xn,)<e

(6.7:1)

is satisfied.

(a) Each Cauchy sequence (xn) is bounded in (X, P), that is the set {xn ( n E N) is bounded in (X, P). (b) If (xn) is a Cauchy sequence in (X, P) and if a subsequence of (xn) converges, say to x E X, then (xn) converges to x. The following theorem tells us, among other things, that the notion of a Cauchy sequence introduced in this way is a topological notion; that is, it depends only on the topology generated by the family P of semi-norms and not on the (non-unique) family P which generates the topology. So

328

Functional analytic basis

we can speak about Cauchy sequences in locally convex spaces without

ambiguity.

Theorem 6.7.2 (Cauchy sequence). Let (X, P) be a locally convex space with topology Tp and let B be a basis of neighbourhoods of zero in (X,Tp). Then for each sequence (xn) in X the following statements are equivalent:

(a) (x,) is a Cauchy sequence in (X, P). (b) (xn) is a Cauchy sequence for each continuous semi-norm q on (X, P) in the semi-normed space (X, q).

(c) VUEB 3noEN bk,n>no : xk-xOEU. Moreover, if (X, P) is semi-metrizable (cf. 6.4.25) and d is a translation invariant semi-metric which generates Tp, then the following statement is equivalent to (a)(-(c)): (d) (xn) is a Cauchy sequence in the semi-metric space (X, d).

Proof. (a) 4* (b) : Statement (b) implies (a) since each p E P is continuous by 6.4.11. To prove the converse statement let (xn) be a Cauchy sequence in (X, P) and q be a continuous semi-norm on (X, P). The latter implies, by 6.4.11, the existence of an M > 0 and of pi, ... , pr E P with

q(x) < M E pj(x)

(x E X).

j=1

Now, for a given 5 > 0, for a := *6, j E N,. and pi (instead of p) we choose, according to (6.7:1), numbers ni (instead of no) and put no max{ni,... , nr}. Then we obtain for all k, n > no the estimation r

q(xk - xn) < M

pi (xk - xn) < 5. j=;

This proves that (xn) is a Cauchy sequence in (X, q). (c) : Let (a) be valid, and let U E B be given. Then there exists (a)

a V E Bp with V C U. By the definition of Bp there exist r E N and El, ..

, E'r

and pr,..

, pr

EP with V= n;= I UP,'. Now, if we choose-

similar to the last part of the proof-natural numbers ni, according to (6.7:1) for sj > 0 and pi (j E N,), and put no := max{nl,... , nr}, then we obtain for all k, n > no the desired statement pj (xk - xn) < ej. That is, xk - xn E Ue' C U. (a) : Let (c) be satisfied, p E P, and let e > 0 be given. Then, for (c) UP there exists a U E B with U C UP. In accordance with (c) we may choose an no such that for all k, n > no the statement xk - xn E U, and hence xk - xn E UP, holds. Therefore, (xn) is a Cauchy sequence in (X, p) for every p E P, and thus in (X, P).

Fr6chet spaces

329

Now let (X, P) be semi-metrizable and let d be a translation invariant semi-metric which generates 7p. In this situation we have to prove the equivalence of (c) and (d). We choose as a basis of neighbourhoods of zero the system

B:_ {UE Ir>o} with U f ={yEX I d(y,0)<e}. On account of the translation invariance of d we have d(xk - xn, 0) _ d(xk, xn), which implies immediately that (c) is equivalent to

`de>0 3noE.N bk,n>no : d(xk,xn)<e. Thus (c) is equivalent to (d).

Theorem 6.7.3. Let X be a linear space and let T,t and T2 be locally convex Hausdorff topologies on X. Then the following statements are equivalent: (a) (X, TI) and (X,72) have the same convergent sequences. (b) (X, T1) and (X,72) have the same sequences converging to zero. (c) (X,71) and (X,72) have the same Cauchy sequences.

Proof. (a)

(b) : If (a) holds, xn --I a in (X, rl) and (xn) -} b

in (X, 72), then a = b since otherwise the sequence (XI, a, x2: a, x3, ...) would converge to a in (X,71) and obviously diverge in the Hausdorff space (X,T2), which contradicts statement (a). Consequently, (b) is true. (c) : This is an immediate consequence of Exercise 6.7.22. (b) (c) . (a) : Let (c) hold and suppose xn -+ a in (X,Tl). Then the sequence y := (x1 , a, x2: a, x3, ...) is also convergent in (X, 71). In particular, y is a Cauchy sequence in (X, 71), and thus in (X, 72). Since it has a convergent (constant) subsequence it has to converge (to a) in (X, 72). Statement (a) now follows by symmetry. As we have already stated, the notion of a Cauchy sequence in a locally convex space is a topological notion and justifies the following definition of sequential completeness and-in the case of semi-metrizability-of completeness. Let us remark that for a proper definition of the completeness in the case of (general) locally convex spaces we need to generalize the notion of a Cauchy sequence to the notion of a Cauchy net or a Cauchy filter, and their convergence. We elect not to do this.

Definitions and Remarks 6.7.4 (F-space). We call a locally convex space (X, r) sequentially complete if every Cauchy sequence in (X, 7) converges; if (X, 7) is semi-metrizable, then it is called complete. A complete metrizable locally convex space is called a Frechet space, or an F-space. Note, a semi-metrizable locally convex space is complete if and only if it is complete as a semi-metric space; that is, the corresponding notions

330

Functional analytic basis

of completeness are, in the case of semi-metrizable locally convex spaces, consistent. Thus each Banach space is an F-space. Ll

Examples 6.7.5. (a)

is an F-space but not a Banach space, as we now verify. It is obviously a metrizable locally convex space (cf. 6.4.26) and r,, is generated by (q3 ! j E NO) where q? (x) := I x? I for x = (Xk) E W. However, it is not semi-normable by 6.4.23, and thus not a Banach space.

To prove the completeness of (w,r,,,) we could use the fact that r, is generated by the translation invariant metric du, and the argument in 6.2.21(c) that an at most countable product of complete metric spaces is complete with respect to the product metric. We now prove it on the basis of 6.4.16.

Let (x(n)) = ((xkn})k) be a Cauchy sequence in (w, T,,,). By Definition 6.7.1 this means that for each j E N° it is a Cauchy sequence in (o., qj), which implies that (xjn})n is a Cauchy sequence, thus a convergent sequence, in (K, I I) for every j E N° because is complete. But this means by 6.4.16 that (x(n)) converges in Thus, (w, r,,,) is an Fspace.

(b) The locally convex space (C(R), P) in 6.4.3(c) is an F-space, but not a Banach space (cf. 6.7.23). (c) The space (co, a(coi $)) is not sequentially complete. To prove this statement we consider the sequence (elnI) where elnl = (ekn1) k with eknl

j 1 if k - n j 0 otherwise

(k,n E N°).

It is a Cauchy sequence in (co, a(co, 2)) because k

Py (elk!

- elnl)

=

00

E yrj

Iy, I -} 0 (n -* oo)

/j=n-1.1

(k > n, y = (y,,) E P). Trivially, it is coordinatewise convergent to e. Consequently, since r;, I CO is Hausdorff and weaker than o(co,1) and because e co, it cannot converge in (co, a(co, f)).

(d) The locally convex space (f, a(f, m)) is sequentially complete. This is an immediate consequence of the (reformulated) theorem of Schur 6.6.24, of 6.7.3 and of the completeness of (e, 11 111).

For the present we leave off consideration of these examples of F-spaces and sequentially complete locally convex spaces. In subsequent chapters we will give further examples. In this connection see also Exercise 7.2.23.

We now formulate the open mapping theorem after first defining the notion of an open map.

Frecbet spaces

331

Definition and Example 6.7.6 (open map). Let (X,rx) and (Y,ry) be topological spaces. A map T : X -* Y is called open if the image of each open set in (X,rx) is open in (T(X),ryIT(x)) That is, if

VQErx 3SEry : T(Q)=SnT(X). The projections irk : (w, rw) -+ (1K,11) are open as follows easily from the definition or from the following theorem.

Theorem 6.7.7 (open mapping). Let (X, P) and (Y, Q) be F-spaces. If T : X -+ Y is linear, continuous and onto, then T is open. The proof of the open mapping theorem is based on the Baire category theorem. Since the proof is not important for understanding the subsequent chapters, we omit it and refer the reader to [250].

As an immediate corollary of the open mapping theorem we get the following result about an inverse operator.

Corollary 6.7.8 (inverse operator). If in addition to the assumptions in 6.7.7 the map T is bijective, then T-' is also continuous. Proof. The statement follows directly from 6.7.7 since the inverse T-' of 0 a bijective map T is obviously continuous if and only if T is open. From the theorem on the inverse operator we now deduce the closed graph theorem. First we recall the relevant notions.

Definition 6.7.9 (graph of a map). If T : X -> Y is any map, then the set G (T) := { (x, T (x)) E X X Y ! x E X } is called the graph of T.

Definitions and Remarks 6.7.10 (closed map). Let (X, P) and (Y, Q) be locally convex spaces, and let (X x Y, r) be their product according to 6.4.9. A map T : X -+ Y is called closed if the graph G(T) is closed in (X x Y, r). (a) If in addition both spaces are semi-metrizable, then T is closed if and only if for each sequence (xn) in X, and for each x E X and y E Y, the implication

xn -+ x in (X, P) T (xn) ---9 y in (Y, Q)

y = T (x)

holds.

(b) If (Y, Q) is Hausdorff, then every continuous map T : X -3 Y is closed.

Proof. (a) Let both spaces be semi-metrizable. Then, since convergence of sequences in the product space is equivalent to componentwise convergence, closedness of G(T) is equivalent to the following statement from which the desired equivalence follows easily. If (xc,T(xn)) converges in (X x Y, r)

332

Functional analytic basis

to (x, y) E X X Y, then (x, y) E G(T).

(b) Using the sequential continuity and applying (a), the statement is trivial in the case of metrizable spaces. The general case is based on the same idea; one has to consider nets instead of sequences. We omit the details.

0

Now, the question arises whether the last statement is true in general, that is whether closed maps are continuous. That fails in general as the following example proves.

Example 6.7.11. The identity map i : (m, r,) -3 (m, T11 Ii_) , x x is closed, but it is not continuous. It is not continuous since the sequence (ek) converges to zero in (m, ru,) and is not convergent in (m, rll II ). That the identity map id is closed is an immediate consequence of the following closed graph lemma.

Lemma 6.7.12 (closed graph lemma). Let (X, P) and (Y, Q) be locally convex spaces and T : X -+ Y be a closed map. If X and Y are both given stronger (locally convex) topologies, T remains closed.

Proof. The proof is quite easy. If we refine both topologies rp and rQ, the product topology becomes finer too (cf. 6.4.9 and 6.4.12), which means O in particular that each closed subset in the product remains closed.

In comparison to closed maps the situation in the case of continuous maps T : X -+ Y is more difficult. In general, T remains continuous if the topology of the domain space becomes finer or that of the range space becomes coarser. Thus, the advantage of the closedness of maps lies in hands: one checks closedness for coarse and `handy' topologies and knows by 6.7.12 that T is closed with respect to all finer topologies on the domain

and range. In the case of Example 6.7.11 this means that the identity map i : (m, r,,,) -3 (m, ry,) is obviously continuous, and thus closed. Consequently, the identity map i : (m, r,,,) -+ (m, rll II_) is closed too.

Theorem 6.7.13 (closed graph theorem). Let (X, P) and (Y, Q) be F-spaces and T : X -p Y be a closed linear map. Then T is continuous.

Proof. Let T : X --* Y be a closed linear map from (X, P) into (Y, Q). Then the map T : G(T) -+ X, (x,Tx) -4 x is obviously bijective and also linear since T is. Furthermore, T is continuous relative to the topology

r induced by the product topology of X x Y since it is the restriction of the continuous map (projection) lrX : X x Y -+ X, (x, y) -+ x (cf. (a)' in 6.5.1). Because T is closed, G(T) is a closed linear subspace `(c) of the topological product (X x Y, r), that is (G(T), rI G(T)) is an F-space by Exercise 6.7.24(a) and (b). Applying the theorem on the inverse operator 6.7.8, we get the continuity of the inverse T-1 of T. Thus the map T = Try o i o T-1 is continuous as the composition of the continuous map T-1,

Fr*chet spaces

333

the inclusion map i : G(T) - X x Y and of the projection zry : X x Y --+ o Y,(x,y) --+ YIn the case of families of continuous linear maps of a complete seminormed space into a semi-normed space the uniform boundedness principle 6.3.35 tells us that pointwise boundedness of such families implies uniform boundedness. If we consider more generally (semi-metrizable) locally convex spaces we may easily generalize the notion of pointwise boundedness of such families, but the notion of uniform boundedness does not make sense in this situation. However, as we have stated in Theorem 6.3.33 uniform boundedness of such families is equivalent to their being equicontinuous. We now state that, using equicontinuity as a substitute of uniform boundedness, the uniform boundedness principle can be generalized to the situation of semi-metrizable locally convex spaces. We also adapt the notion of pointwise boundedness from semi-normed spaces so that it applies to locally convex spaces.

Definition and Remarks 6.7.14 (pointwise boundedness).

Let (X, P) and (Y, Q) be locally convex spaces and 4 be a non-empty family of linear maps from X into Y. Then 4> is called pointwise bounded if {Tx I T E t} is bounded in (Y,Q) for each x E X. Accordingly, as with the notion of boundedness of sets, the notion of pointwise boundedness of families of linear maps is consistent with the corresponding notion in semi-normed spaces. If 4 is equicontinuous, then ' is also pointwise bounded. This follows from the definition of the boundedness and from 6.5.16(c). A

Theorem 6.7.15 (Banach). Let (X, P) and (Y, Q) be semi-metrizable locally convex spaces, (X, P) be complete and 41 be a non-empty family of continuous linear maps from X into Y. Then pointwise boundedness of implies that 1 is equicontinuous.

As in the case of the open mapping theorem the proof of Banach's theorem is based on the Baire category theorem. Since the proof is not important for understanding subsequent chapters, we omit it and turn to two applications of Banach's theorem. The first one is the Banach-Steinhaus theorem for pointwise convergent sequences of continuous linear maps between an I+-space and a metrizable locally convex space which we have already encountered in 6.3.38 in the case of normed spaces.

Definition and Theorem 6.7.16 (Banach-Steinhaus). Let (X, P) be an F-space, (Y, Q) be a metrizable locally convex space, and (T,) be a sequence of continuous linear maps from X into Y. If (T") is pointwise convergent (that is, if (T, (x)) converges in (Y, Q) for every x E X), then

T(x) := limoTn(x) defines a continuous linear map T : X -* Y.

(x E X)

334

Functional analytic basis

Proof. Obviously, T is well-defined and linear. Furthermore, the family 4P :_ {Tn I n E No } is pointwise bounded as a pointwise convergent sequence. Therefore 'I is equicontinuous by 6.7.15. Thus we obtain by 6.5.16 (for a given q E Q and some k E N, p ? EP and M> 0) f o r each X E X the inequalities k

q(T(x)) = lim r q(TT(x)) < sup q(Tn(x)) < MEpj(x). nEN°

j=1

0

Now, the continuity of T follows from 6.5.1.

The second application of Banach's theorem is a completion of the state-

ment in 6.6.21 that metrizable locally convex spaces carry the Mackey topology. We now prove that in the case of F-spaces the Mackey topology and the strong topology coincide.

Theorem 6.7.17 (F-space). If (X, r) is an F-space and X' is its dual, then r = r(X, X') = ,B(X, X). Proof. On account of 6.6.21 it is sufficient to verify r = /3(X, X'). If we can

show that the non-empty bounded subsets in (X',o(X',X)) are, relative to (X, r), the equicontinuous non-empty subsets of X', r = 6(X, X') follows from 6.6.16. In the proof of 6.6.16 we have already verified that each equicontinuous

subset of X' is weakly bounded. On the other hand, if weakly bounded subset of X', then (cf. 6.6.14(a))

is a non-empty

b'zEX 3M>0: suplf(x)j<M
The latter means that 41 is a pointwise bounded family of continuous linear

functionals (cf. 6.7.14). Because (X, r) is assumed to be an F-space, we may conclude the equicontinuity of P by Banach's theorem 6.7.15. This

0

completes the proof.

Next we formulate a further well-known closed graph theorem which goes back to N. J. Kalton. This theorem will prove useful in subsequent chapters. We precede it with a sufficient condition for separability in the case of locally convex spaces.

Remark 6.7.18 (separable lcs). A locally convex space (X, P) is separable if it contains a dense linear subspace Y which has an at most countfor some A E N°. able algebraic basis (Yk ( k E I) with I = NO or I =

Proof. We sketch the proof leaving the details to the reader. Put Q<

and

ifK=R Q {Q+iQ iflK=tC

Barrelled spaces

Prat :=

{xkYk

335

I n E I, Xk E QS (0 < k< n)

k=0

and prove that Y C Yrat.

Theorem 6.7.19 (Kalton's closed graph theorem). Let (X, P) be a locally convex space with the Mackey topology r(X, X') and which has a sequentially complete weak dual (X', o(X', X)). If (Y, Q) is a separable F-space and T : X -3 Y is a closed linear map, then T is continuous. Proof. For a proof of this theorem we refer to [121] and [252]. In closing this section we give two further theorems.

Theorem 6.7.20. Every F -space is of second category. Proof. See the Baire category theorem 6.2.28.

Theorem 6.7.21. If (X, rx) and (Y, ry) are F-spaces and T : X -+ Y is linear and continuous, then either T(X) is of first category as a subset of (Y, ry) or T is surjective (and thus open).

Proof. The proof is essentially the same as that of the open mapping theorem. We leave it to the reader.

Exercise 6.7.22. Let (X, P) be a locally convex space. Prove that a sequence (x,,) in X is a Cauchy sequence if and only if xm,. (k -4 oo) holds for all index sequences (mk) and (nk).

40

Exercise 6.7.23. Show that the locally convex space (C(R), P) in 6.4.3(c) is an F-space (cf. 6.7.5(b)) but not a Banach space. (Make use of the fact that C[a, b] with the supremum norm is a Banach space.)

Exercise 6.7.24. Verify the following statements: (a) Each closed subspace of an F-space (endowed with the induced topology) is an F-space. (b) An at most countable topological product of F-spaces is an F-space.

Exercise 6.7.25. Let X be an F-space with two topologies rl and r2. Prove that rl C r2 implies rF = r2. Bibliography: [209], [250]

6.8 Barrelled spaces In Section 10.1, more precisely, in a statement in Theorem 10.1.3, and mainly in Section 11.4 we make use of the notion of a barrelled space. For readers who wish to study Section 11.4 and are not familiar with barrelled spaces we introduce in this section the notion of a barrelled space and present some main results which are applied in Section 11.4. These are

336

Functional analytic basis

results like the Banach-Steinhaus theorem and the closed graph theorem which we encountered in the special case of Banach spaces and, more generally, in the case of F-spaces. We present these theorems in these three steps of generality because we do not need their full generality in every application, so readers with a little functional analytic background, who are not interested in all of the applications, do not have to deal, for instance, with barrelled spaces. Motivated by the fact that for every locally convex space there exists a neighbourhood basis of zero consisting of absolutely convex, absorbing and closed sets (cf. 6.4.5(c)), we introduce the notion of barrelled spaces.

Definition 6.8.1 (barrelled space). Let (X, r) be a locally convex space. Then a subset is called a barrel if it is absolutely convex, absorbing and closed in (X, r). Moreover, (X, r) is called a barrelled space if each barrel is a neighbourhood of zero. Without proof we give a useful characterization of barrelled spaces.

Proposition 6.8.2. A locally convex Hausdorff space (X, r) is barrelled if and only if it carries the strong topology, that is r = 8(X, X'). Consequently, a barrelled Hausdorff space is a Mackey space.

The next theorem gives us a large class of barrelled spaces.

Theorem 6.8.3. Every F-space, and thus each Banach space, is barrelled. The notion of barrelled spaces enables us to generalize (cf. 6.8.3) Bar nach's theorem 6.7.15.

Theorem 6.8.4. Let (X, P) be a barrelled space, (Y, Q) be a locally convex space, and 4 be a non-empty family of continuous linear maps from X into Y. Then pointwise boundedness of implies that -t is equicontinuous. Now, we generalize the Banach-Steinhaus theorem 6.7.16 to the situation of barrelled spaces. Corollary 6.8.5 (Banach-Steinhaus). Let (X, P) be a barrelled space, (Y, Q) be a locally convex Hausdorff space, and (To) be a sequence of continuous linear maps from X into Y. If (To) is pointwise convergent, then T : X -* Y, x -> T (x) := limo T (x) is (linear and) continuous. Proof. We adapt the proof of 6.3.38 to the present situation. The sequence (To) is assumed to be pointwise convergent, thus pointwise bounded and consequently equicontinuous by 6.8.4. Hence, for any q E Q there exist by 6.5.16(c) an M > 0 and pt, ... , pr E P such that

q(TT(x)) < M Epj(x) for all x E X and n E N.

(6.8:1)

j-1 This gives q(T(x)) = limo....... M F,J-1 pj (x) for each x E X 0 and therefore by 6.5.1(d) the continuity of T.

Barrelled spaces

337

The next theorem is a generalization of the general version of the Banach Steinhaus theorem in 6.3.39.

Theorem 6.8.6 (Banach-Steinhaus, general version). Let (X, P) be a barrelled space, (Y, Q) be an F-space, and let H be a dense subset in (X, P). Furthermore, let (Tn) be a sequence in B (X, Y). If (T,) is pointwise bounded and (TnI H) is a pointwise Cauchy sequence, then (T,a)

is pointwise convergent and the pointwise limit T of (Tn) is linear and continuous.

Proof. The assertion is proved by applying 6.8.5, if we can show that (T,a) is pointwise convergent. For this, since (Y, Q) is complete, it is sufficient

to prove that (T.) is a pointwise Cauchy sequence. Now, let x E X `H and q E Q and e > 0 be arbitrarily given. We have to show the existence of an no E N such that

q(Tn(x) - Tk(x)) < e for all n, k > no. Since (Tn) is equicontinuous, we can choose an M > 0 and pi, ... , p,. E P

such that (6.8:1) is satisfied. Since H is dense in (X, P) there exists a y E H with p(x - y) < 3' where p pi. Then we choose for y an no with q(T,a (y) - Tk (y)) < 3 for all n, k > no. Altogether we obtain for all n, k > no the inequality

q(Tn(x) -Tk(x)) < q(Tn(x - y)) + q(Tk(y) -Tk(y)) + q(Tk(y - x)) < MAX - y) + 3 + M AX - y) < e. Thus, (Tn) is a Cauchy sequence and hence pointwise convergent.

0

Next we give a very common version of the closed graph theorem in the case that the domain space is barrelled. We omit the proof.

Theorem 6.8.7 (closed graph theorem). Let (X, P) be a barrelled Hausdorff space, (Y, Q) be an F-space and T : X -+ Y be a closed linear map. Then T is continuous.

The hypothesis in the foregoing theorem, that (X, P) is" barrelled, is the best possible for the validity of this theorem:

Theorem 6.8.8 (M. Mahowald, cf. [164, 2.2]). For any locally convex Hausdorff space (X, P) the following statements are equivalent:

(a) (X, P) is barrelled. (b) If (Y, Q) is any F-space, then each closed linear map T : X --f Y is continuous.

(c) If (Y, 1111) is a Banach space, then each closed linear map T : X -} Y is continuous.

Bibliography: [209], [250]

7

Topological sequence spaces: K- and FK-spaces With the aim of topologizing domains of matrices with natural topologies,

we mainly study FK-spaces in this chapter. These are sequence spaces carrying a metrizable locally convex topology which is complete (F-space) such that convergence implies coordinatewise convergence (K-space). FKspace theory was initiated by K. Zeller in 1949. On the one hand it makes possible the application of functional analytic methods to a number of ma-

jor problems in summability and, on the other hand, it has proved very fruitful in the development of some topics in functional analysis, for example that of topological sequence spaces. Definitely, the inspiration for FK-space theory came from the Polish school around S. Banach, S. Mazur and W. Orlicz in which functional analytic methods were applied, for instance, to prove the bounded consistency theorem. Around 1949, K. Zeller published some seminal papers, for example, [261], [260], [262]. The subject was then further developed by Zeller and many other mathematicians. Wilansky's book, [254], traces the development of the subject up to 1984. In connection with inclusion theorems, like the Toeplitz-Silverman theorem and the Schur theorem discussed in Chapter 2, the a-dual of certain

sequence spaces played an important role. In addition to the 8-dual, we introduce further Kothe-Toeplitz duals in Section 7.1, namely the (-dual where ( E {a,13, y}, and calculate them for certain sequence spaces. The Kothe-Toeplitz duals will prove very useful in topologizing sequence spaces with locally convex topologies where convergence implies coordinatewise convergence (K-topologies). In studying the structure of sequence spaces endowed with certain locally convex topologies, it is obvious and useful to consider only such topologies for which convergence implies coordinatewise convergence. Locally convex spaces of that type are called K-spaces and are the subject of Section 7.2. Besides examples of K-spaces we study their structure by considering

`distinguished subspaces' that were introduced and are important in the literature. The `distinguished' subspaces flow from ideas such as section

convergence, weak section convergence, functional section convergence, section boundedness and section density.

Sequence spaces and their S-duals

339

In Section 7.3 we study FK-spaces-these are K-spaces which are also F-spaces. We characterize continuous linear maps between F- and FKspaces and establish the `monotonicity' and `uniqueness' of the topologies of FK-spaces. Moreover, we present methods-like taking the pre-image of FK-

spaces under certain continuous linear maps-to generate new FK-spaces from known FK-spaces. One of the main results of this section tells us that matrix maps between FK-spaces are continuous. In Chapter 2 the Silverman-Toeplitz theorem 2.3.7 is proved by gliding hump arguments which are quite technical. As promised in Section 2.3, in Section 7.4 we give functional analytic proofs of some Silverman-Toeplitztype theorems based, for example, on the uniform boundedness principle and the Banach-Steinhaus theorem. However, it should be noted that there

is no known, completely functional analytic proof of the Schur theorem 2.4.1.

Section 7.5 contains essentially the determination of the dual space of `standard' FK-spaces and of `generated' FK-spaces (in the sense of 7.3) in terms of the dual spaces of the `generating' FK-spaces. We complete in Section 7.6 the investigation of `distinguished subsets', started in Section 7.2 in the particular case of FK-spaces. Among other results, we show that the `distinguished subsets' under consideration are FK-spaces. It is then easy to characterize FK-AK-spaces which are, by definition, FK-spaces with section convergence.

7.1 Sequence spaces and their cc-duals In the present section we introduce more sequence spaces as well as some (algebraic) properties of sequence spaces. Besides the 8-dual (cf. 2.3.1) of a sequence space we will study the a-dual, due to KSthe and Toeplitz, and the -y-dual. That enables us to endow sequence spaces in a natural way with suitable semi-norms and, using duality theory, with locally convex topologies.

Besides the sequence spaces introduced in preceding chapters the following sequence spaces play an essential role in summability as well as in functional analysis and other areas where summability is applied.

Notation 7.1.1. K :=

{x=(xk)Ew 13KENo Vk>K : xk=ZK} = cp®e (ultimately constant sequences)'.

d :=

H,

I

kEN

{ x = (xk) E W I Jim sup Ixk I r ll

(analytic sequences).

x = (xk) E w I sup Ixk I k < o0

k

1 We write sp @ e instead of V @ ({e}).

))))))

1

-r

(0 < r < ob).

340

Topological sequence spaces: K- and FK-spaces

{x=(xk)Ew limsuplxkI' < 1} (l k r

dr

b :=

(0
limsup,xkI' = 0} (entire sequences).

S x = (xk) E w ll

k

0

We now state a number of important inclusion relations among some common sequence spaces.

Remarks 7.1.2. (a)
W C S : If x = (xk) is defined by Xk :_ (k + 1)-k (k E NO), then x but x E b since lim supk Ixk i = lim supk k+i = 0. S C d,.: For x:= ((2r) -k) we have Iimsup((2r)-kj'

k

=

1

2r

<

r that is x E d, \S.

d,. C II,.: Obviously, x:= (r-k) E IIr \ dr II,.

.

d: If x:= ((2)-k), then =

2

kEN `2T sup

= llmk sup

1(r)-k 0

and thus fIr 96 d.

In the following remarks we point out the relationship of the sequence spaces d, dr, 11r and b with certain function spaces.

Remarks 7.1.3. If we consider the (formal) map

x = (xk) -i Tx := f with f (z) _ E xkzk k

(where in the case x E d the function f may be analytically continued into the Mittag-Lefer star S[f] ), we get

T (d) =

If : S[f] -+ C I f is holomorphic at 0 } (set of all (at zero) analytic functions),

Sequence spaces and their (-duals

341

T(S) = If : S(fJ -+ C I f is holomorphic on C, that is C = S[f] } (set of all entire functions), and for all r E 10,1[ we get

T(fr) = If S[f] -+ C A. C S[f]} , T (dr) = If : S[f] -.+ C I Dr C S[fj} . These results are immediate consequences of Hadamard's criterion for the determination of the radius of convergence of power series. A

Up to isomorphism, w is the algebraic dual of W. Moreover, as one may prove, V is the unique sequence space containing W and such that the algebraic dual is-up to isomorphism-a sequence space. Now, if we endow sequence spaces with certain semi-norms, or with a locally convex topology (as we do later), then, in general, the topological dual (that is, the set of all continuous linear functionals) is not a sequence space. Following Kothe and Toeplitz (1934), for a given sequence space X one may choose a sequence space Y such that X endowed by a topology which is `naturally generated by Y' has Y (up to isomorphism) as the topological dual. (Compare this with the discussion of duality theory in Section 6.6.) We now give the definitions of the a- and 7-duals and recall the definition of the fl-dual.

Definition 7.1.4 (a-, 0- and -y-dual). If X is any sequence space, then we define

X"

{yEw) vxEX: xyEe}

XR

{yEw I V xEX: xyEcs}

(,B-dual of X),

X,

{yEw I `d xEX: xyEbsl

(-t-dual of X).

(a-dual of X),

Further to the statements in 2.3.2 we make the following remarks which are immediate consequences of the definition of the (-duals ((E {a, f3, y} ).

Remarks 7.1.5. (a) The Definition 7.1.4 also makes sense in the case of non-empty subsets X of w (cf. 2.3.1).

(b) cp < X' < X,6 < X-1 < w; in particular, X' (( E {a, fl, y}) is a sequence space.

Y( < XS ((E {a,,6, y}). (c) X < Y < w (d) If I is an index set, if X2 (i E I) are sequence spaces and if X UiEI X=, then2 2 Note, X is not necessarily a linear space.

342

Topological sequence spaces: K- and FK-spaces

(X)s = n Xi,

( E {a, / , y}).

iEl

(e) X < X(s :_ (X()S

((E {a,,t3,y}).

Proof. (a) and (c) are obviously true, and (b) follows from f < cs < be.

(d) Let C = a. (If C _ 6 and S = y, then we only have to replace t by cs and be, respectively.) Because X2 C (X) (i E I) we get

(x) " C X,' (i E I ),

and thus

(X )" C n Xi", iEI

by (c). On the other hand, if y E IiEI Xi", that is y E Xi" (i E I), then xy E e (x E X) and therefore y E X". (e) Again we consider the case C = a and prove X < X"". Let X E X. I

Then xy E 2 (y E X"), that is x E X", and thus X < X" by (b). 0

In general, X # XSS as we get from 2.3.3 in the case of { = d and X := co. We have cr = m # co. This remark gives rise to the following definition.

Definition 7.1.6 (c-space, Kothe space). Let C E and let X be a sequence space. X is called a {-space if X = X. Further, an a-space is also called a Kothe space or perfect sequence space. From 7.1.5(e) and (c) we obtain immediately the following remark.

Remark 7.1.7. If X is a sequence space and C E {a,,3, y}, then X is a (-space, that is XS = X«' (= (XCC)s). is Since, as we will see in 7.1.11(f), there exist sequence spaces X with

X" # Xo 34 X7, we look for sufficient conditions for X" = X0 = X. This gives rise to the notion of solid sequence spaces.

Definition and Theorem 7.1.8 (solid sequence space). Let X be a sequence space. Then X is called solid if

{(uk)Ew 3(xk)EXVkEloo : jukt
(b) If K = C, then X is solid if and only if {(yk) E w 3 (xk) E X b k E No : jykI = ixkl} C X. 3If A, B C w then we put AB := {(akbk) I (ak) E A, (bk) E B} .

(7.1:1)

Sequence spaces and their (-duals

343

Remark 7.1.9. Statement (b) in 7.1.8 is not true in general if K = R. For example, let X := no. Because the sequences u = (Uk) := (k+1) and

x = (xk) := e satisfy Iukl < IxkI (k E N°) and e E mo, but u f mo, the space mo is not solid. However, (7.1:1) is satisfied as we now check. If y = (Yk) is a sequence such that there exists an x = (xk) E mo with Iyk I = IxkI (k E NO), then { xk I k E N° } ; thus {IxkI I k E Nl° } and therefore { Iyk I I k E N° } is a finite set. Consequently, since yk E R (k E NO), the set {yk i k E No) is finite too, that is y E mo. A

Proof of 7.1.8. (a) This simple proof is left to the reader (cf. 7.1.13). is trivial. (b) The implication Now, let u = (Uk) E w and x = (xk) E X with Iukl <- IxkI (k E N°) be given. Further, let xk = k + i77k (k E N1° and G, 77k E R). Then i77k) E X holds because IxkI =IxkI (k E NO). Let x = (xk) = Uk = ak + ifik (k E NO and ak, (3k E R). We obtain

ak +/k2

< k2 +77k2

since

IukI 5 IxkI (k E N°).

We now choose yk E R with ak + yk2 = tk2 + 77k2 (k E NO) and consider

w = (wk) defined by Wk := ak + iyk. Then IWkI = IxkI (k E N°); thus

w E X and w = (wk) E X. From this we get a:= (ak) = (w + W') E X. 2 it follows Determining Sk E R such that Sk2 + ,dk2 = k2 + 77k2 (k E NO), similarly that v = (Vk) := (bk +i13k) E X and v = (vk) E X, which imply ,d := (/3k) = z= (v - v) E X. Thus, as required, u = a + i/3 E X.

Theorem 7.1.10. If X < w, then the following statements hold: (a) If X is a Kothe space, then X is solid.

(b) If X is solid, then X° = Xa = X. (c) If X is a Kothe space, then X is a (-space (c E {a, J3, 'y} ).

(d) If X is solid and e E X, then m C X and X° = X' = X" C . Proof. (a) If X is a Kothe space and x E w, then

x E X . xEXO'a 4-- VyEX' : xyEt. From this we get for x = (xk) E X and u = (Uk) E w with IukI < IxkI (k E N0) the statement -

1: Iukyk1 < 1: IxkykI < 00

(y = (yk) E X °).

k

k

Therefore uy E P for each y E Xa, that is u E X.

(b) Let X be a solid sequence space. To prove X' = X'3 = X7, it is sufficient to verify X7 C X° (cf. 7.1.5(b)). So let y = (Yk) E X'', that is sup 11: xkykj < oo for every x = (xk) E X. n

k=0

344

Topological sequence spaces: K- and FK-spaces

If x = (xk) E X is given, then we put zk := xk sgn (xkyk) (k E N°) and get z = (zk) E X since X is solid and jzkI < jxkI. This implies N

N

IxkykI

ykxk sgn (xkyk) = i xkyk

_ k=0

k=0

<

k=0

sup

xkyk! < oo

(N E NO);

k=0

thus xy E e, that is y E X*, because x E X was arbitrarily given. (c) This is an obvious consequence of (a), (b) and 7.1.7. (d) If X is solid and e E X, then m c X follows from 7.1.8. Moreover, 0 XcX = V = X' C e is a corollary of (b), 7.1.5(c) and 2.3.3.

In particular, the statements in (a) and (b) imply that in the case of solid sequence spaces, especially in the case of KOthe spaces, one has to determine only one of the duals (usually the a-dual). In the remaining part of this section we will determine the (-dual ((E {a, 0, y}) of certain sequence spaces and examine them to see whether they are `solid' or a `Kothe space'.

Theorem 7.1.11. (a) cp, w, PP (0 < p < oo), co, m, d, d,., IIr and b are solid sequence spaces.

(b) rc, c, m0i f, f0 and by are not solid, therefore none of them is a Kothe space. (c) For each C E {a, 0, y} the following statements hold: wC _ co

and {p(= w,

es=m and m(=e, do=d and dt=d, =d

and d,(=II1 for everyr>0,

(ep)C = eq

(1 < p, q < oo and p + q =1).

II

In particular, cp, w, eP (1 < p < oo), m, 5, d, d,. (r > 0) and II,. (r > 0) are Kothe spaces and therefore (-spaces (( E {a,,3, y} ). (d) If (E {a, (3, y} and Co < Y < m, then Y S = e and Y C Yss =m. In

particular, ca = cc = f0 = f( = e, and each of co, c, f° and f is not a (-space.

(e) ma = e (( E {a, 0,,y}), and m° is not a (-space. (f) rca = e, rcf = cs and rc" = bs. Proof. (a) and (b) That the specified spaces are solid is an immedi-

ate consequence of their definition. On the other hand, rc, c, m°, f and by are not solid since they include e and are strict subspaces of m (cf.

Sequence spaces and their (-duals 345

7.1.10(d) and 7.1.2). Further, fo is not solid because (1, -1, 1, ...) E fo and (1, 0,1, ...) o fo. (c) By 2.3.2(d) we have cpF = w and wy = cp, and we know Ba = m and my = B from 2.3.3. Since the spaces w, cp, a and m are solid we get (Po = w, ws = cp, es = m, ms = t by 71.10(b). The proof of the statements on the (-dual of II,. and d, is left to the reader (cf. 7.1.14).

Now, we verify 8s = d ((E {a, Since b is solid, it is sufficient to show that 8" = d (cf. 7.1.10(b)). For this let y = (Yk) E d be given. We choose an M > 0 such that jykj < Mk (k E N°) and an £ > 0 with £M < 1. Moreover, for any given

x = (xk) E b we choose a K E N with jxkj < £k (k > K). Thus, we obtain 57,1XkYkj

K E I xkykl +

(£M)k < 00, k=K+1

k=O

k

CO

that is xy E 2. Thus, y E b", that is, d C 5". If on the other hand y = (yk) f d, then we may choose an index sequence

with

jykj > 12k.,

(v E N).

Putting Xk

_ Jv-k- if k = k,, andvEN

l0

otherwise

we get x = (xk) E S and Ek jxkykl > Ev vk° therefore xy 0 e, which implies y f &" and hence d = S". The proof of ds = b is similar to that of d = bs and we leave it to the reader (cf. 7.1.14). It remains to prove (eP)C = 24 (1 < p, q < oo and 1. + q = 1). Since eP is solid, it is sufficient to consider the case $. In contrast to the cases p = 1 and p = oo we will apply functional analytic tools that simplify the proof.

Let y = (yk) E w and p, q E R with 1 < p, q < oo and v + a = 1 be arbitrarily given. We now prove (Yk) E (tP)1

(yk) E Q4.

Let y = (yk) E f9. Then it follows from Holder's inequality that Ek ykxk converges for each x = (xk) E £P. =*.: Let y = (Yk) E (eP)O. For every n E No we consider the linear map

fn : eP -- * 1K, x = (xk) -4 fn(x) := E vkxk. k=o

Define yinl :_ (ykn) :_ (yo, ... , yn, 0,.. .). Then, by Holder's inequality and since y1nl E V C e9, we obtain

346

Topological sequence spaces: K- and FK-spaces n

fn(x)I s

00

Iykxkl =

Iykn'xkI < IIyln'IIq

IIxIIP

for each x = (xk) E Pp and thus the continuity of fn. In particular, we have IIfnII

for each n E N°.

IIyln'IIq

We now prove the reverse inequality for each n E NO.

IIfnII

IIyln'IIq

Without loss of generality we assume yln) # 0 since the case yln) = 0 is trivial. We consider the sequence X(n) = (xjn))k defined by

if yk#Oand k
31k 9

Xkk

:= f 0

91k

(kEN°).

otherwise

Then X(n) E 2p and, because q = (q - 1)g, we have IIx(n)

(Iyk_1t) °

ilp =

p

_

(IVkI)

k=0

= (IIyln' IIq) P

k=0

and, in particular, IIx(n)IIP j4 0. This implies n

If

(n}n))I

_

Ip

II

E(Iyklq II

-

IIp

llgg

=

Ilyln'Ilgq(1p)

-

IIp

II

hence (cf. 6.3.19(d)) Ilyln'IIq : IIfnII Thus IIfnI) = llyln'IIq for n E No. We now apply the Banach-Steinhaus theorem (cf. 6.3.38). By our hypothesis, the sequence (fn) converges pointwise. Since (Pp, II IIp) and (1K, 11)

are Banach spaces, the (well-defined and linear) map fy

IP -i K, (xk) --+ fv(x) M fn(x) = E ykxk k

is continuous, and IIffII

sup IIfnII = sup Ilyln'Ilq < 00

nEN

nEN°

holds. Because E SUP Ilyln'IIq NO

EGO

(IykIQ1 k=0

_ (ko=00 IykIq)

we have (yk) E 0. Thus, we have proved (2P)O = fg.

Sequence spaces and their (-duals

347

(d) Since co and m are solid, we know already from 2.3.3 that co; = ms = f ((E {a, j3, y}). So the statements in (d) follow from 7.1.5(c). (e) Since m< = P and mo C m we obviously have e C mo( (( E {a,)8, y}).

If we could verify that m0 C t, then mo; = f would be proved for ( E {a, f6, y} (cf. 7.1.5(b)). Let y = (yk) 0 1, that is (?yk) 0 f without loss of generality, and let xk := sgn tyk (k E N°). Then (xk) E mo and n sup n

n

E xkyk

> sup

k=o

= sup E I?yk l = 00, n

n

k=°

which implies y ma . Thus we have proved m°7 C 1. (f) By the definition of r£s and since e E >£ we get rc" C P, rc,6 C cs and

0 C bs. Furthermore, t C rc" because >£ C c and ca = 2. To see that cs C rc1 and bs C rc7, note that xy = (xoyo, , xnyn, ayn+l, ayn+2, ) for x = (xo,... , xn, a, a.... ) E rc and y = (Yk) E w so that xy E cs 0 whenever y E cs and xy E bs whenever y E bs. With the exception of X = rc all of the sequence spaces X considered in 7.1.11 have the common property X1 = XR = X7, though among them there are non-solid spaces like c, m° and f. Next, we determine the (-duals of the spaces cs, bs, by and bvo. We will find that none of these sequence spaces is solid; in particular, none of them is a Kothe space.

Theorem 7.1.12. (a) cs° = bva = bvoa = bs° = t. (b) csQ = bv, bvO = cs, bvo = be, bs$ = bvo. (c) cs" = by, bv7 = bs, bvo' = bs, bs" = by. In particular, cs, bs, by and bvo are J3-spaces, but they are not Kothe spaces. Moreover, by and bs are -y-spaces, whereas both cs and bvo are not -y-spaces. None of the spaces cs, bs, by and bvo is solid.

Proof. (a) For each X E {cs, bs, by, bvo} we have 2 C X° since X C m and ma = 2. Because e E by we also get bv' C t, that is bva = t. We now prove cs' C e, which implies csa = bs' = 1. Let y = (Yk) P (and let us prove y cs° ). Then we may choose an index sequence such that IykI > 4°

(v E N°).

If we define x = (Xk) by

xk:=f(0

k,

1)k2-L if


(k ENO),

then x E cs by the Leibniz alternating series test. According to the choice of the inequalities

348

Topological sequence spaces: K- and FK-spaces

Ixkykl ? E k

2_" E

Iykl >_ E2' v

v

hold; thus xy 0 f, which implies y 0 cs°. As well we get bv01 C 2, if we put in the preceding proof

xk:= 12-" 0 if0
vEN0

(kEN°).

(b) Here we prove cs3 = by. The proofs of the other statements of (b) are quite similar and are left to the reader in Exercise 7.1.15. If x E cs and y E by, then xy E cs by the Du Bois-Reymond test (cf. 2.1.4(a)), therefore by C cso. To prove cs-6 C by we make use of the fact that

E:CS -iC,x=(xk)-*EX := (t0) is an isomorphism. Namely, if y = (yk) E cs13 (C Ba = m) and x = (xk) E cs with E Xk = 0 k

holds and if z = (zk) := Ex, then zNyN+1 -3 0 (N -3 oo), which, in view of

N

N

N

k

E xkyk = E(Yk - Yk+1) E x" + yN+1 E x" k=Q

v=°

I

(7.1:2)

v=°

(Abel's partial summation formula, see 2.1.1) and y E c813, implies (zk(yk - yk+1)) E cs.

(7.1:3)

Now, if x runs through all members in cs with Ek Xk = 0, then z := Ex runs through co, that is (7.1:3) implies (Yk - yk+1) E co = 1. Therefore y E by and cs3 C by is proved. (c) We now prove cs" = by. (As in part (b) we leave the proofs of the other identities in (c) to the reader (cf. 7.1.15).) Obviously, by = csF C cs" (cf. 7.1.5(b)). Now, if

y = (yk) E cs' (C e" = m), x E cs and z = (zk) := Ex, then, because (zNyN+1) E m and y E cs", the statement N sup

N

E zk(yk - yk+1)

< 00

k=O

follows from (7.1:2). In view of E(cs) = c, we have (yk - yk+1) E c7 = E. O Thus, Y E by and therefore cs" C by.

K-spaces

349

We will now leave the discussion of these results on (-duals of the spe-

cial sequence spaces considered in the present section. In particular, in later chapters we will make use of these results, for example in connection with the topologization of sequence spaces and the determination of the topological dual of certain sequence spaces.

Exercise 7.1.13. Prove that a sequence space X is solid if and only if mX C X (cf. 7.1.8). Exercise 7.1.14. Verify that for all C E {a, /#, y} the following statements hold (cf. 7.1.11(c)): (a) dc = S.

(b) Vr>0: II,.s=d} and d,s=Hi. Exercise 7.1.15. Prove the identities bv,6 = cs, bvo'3 = bs, bs0 = bvo

and

bv" = bs, bv0 = bs, bs" = by

in 7.1.12.

Exercise 7.1.16. Show that X' = X'1'a holds for any sequence space X. Bibliography: [122] 7.2

K-spaces

In Section 6.4 and subsequently we learnt that it is natural and useful to endow the sequence space w with the topology r,,,, which is the topology of coordinatewise convergence. Accordingly, we first collect and complete the properties of r,,,, which we established in earlier sections. From now on we denote exclusively by q,, (j E N°) the semi-norm

qi : w -+ R, x = (xk) -+ Ixi I (or its restriction to a subspace of w) and by r,, the locally convex topology, which is defined by the family (qj j j E N°) , and its relative topology on subsets of w.

Properties 7.2.1 (of r,,). (a) For each j E No the projection map Irj:(w,rw)-}(KI I),

x=(xk)-+xj

is continuous (since I ir? (x) I = q2 (x) for all x E w and j E NO). (b) In view of 6.4.18, a sequence in converges to an x E w if and only if it converges coordinatewise to x. (c) r,,, is the weakest locally convex topology on w such that each projection 7r? (j E N°) is continuous. (To prove this, note qj = I I o irj (j E NO) and recall that since r,, = a(w, cp) (cf. 6.6.8), r, is, by 6.6.5(a), the weakest

350

Topological sequence spaces: K- and FK-spaces

locally convex topology on w such that every qj (j E N°) is continuous.) (d) Let us consider the dual pair (w, gyp) in accordance with 6.6.2(c). By 6.6.8, we have r,, = o(w, cp). By 6.5.12, each f E (w, (w, o(w, cp))' can be represented by

f(x) = Exkf(ek)

(x = (xk) E w),

k

and, by 6.5.12 and 6.6.7, zp is up to the isomorphism

T

y = (yk) -+ f y with fy(x) _ E xkyk (x = (xk) E w) k

the dual of

Further, r,,, = o(w,gyp) = r(w,W) by 6.6.23(a).

A

It is both natural and useful to endow sequence spaces with locally convex topologies that are stronger than ru., (cf. 6.4.15). In such stronger topologies, convergence implies coordinatewise convergence. The results that flow from a consideration of such topologies justify introducing the following terminology.

Definition 7.2.2 (K-space). A locally convex space (X, r) is called a K-space4 if X < w and r,,, C r. In such a case r is called a K-topology on X. Remarks 7.2.3. (a) Obviously, (w, r,,) is a K-space. (b) If (X, r) is a K-space and Y < X, then (Y, rly) is a K-space. (c) If (X, r) is a K-space, then X is also a K-space if it carries any stronger locally convex topology.

(d) For each locally convex sequence space (X, r) the following statements are equivalent: (i) (X, r) is a K-space. (ii) The inclusion map ix : (X, r) -* (w, ru,) is continuous.

(iii) For every j E No the projection map irjIx : (X, r) -* (K, I ) is continuous.

(iv) For each j E N° the semi-norm qj is continuous on (X, r). (The implications (i) e* (ii), (iii) = (iv) and (iv) (ii) are trivial. The implication (ii) ' (iii) holds because 7rj I x = 7rj o ix.) (e) Each K-space is Hausdorff.

Almost all of the sequence spaces which we have considered till nowendowed with their `natural' topology, norm or metric-are K-spaces. 4 The `K' in `K-space' comes from the German word Koordinate which means 'coordinate'.

K-spaces

351

Examples 7.2.4. (a) The (locally convex) sequence spaces introduced in 6.4.15 are K-spaces since-as we stated already-their topology is stronger than Tu,. In particular, if

0, then II, becomes a K-space if we endow it with the family (pj j E N°) of semi-norms defined by

(x = (xk) Ell,., j E N°)

pj(x) := supIxkrjkl k

where (rj) is a sequence with 0 < ro < Ti < ... and supj rj = r. Proof. (a) is already proved. (b) and (c): Obviously, the norm 1111 under consideration satisfies qj(x) _ 1xj1 <_ (1x11 (x = (xk), j E N°). (d) and (e): The K-space property of (by, 11 ((b,,,) and (bs,11 Ilbs) and of the subspaces byo and cs follows from j-i

qj(x)=1xj1=

x0 - E(xk - xk+i)

< Ilxflbv

(x = (xk) E by)

k=0

and

xk-rxk

qj(x)=Ixjl =

kk=moo

< 211x(I68

(x =

E bs),

k=°

respectively.

0

(f) We will give a proof in 7.3.16(a).

Based on the a- and y-dual there are further possibilities to endow sequence spaces with K-topologies.

Definition and Remarks 7.2.5 (normal topology). Let X and Y be sequence spaces with V C Y C X*. Then for every y = (yk) E Y a seminorm qy is defined by ixkyk(

qy(x)

(x = (xk) E X).

k

The locally convex topology rl(X,Y) generated by Qy :_ (qy I y E Y) is

called the normal topology (on X with respect to Y). (a) Because p C Y, X° C V and py(x) < qy(x), the normal topology

352

Topological sequence spaces: K- and FK-spaces

obviously satisfies r,,, (x C rp,, C r7(X, Y) where Py is defined as in 6.4.15. In particular, (X,T7(X,Y)) is a K-space. (b) V x = (Xk) E X : x = Ek xkek (relative to 77(X, Y), cf. 6.4.16); in particular, cp is dense in (X,r7(X,Y)).

(c) Each f E X' has the representation (x = (xk) E X),

P X) _ 1: xk f (ek) k

and T (Y) C (X, v7(X, Y))' by virtue of the well-defined map T in 6.5.12(b).

T is an isomorphism, that is Y = X' (up to isomorphism), if and only if Y is solid. (d) The normal topology i7(X, Y) is a topology of the dual pair (X, Y) (together with the bidual map in 6.6.8) if and only if Y is solid. (e) Further to 6.6.23(a) and (b) we state r, = a(w, gyp) = r(w, gyp) = r7(w, cp),

a(m, e) C 17(m, e) C r(m, 2) C /#(m, t) = ill II- and r7(m, 8) C 'll ll.

ll

denotes the topology generated by 11 lkoo. The proof of these statements is left to the reader in Exercise 7.2.19. A where T)I

In 7.2.5 we stated that a(m,1) C 77(m, 8). However, it is not clear whether the topologies are equal. Using familiar methods we are able to verify that the two topologies have the same convergent sequences. This is contained in the following theorem as a special case.

Theorem 7.2.6. Let X be a sequence space such that V C X, let a = (ak) E X and let (a(')) with a(n) = (ank)k E X be a sequence in X. Then the following statements are equivalent:

(a) a(') -1 a in (X,a(X,Xa)) (b) a(') - + a in (X, i7(X, X')). Proof. It is obviously sufficient to verify the equivalence of (a) and (b) in the particular case a = 0. The implication `(b) = (a)' is trivially satisfied because a(X,X°) C r7(X,X°).

(a) = (b): We assume that (a) holds, but (b) does not, that is

Vy=(yk)EX' :

E ankyk

(7.2:1)

k

and there exist z = (zk) E X° and 8 > 0 such that b' n E No : E {ankzkI > 8,

(7.2:2)

k

where, if necessary, we consider, without loss of generality, a suitable subsequence of (a(')) which we again denote by (a(n)). Using gliding hump

K-spaces

353

arguments we construct a y E X° for which (7.2:1) fails and so (a) is contradicted.

Since o(X,X°) is stronger than r,,Ix (cf. 6.4.15), (a) implies coordinatewise convergence, that is

V k E to : ank n Ztf 0.

(7.2:3)

Now, noting (7.2:2) and (7.2:3), we inductively choose index sequences (nv)

and (kv), ko := 0, with the following properties:

k (v E N°)

E Iani kzk I < 2-1'

(7.2:4)

k=0

and, remembering that z E Xa and a(n) E X, 00

E

2-"

(v E N°).

(7.2:5)

With (7.2:2), (7.2:4), (7.2:5) and the second triangle inequality we get

(vEN°).

L,

(7.2:6)

We now define the desired sequence y = (yk) E X° by setting

ifk,
(kENO)

yk := {zksgn(azk) if k = 0 zo

and prove that (7.2:1) fails in the case of this y. For each v E N° we obtain from (7.2:4)-(7.2:6) that

k

E

00

ankyk - Elankykl - : Iankykl

ankyk

k=k+1 k+1

k

k=0

k=k+I+1

k

CO

E IankzkI - L Iankzkl - E lankzkI

k=k+1

k=0

>

6-2-v-2-"-2-"-2-v

>

2 if v is sufficiently large.

k=k+i+i

0 Remark 7.2.7 (strong summability). The result in 7.2.6 can be inThis completes the proof.

terpreted to yield an important statement connecting a statement about `summability' with one about `strong summability'. More precisely, if n:±

0. ank >- 0 and A = (ank), then X° C COA if and only if Ek ankIxkI That is to say, each element of X* is A-summable to zero if and only if

each x E X" is strongly A-summable to zero.

354

Topological sequence spaces: K- and FK-spaces

Proof. We have the following equivalences: X' C COA

E'*

V (xk) E Xa :

ankxk

n

0

k

4=*

IankxkI n

V (xk) E Xa :

0

n°0 11

f tlxEXa

.. 4=>

{

px (an)

[ps as in 6.4.15 and 6.6.4]

a(') -+ 0 in (X, o(X, Xa)) alni --3 0 in (X,77(X,Xa)) bxEXa

:

qx (a(n)) n2!r 0

[by 7.2.6]

[q,, as in 7.2.5]

n 40 Iankxkl

V (xk) E Xa : k

4=f

V (xk) E Xa : E ank IxkI

0 [because ank >_ 0].

k

0

Thus, the remark is proved.

Definition and Remark 7.2.8 (weak 7-dual topology). Let X, Y be sequence spaces with cp C Y C X''. Then the locally convex topology oy(X,Y) generated by the family of semi-norms rb (y = (yk) E Y) with rj'(x) := sup n

E xkyk

(x = (xk) E X)

k=0

is called the weak 7-dual topology, and because Ix? I < rei (x) (x = (xk) E X), it is a K-topology on X. For an example of a weak y-dual topology see Exercise 7.2.20 where we consider the space X := bs. Further to 7.2.3(d) we now characterize the `K-property' of a locally convex sequence space in terms of its weak topology.

Theorem 7.2.9 (K-space). If (X, r) is any locally convex sequence space with cp C X and dual space X', then the following statements are equivalent: (a) (X, r) is a K-space. (b) o(X, cp) C o(X, X'). (c) (X, a(X, X')) is a K-space.

In particular, if (X, r) is a K-space, then (X, r*) is also a K-space for every locally convex topology r* which is stronger than a(X, X').

Note, in (b) we consider both the dual pair (X, gyp) with the bilinear map defined in 6.6.2(c) and the dual pair (X, X') considered in 6.6.2(a).

K-spaces

Proof of 7.2.9. (a)

355

(b) : If (X,T) is a K-space, then fy E X' (y =

(Yk) E sp) where fy is given by

f (x)

xkyk = 1: yk7rk(x) k

(x = (xk) E X)

k

(cf. 7.2.3(d)(iii)). Thus a(X, gyp) C a(X, X'). (b) . (c) : Note o(X,V) = T, Ix. (c) . (a) : By (c) we have r,,,Ix C o(X, X') C T, and thus (a) holds.

0 In the remaining part of this section we examine, under certain hypotheses, the dual spaces of K-spaces. For this we recall the proof of 6.6.9 where we get a simple and useful representation of continuous linear functionals on (co, 1111.) by using the fact that for every x = (Xk) E co we have n X(n)

:_ E xkek - x (n . oo)

relative to II lao,

(7.2:7)

k=0

so that f (x) = E xk f (ek)

for every f E co'.

(7.2:8)

k

Obviously, we also get (7.2:8) if we replace in (7.2:7) norm convergence by the convergence in the weak topology. We made a similar observation in 6.5.12 in the case of the K-space (X, v(X, Y)) where X and Y are sequence

spaces with the properties p C X and cp C Y C V. These observations suggest that we consider in K-spaces (X, T) the set of all members x in X which satisfy the condition (7.2:7) for r and a(X, X'), respectively. In such a case x is said to be sectionally convergent and weakly sectionally convergent, respectively. Further we introduce the notions of section density and section boundedness. All of these notions were introduced and examined in the early papers of K. Zeller.

Definition and Remarks 7.2.10. Let (X, T) be a K-space with cp C X and dual space X', and let x = (xk) E X be arbitrarily given. Then [n1._

n k k=0

is called the n" section of x. We define the following properties: x has AK (sectional convergence) if x[n] -+ x in (X, T). x has SAK (weak sectional convergence) if x(n) -3 x in (X, a(X, X')). (Note, x has SAK if and only if AX) = Ek xk f (ek) for all f E X'.)

x has FAK (functional sectional convergence) if Ek xk f (ek) converges for all f E X'.

356

Topological sequence spaces: K- and FK-spaces

x has AB (sectional boundedness) if {x[n] I n E NO) is bounded in (X, 7), which is equivalent to weak boundedness by Mackey's theorem (cf. 6.6.18). That is, for every f E X' the sequence (Ek=° xk f (ek))n is bounded.' Corresponding to this notation we introduce distinguished subspaces of (X, 7) :

Sx Wx

Fx Bx

f x E X I x has AK} {x E X I x has SAK} {x E X I x has FAK}, {x E X J x has AB}.

('S' stands for 'strong), ('W' stands for 'weak),

We have

cc C Sx C Wx C Fx C Bx and W x C ip

(7.2:9)

where iP denotes the closure of cP in (X, 7). We get cP C Wx since every x E {p can be represented by x = Ek=oxkek for some n E NO. The other inclusions on the left hand side of (7.2:9) follow from the definitions of the distinguished subspaces and the corresponding remarks. To prove Wx C
quential dual Xf := {(g(ek)) g E X'} . Obviously, cp C Xf since (X, -r) is assumed to be a K-space (cf. also the proof of '(a) (b)' in 7.2.9). Now, the question arises whether

f -+ (f (ek))

(f E X')

gives an isomorphism from X' to X1. Such an isomorphism would allow

us to identify X' and Xf. In general, that idea fails as (c, 11 II() and f := lim E c' prove. However we have the following useful statements.

Theorem and Notation 7.2.11. If (X, T) is a K-space with cp C X and dual space X', then the following statements hold: (a) Xf = (ip)f and (gyp)'=Xf by virtue of f (f(ek)). (b) X' = X f by virtue of f -+ (f (ek)) X = tom: (X,7) is an AD-space6.

'The abbreviations AK, SAK, FAK and AB come from the German

Abschnittskonvergenz, schwache Abschnittskonvergenz, Abschnittskonvergenz and AbschnittsbeschrSaktheit, respectively. words

funktionale

6cAD' comes from for the German word Abschnittsdichte which means 'section

density'.

K-spaces

Proof. (a) Obviously, Xf C (ilp)f. . Conversely, if f E

357

that is

(f (ek)) E (T)f, then, by the Hahn-Banach theorem (cf. 6.5.4) there ex-

ists a g E X' with g1v = f, that is (g(ek)) = (f (ek)), which implies (f (ek)) E X f . Obviously, the map T : (T)' --+ (ip)f , f -a (f (ek)) is an isomorphism if and only if it is injective. However, the latter is the case since by definition W is dense in (ip, rI7). (b) The implication ` <--' follows immediately from (a). The converse holds,

since X 34implies that by 6.5.5 there exists an f E X' with V C Kern f and f $ 0 because, by hypothesis, 7 is a strict closed subspace of (X,r). Thus, the map T : X' --> Xf , f -> (f (ek)) is not injective. Using the definition of the weak sectional convergence we get immediately (cf. also (7.2:7) and (7.2:8)) that in the case WX = X the continuous linear functionals have a very simple representation. That observation suggests singling out the K-spaces with `maximal' distinguished subspaces.

Definition and Remark 7.2.12. A K-space (X, r) satisfying 'p C X is called

FAK-space :4=f Fx = X, AK-space :4-- Sx = X. In such cases we also say that (X, r) has AB, FAK, SAK and AK, respectively. On account of (7.2:9) each of these properties implies the

AB-space :4=* Bx = X, SAK-space :4=* Wx = X,

previous property, and each SAK-space is an AD-space.

6

The AD-spaces are separable spaces as we may easily deduce from Remark 6.7.18.

Remark 7.2.13. Each AD-space and each K-space (X, r) with X = X is separable, since (ek I k E N°) and {e} U {ek I k E NO } are algebraic bases of ip and ,c, respectively. A We complete the statement 7.2.11(b) in the case that the K-space under consideration is not only an AD-space, but also an SAK-space.

Theorem 7.2.14. Let (X, r) be an SAK-space with dual space X'. Then in addition to X`= X f the inclusion X f C X16 holds and each f E X' has the representation

f (x) = E xk f (ek) for each x = (xk) E X. k

The inclusion Xf C XR holds even for FAK-spaces.

Proof. If (X, r) is an SAK-space, then it is also an FAK-space, and by definition of FAK we get Xf C V. The second statement follows from the definition and the given reformulation of SAK in 7.2.10. As an application of the ideas considered above we list the distinguished subsets for some particular K-spaces. The easy proofs are left to the reader in Exercise 7.2.21.

358

Topological sequence spaces: K- and FK-spaces

Examples 7.2.15. (a) (w, Tw) is an AK-space which implies in particular

w=Sw=Ww=Fu,=B., and wf =gyp. (b) (2P, 11 11P) is an AK-space for each p E [1, oo[. Thus Sen = Wen = Ft, =

Btp = P. (c) (co, 1111.) is an AK-space. (d) In the case of (m, I1 1{,,) we have Sm = Wm. = co and Fm = Bm = m. In particular, (m, 11 l(am) is an FAK-space.

(e) If co C X < m, then for (X, II III) we have Sx = WX = co and Fx = Bx = X. In particular, (X,11 lk) is an FAK-space. (f) (bvo, ({ (Ibv) is an AK-space. (g) (bv, fl llb.,,) is an AB-space with Sb,, = bvO.

In closing this section, we examine the K-spaces introduced in 6.4.15, 7.2.5 and 7.2.8 and their relationship with the AK-space property.

Theorem 7.2.16. If X is a sequence space cp C X and Yc,, Yp, Yy are sequence spaces such that cp C Y( < XS (C E {a, f3, y}), then (c£ 6.4.15, 7.2.5 and 7.2.8): (a) (X, o(X, Yp)) and (X, rl(X, Y.)) are AK-spaces. Y.,. C V. (b) (X, oy(X, Y-,)) is anAK-space

Proof. The first part in (a) is contained in 6.5.12(a) whereas the proof of the second part and of the statement in (b) is simple and is left to the reader (cf. Exercise 7.2.22).

Exercise 7.2.17. Show that (bs, 1111b8) is an AB-space with Sb8 = cs and that (cs, 11 114) is an AK-space.

Exercise 7.2.18. Let X := {x = (xk) E w 13 f E C2 .

f is even, V k E No : xk = ak (f)

where ak(f) is defined as in 5.4.1. Prove that 1111 : X ---+ R, x ---+ llxll := IE f 11. (f as in the definition of X)

is a well-defined norm on X and that (X, {l II) is an AD-space, but not an AB-space.

Exercise 7.2.19. Verify the statements in 7.2.5. Exercise 7.2.20. Verify that the topology r11 11b. generated by 11 bs satisfies r1l I,,. = oy(bs, rc) = oy(bs, bv).

Exercise 7.2.21. Verify the statements in 7.2.15. Exercise 7.2.22. Prove 7.2.16(b) and the second part of 7.2.16(a).

11bs on

FK-spaces

359

Exercise 7.2.23. For each sequence space X containing W the following statements are equivalent: (a) X is a K6the space. (b) (X,ij(X,X°)) is sequentially complete where i7(X,X°) denotes the normal topology on X. Bibliography: [122], [211]

7.3

FK-spaces

Most of the sequence spaces hitherto introduced are, together with their `natural' topology (norm or metric), K-spaces (cf. Section 7.2). Moreover, some of them are even Banach spaces or F-spaces (cf. 6.3 and 6.7). With K-spaces that are also complete we may apply the essential tools of functional analysis like the open mapping theorem, the closed graph theorem, the uniform boundedness principle (theorem of Banach), and the BanachSteinhaus theorem. We will see below that K-spaces (X, r) that are also F-spaces have the advantage that their topologies are unique. We thus make the following definition.

Definition 7.3.1 (FK- and BK-space). A locally convex space (X, r) is called an FK-space and r is called an FK-topology if (X, r) is both, a K-space and an F-space. By definition, a BK-space is a normable FKspace and its topology is called a BK-topology. Now, we revisit the examples of K-spaces given in 7.2.3(a) and 7.2.4 to see whether they are FK- or BK-spaces. The references given below are to the places where the property of being a Frechet space or a Banach space was discussed for the spaces in question.

Examples 7.3.2 (FK- and BK-spaces). (a)

is an FK-space,

but no BK-space (cf. 6.7.5(a)).

(b) m, c, co, fo and f endowed with II III are BK-spaces (cf. 6.3.8(i) and 7.3.19). (c) (PP, II IIP) is a BK-space for every p E [1, oo[ (cf. 6.3.8(i)). (d) by and bvo endowed with II IIb are BK-spaces (cf. 6.3.42(b)). (e) bs and cs endowed with II Ilb. are BK-spaces (cf. 6.3.42(a)).

0

The remaining part of the present chapter is devoted to FK-space theory which was mainly established by K. Zeller (1949/50). However, within the confines of the present book we cannot exhaustively discuss the theory. We will provide a basis for further investigations in later chapters. In particular, we will consider in Chapter 8 domains of matrix methods as FK-spaces and study their topological structure. First we give a characterization of continuous linear maps between FKspaces from which we can deduce the monotonicity and uniqueness of their

360

Topological sequence spaces: K- and FK-spaces

topologies. After that we will verify how one can generate new FK-spaces from known FK-spaces.

Theorem 7.3.3 (continuous linear maps). Let (X, rX) and (Y, -ry) be an F- and an FK-space, respectively, and let T : X - Y be a linear map. Then the following statements are equivalent: (a) T is continuous. (b) iy oT is continuous where iy : (Y, -ry) -+ (w, r,,) is the inclusion map. (c) Ti := irk o iy o T : (X, rX) --- (K, 11) , x -+ [T (x)]1 is continuous for each j E N°. Here irk : w --* K, (xk) -3 xi is the projection map and [T(x)]j is the jth coordinate of T(x).

Proof. `(a)

(b)' is trivial since (Y, Ty) is a K-space and iy is consequently continuous (cf. 7.2.3(d)). Further, `(b) = (c)' obviously holds because 7ri (j E N°) is continuous (cf. 7.2.1(a)). (c) (a) : Now, let T? (j E N°) be continuous. To prove the continuity of T, by the closed graph theorem (cf. 6.7.13), it is sufficient to verify that T is a closed map. Because the topologies rX and ry are metrizable it is again sufficient (cf. 6.7.10) to show the validity of the implication

x(n) -> x in (X, rX) T (x(')) --a y in (Y, ry)

y = T(x)

}

for (x(°)) in X, x E X and y E Y. For this let x(n) -3 x in (X,rx) and T (x(n)) -3 y in (Y, Ty). Since convergence implies coordinatewise convergence in (Y,Ty), we obtain from the continuity of T? (j E N°) the identities

yi = [limT(x(n))]'

=

lim [T(x(n))]?

=

limTj (x(n))

=

T? (lim x(n))

n

[(Y,ry) is a K-space] [by the definition of T3 ]

n

[by the continuity of T3 ]

T?(x) = [T(x)]i, which imply y = T(x).

0

As an immediate consequence of the last theorem we get the monotonicity and the uniqueness of the topologies of FK-spaces in the following sense.

Corollary 7.3.4. (a) Monotonicity: Let (X, rX) and (Y, ry) be FKspaces and let X C Y. Then Ty [X is weaker than -TX.

(b) Uniqueness: Let Z be a sequence space and let r and r` be locally convex topologies on Z. If both (Z, r) and (Z, r*) are FK-spaces, then

r=r'.

FK-spaces

361

Proof. We get statement (a) by applying `(b) (a)' in 7.3.3 to the inclusion map iXY : X -+ Y noting ix = iY o ixy. Now (b) is a simple corollary of (a).

Remark and Notation 7.3.5. Statement 7.3.4(b) tells us that for every sequence space X there exists at most one topology r on it such that (X, r) is an FK-space; that is, the property of a locally convex sequence space (X, r) to be an FK-space depends only on the set X. For that reason we speak, in the case of an FK-space, about the FK-topology and about the FK-space X (instead of (X, r)). tl

As a further corollary of Theorem 7.3.3 we obtain the continuity of those linear maps which are important in summability theory as well as in the theory of topological sequence spaces, namely the continuity of matrix maps between FK-spaces. First we define the notion of a matrix map.

Definition 7.3.6 (matrix map). Let X and Y be sequence spaces over K, and let T : X -> Y be a linear map. Then T is called a matrix map if there exists a matrix A = (ank) such that X C WA and T(x) = Ax for all x E X. (Where no confusion can arise, we denote both the matrix map and the matrix by the same letter.)

Corollary 7.3.7. Matrix maps between FK-spaces are continuous.

Proof. Let X and Y be FK-spaces, and let A : X -+ Y, x --> Ax be a matrix map. To prove the continuity of A, by 7.3.3, it is sufficient to verify that for each n E N° the nth component map

An :=xnoiyoA:X -+K, x=(xk)-->Eankxk k

is continuous. However, the continuity of An follows from consideration of the maps

A,,,,: X -i K, x = (xk)

ankxk

(vEN°)

k=0

and from the Banach-Steinhaus theorem 6.7.16, since the continuous linear maps Anv = Ek=O ankIrk IX converge pointwise to An if v -4 oo.

In the second part of this section we extend the number of examples of FK-spaces. For this we consider methods to generate further FK-spaces using known FK-spaces.

Theorem 7.3.8 (subspace). Every closed subspace of an FK-space (endowed with the subspace topology) is an FK-space.

Proof. Apply 6.7.24(a) and 7.2.3(b).

362

Topological sequence spaces: K- and FK-spaces

Theorem 7.3.9 (intersection of FK-spaces). Suppose that (X,,, Pa), n = 1, 2, ... , are at most countably many FK-spaces. Then the intersection

X := nn=1,2 norms is an

...

Xn together with the family P := Un=1,2 ... P of semi-

FK-space7.

Proof. Since (Xn, Pn) is metrizable, by 6.4.25 we can choose for each n = 1,2.... a countable family Qn such that rp, = rqn. Now, it is easy to verify that P and Q := Un-1,2 Qn generate on X the same topology, say rp. Thus, because Q is a countable family, rp is semi-metrizable by 6.4.25 and it is metrizable since P is total because Pn is. Further, (X, P)

is a K-space since rP Ix C rp. Thus, (X, P) is proved to be an FKspace if we can show its completeness. For this let (x(")) be a Cauchy sequence in (X, P), that is (x(")) is a Cauchy sequence in (Xn, Pn) for each n = 1, 2, .... Therefore, since (Xn, Pn) is complete, we can choose an x E Xn with

x(") --4 x in (Xn, Pn) (n = 1, 2, ...).

(7.3:1)

Because in K-spaces convergence implies coordinatewise convergence x

does not depend on n, that is x E X. Obviously, by (7.3:1) we get x(") -* x in (X, P), and the completeness of (X, P) is proved. In contrast to the at most countable intersection, the union of at most countably many FK-spaces, even if it is a linear space, is not an FK-space in general, as we prove in Theorem 7.3.12. First we prove that the finite sum of FK-spaces is again an FK-space. Therefore any at most countable union of FK-spaces (if it is a linear space) can obviously be represented as the union of an increasing sequence of FK-spaces.

Theorem 7.3.10 (finite sum of FK-spaces). Let n E N be given, and suppose that (Xi, Pi) (i = 1, ... , n) are FK-spaces where Pi is an at most countable family of semi-norms. Then X := Ei= 1 X, is an FK-space and its FK-topology is generated by the family of semi-norms gp(i)...p(.) defined by n gp(1) ...p(n)

(x)

:= inf tP(1)(x(t)) a-1

n

I

X = E x{i} i=1

,

xiii E Xi }

(7.3:2)

J

where p(2) E Pi (i = 1, ... , n) are arbitrarily given.

Proof. Without loss of generality we assume n = 2, and, in accordance with (7.3:2), we consider on X = X1+X2 the maps gp(l)p(2) where p(i) E Pi (i = 1, 2) are arbitrarily given. Obviously, by (7.3:2) an at most countable 7 More exactly, P := Un=1,2,... P Ix. Here and subsequently we use similar abuses of notation. 8 We do not make use of the notion of para-norm (cf. 6.4.29) which would simplify the proof.

FK-spaces

363

family Q of semi-norms on X is d e f i n e d , s a y Q := (qk I k = 1, 2, ...). In

particular, (X, Q) is metrizable. First we proof that (X, Q) is a K-space. For this let k E NO be given. Then, on account of the K-property of (X1, P1) and (X2, P2) there exist

constants M(1), M(2) > 0, p E N, semi-norms p12) ... , pµ2) E P2 such that XI(')

I < M(=)

p(jl)

pµ1)

E P1 and

(x(j) = (x(')) E Xi , i = 1, 2).

p?=) WO) j=1

Now, we put M := max{M{1}, M(2)}. Then for any x = (xk) E X and arbitrarily given x(i) = (xk')) E Xi with x = x(1) + x(2) we obtain p

r2

Ixk I

<

Ixkl) I + Ixk2) I

<

L M(i) E poi) (x(i) i=1 j=1

A

< ME

{p(.1) (x(1)) ..gyp?2)(x(2)))

j=1

which holds for any representation x = x(1} +x(2) with x(i) E Xi. Therefore we get P

(x)

I PkI X(x)I = IxkI -< M

(x E X),

hence the K-property of (X, Q) is established by 7.2.3. Thus, (X, Q) will be proved to be an FK-space when we verify the completeness of (X, Q). For this we give without proof the following statement: If di (i = 1, 2) denotes the Frechet combination defined by

di (xW yW :=

2-j

(x(i)

y(i)) 1 + 0 ((x(i) _ (i)

P0

(

x(i) y(i) E Xi) ,

ifd:XxX -* R is given by d(x, y) := inf {dl (x(1), 0) + d2 (x(2), 0)

x - y = x(') + x(2) and

x(i) E Xi (i = 1, 2) }

(x, y E X)

and if d is the Frechet combination qk (x - y) d(x, y) := E 2-k 1+qk(x-y) k=1,2,...

(X, y E

X)

on X, then d is a translation invariant semi-metric and the topologies rrd and Td on X generated by d and d are equal. (To prove this statement,

364

Topological sequence spaces: K- and FK-spaces

it is sufficient to verify that in (X, d) and (X, d) the same sequences are convergent.)

Now, by the last remark, (X, Q) is complete if and only if (X, d) is. Moreover, by 6.2.21, the space (X, d) is complete if and only if each sequence (xn) in X with E. d(xn+1, xn) < oo converges. Thus, let (xn) be a sequence in X with n d(xn+1, xn) < oo, and, without loss of generality, let x° = 0. Then there exist sequences (z,()) in Xi (i = 1, 2) with (n E N°)

xn+1 - xn = z*, 1) + zn2) and

Because n-1

n zv=)

d2

n

zv=)

v=0

v=0

di (zn }, 0) < n

d(xn+1, xn) + n

En

1

T7 T 1}2

< 00

the series E. z,(°) converges in Xi (i = 1, 2) by 6.2.21(e). Consequently there exist x(2) E Xt (i = 1, 2) with n

d2 ( v-0

z(2)

- x(2), 0)

0

(n -+ oo).

Further, for each n E N° the identities (note x° = 0) n

n

xv) _

xn+1 v=0

z(1) +

n

E

zv2}

v=0

v=0

hold and we get d(xn+1,x(1) + x(2)) -+ 0 (n -+ oo); in particular, the sequence (xn) converges in (X, d). Thus, (X, d) is complete.

13

We apply the foregoing theorem to the direct sum of finitely many BKspaces.

Corollary 7.3.11. Let n E N be given, and suppose that (X2, p2) (i E N) are BK-spaces. Further, suppose X2 f1 Xj = {0} for i, j E Nn , i # j,

that is X := E 1 X2 is a direct sum. Then X is a BK-space and its BK-topology is generated by the norm X

p : X --3 R,

n

n

= E x(2) ---+ p(x) := Ep2(x(2)). 2=1

2=1

(7.3:3)

FK-spaces

365

Proof. This is an immediate corollary of 7,3.10 since in the direct sum X 1 xtzi. each x E X has a unique representation x=

The next theorem proves that the union of a countable increasing sequence of FK-spaces does not lead to further examples of FK-spaces.

Theorem 7.3.12 (union of FK-spaces). Let Xn (n E N°) be FKspaces with Xn C Xn}1 (n E N°). Then X := Un X is an FK-space if and only if there exists an n E No such that X = Xn. To prove this theorem we need the following theorem which is itself of independent mathematical interest and is an immediate corollary of 6.7.21.

Theorem 7.3.13. If X and Y are FK-spaces with X C Y, then X is a meagre subset of Y.

Proof. Apply 6.7.21 to the inclusion map ixy : X --* Y.

Proof of 7.3.12. The implication is trivial. On the other hand, if X. C X (n E N°) and X is an FK-space, then Xn is a meagre subset of X by 7.3.13 for every n E N. Consequently, X is meagre in itself, which contradicts the fact that F-spaces are of second category (cf. 6.7.20).

We get further examples of FK-spaces by consideration of the inverse image of already-known FK-spaces under certain continuous linear maps. This method of generation of FK-spaces plays an important role in summability as well as in topological sequence spaces.

Theorem 7.3.14 (generation of FK-spaces by linear maps).

Let

(X, P) and (Y, Q) be FK-spaces, T : X --} w be continuous and linear, and let

YT:=T (Y) = {xEX I T(x) EY} and T: YT ---pY, x-pT(x). Then the following statements hold: (a) YT is an FK-space and the FK-topology is generated by the family of semi-norms P* := P U {q o T 1 q E Q} .

(b) The map T : YT --> Y is linear and continuous (as a map between the FK-spaces YT and Y).

(c) If t is bijective, then Q* := (q o T I q E Q) generates the FKtopology of YT.

Proof. (a) The locally convex topology generated by P* is stronger than the relative topology of (X, P) on YT; in particular, (YT, P*) is a K-space which is by definition Hausdorff. Furthermore, (YT, P*) is semi-metrizable, and thus metrizable, since we may assume P and Q are at most countable

so that P* is at most countable. Thus, (YT, P*) will be proved to be an FK-space when we prove its completeness. For this let (x(' )) be a Cauchy

366

Topological sequence spaces: K- and FK-spaces

sequence in (YT, P*); that is, (x(n)) is a Cauchy sequence in (X, P) and (T (x(n))) is a Cauchy sequence in (Y, Q). Because (X, P) and (Y, Q) are complete we can choose an x E X and a y E Y such that

xini _+ x in (X, P)

and T (xlni) -+ y in (Y, Q).

Then T(x) = y, since T(x(n}) --> T(x) in (w, -r,,) on account of the con-

tinuity of T, and since T(x(n)) = iy(T(x(n))) -* iy(y) = y in (w, -r,,,) because the inclusion map iy is continuous (cf. 7.2.3(d)) and the limit of sequences in Hausdorff spaces is uniquely determined. Thus, by the defini-1 tion of P* and 6.4.16, we have verified x E YT =T (Y) and X(n) _,y x in (YT, P*).

(b) The map f is linear since T is. Moreover, the continuity of T is trivial by 6.5.1 because q o f is continuous on (YT, P*) for each q E Q.

(c) Now, let T be bijective. We prove that P* and Q* generate on YT the same topology. Noting Q* C P*, by 6.7.25 it is sufficient to show that (YT, Q*) is an F-space. Obviously, (YT, Q*) is metrizable since (Y, Q) is. To prove that (YT, Q*) is complete, let (x(n)) be a Cauchy sequence in

(YT,Q*). Then (T(x(n))) is a Cauchy sequence in (Y, Q), and we can choose a y E Y with T(x(')) -+ y in (Y, Q). Now, because T is bijective, we get x(n) ---i x := T-1(y) in (YT, Q*).

Remark 7.3.15. Checking the proof, we see that the statements in Theorem 7.3.14 remain true if we replace w with any linear space H, carrying a Hausdorff topology TH, and if (Y, Q) is an F-space with Y < H and THIQ C TQ.

a

We now give some more examples of FK-spaces and of sequence spaces for which no FK-topology exists.

Examples 7.3.16. (a) IT,. together with the set (pj ! j E N°) of seminorms defined in 7.2.4(f) is an FK-space for every r > 0. (b) 6 (endowed with a certain family of semi-norms) is an FK-space. (c) None of the spaces cp, rc or d,. (r > 0), endowed with any locally convex topology, is an FK-space.

Proof. (a) Let r > 0 be given, let (rj) be any sequence with 0 < r° < r1 < ... and supj rj = r, and pj be defined as in 7.2.4(f). We prove that IIr endowed with the family (Pi ( j E N°) is an FK-space. For this we consider, for every j E No, the matrix Aj := diag (r?). By 7.3.7 the matrix map Aj : (w, T,,,) -* (w, r ) is continuous and, by 7.3.14(c), the sequence space MA, = {x E w I Ajx E m} is a BK-space with the norm Pj

II II,,,)oA5, that is pj(x) = IiAjxll,,,, = suPk Ixkrr I (x = (xk) E MA,)-

Therefore, the intersection x := nj MA, endowed with (pj I j E N°) is

FK-spaces

367

an FK-space (cf. 7.3.9). Thus, (a) is proved if the identity X = II, holds. For each x = (xk) E IIr and each j E No there exists a ko E NO such that

bk>ko: Ixkrilb=IxkI' rj <-rj=1. r, 1

Therefore, Ixkrj [ < 1 for every k > ko, that is x E X. Conversely, for every x = (xk) E X and j E No the statement lim supk I xk rjkI k < 1 is true because p3 (x) = supkIxkr?I < oo. We consequently get

lim sup lxk1 < 1 k

-r

[since sup, r,, = r

that is x E II,. (b) The statement that 8 is an FK-space follows immediately from part (a) and 7.3.9 since for rn := n (n E N°) we have 5Z fl 1 Hr. . (c) The statements are obvious applications of 7.3.12:

cp = Un Xn where Xn := ({e°, ... , en}) (n E N°) are BK-spaces with

XnCXn+1Cco (nEN°). K = Un Yn where Yo :_ ({e}) and Yn := ({e, a°, ... , en-1 }) (n E N) are BK-spaces with Yn C Yn+1 C K (n E NO).

dr = U° Zn where Zn := IIr+1 (n E N) are FK-spaces by part (a) with 1

ZnCZn+1Cdr (nEN). In closing this section we mention two problems in the following remarks.

Remark 7.3.17 (union of FK-spaces). In 7.3.12 we stated that the union X of a strictly increasing sequence of FK-spaces Xn is not an FKspace. Now, the problem is whether-as a generalization of FK-topologies-

there exists a locally convex topology on X such that X retains certain important properties of FK-spaces, for example the monotonicity and the uniqueness of the topology, and such that important tools like the closed graph theorem, Banach's theorem and the Banach-Steinhaus theorem are still applicable. Boos showed in [32] that the locally convex inductive limit topology on X of the FK-topologies of Xn (n E N°) has these properties,

and he introduced the notion of IFK-spaces. (Note, in this sense, the spaces cp, K and d,. (r > 0) are not FK-spaces (cf. 7.3.16(c)), but each of them is an IFK-space.) Later, Grolie-Erdmann, who also uses the notation LFK-space instead of IFK-space, pointed out the importance of IFK-space theory by applying it in different fields of topological sequence spaces, for example to `sequence spaces of Maddox' (cf. [99] and [96], see also [96], [100] and [101]) and to the `f-dual' (sequential dual) of FK-spaces (cf. [97]).

Another important application of LFK-spaces was made by Benholz, a student of the author, who investigated `Factor sequence spaces and their LFK-topology' in his thesis (cf. [19]).

0

368

Topological sequence spaces: K- and FK-spaces

Remark.7.3.18 (cp-topology). If X is a sequence space, then, provided that there exists at least one FK-topology on X, the FK-topology is determined by the set X. An obvious problem now is whether for each sequence space X there exists something like a universal topology with properties related to those of FK-spaces. Ruckle approached this problem by considering cp-topologies (cf. [210] and [211]).

Exercise 7.3.19. Prove that (f, II

II

) and (fo,11 11.) are BK-spaces

(which is a statement in 7.3.2(b)).

Exercise 7.3.20. Show that mo (endowed with any locally convex topology) is not an FK-space.

Exercise 7.3.21. Let X (v E N) be FK-AK-spaces. Prove that n,, x-, and Ev 1 X (n E N) are also FK-AK-spaces. Exercise 7.3.22. Let X be a closed subspace of in a BK-space. Verify that X has finite dimension.

which is included

Exercise 7.3.23. Let X and Y be FK-spaces. Show that x f1 Y is closed in Y whenever X is closed in the FK-space X + Y. Bibliography: [254], [250], [261], [211], [267], [252], [32], [264], [263], [256], [223]

7.4

Functional analytic proofs

of some Toeplitz-Silverman-type theorems

The main result in Section 2.3 is the Silverman-Toeplitz theorem (cf. 2.3.7) which gives us a complete characterization of those matrices which sum all convergent sequences. It is essentially based on Theorems 2.3.3 and 2.3.5. To prove the non-trivial parts we used gliding hump arguments which are

very technical. Now, as promised in Section 2.3, we will do these parts of the proofs by applying the uniform boundedness principle (cf. 6.3.35) and the Banach-Steinhaus theorem (cf. 6.3.38)- First we recall Theorems 2.3.3, 2.3.5 and 2.3.6 I. Then, the Toeplitz-Silverman theorem 2.3.7I is an immediate consequence of 2.3.6I.

Theorem 7.4.1 (cf. 2.3.3). cj3 = c3 = mQ = t, and for each y = (Yk) E t and X E {co, c, m} the (well-defined) linear functional

f, : X -* K, (xk) -+

ykxk k

is continuous on (X, 11 llc) with operator-norm II f-mlI = Ilyfli.

Proof. Obviously m'3 C cF C co by 2.3.2(b). Thus the first statement is proved, if we show I C m'3 and co'3 C f.

Functional analytic proofs of some Toeplitz-Silverman-type theorems

369

First, let y = (yk) E e and x = (xk) E m be given. Then n

(n E N°),

IYkxkI <- IIxII.IIyfl1 < 00 k=0

which shows yx E I C cs, thus y E mI3 and therefore e C m 3. Further this shows for any X E {co, c, m} that fy is well-defined, linear and continuous with llfyll 5 Ilylll (cf. 6.3.14 and 6.3.19). To prove the non-trivial parts, cj C e and llfyll > Ilylli, we apply the Banach-Steinhaus theorem (cf. 6.3.38). For this we consider a y = (Yk) E c0 and define for each n E-t' ° the linear functional n

fn : X ---+ K,/(xk) -+ E ykxkk=0

Each fn is linear and continuous with operator norm II frill = Ek=0 lykl Further, trivially, (fn) is pointwise convergent on CO to fy : co -+ K. Therefore, by the Banach-Steinhaus theorem, fy is continuous with llfyll S supra'k=o Iykl < oo, that is y E t. It remains to prove the inequality llfyll > Ilylli. Let e > 0 be given. Then we choose an n E No such that IIylI1- IIf0II =

IYkl < k=n+1

,

that is IIfn11 > IIyll1- 3t

3

and for this n we choose an x E co with Ilxlloo < 1 such that 1fn(x)I

IifnIl - 3

Now, we have

Ify(x)I ? ifn(x)I-Ifn(x)-fy(x)I ? I1fn11-3-3 > IIyII1-3-23 = IIyII1-e and thus 11f,11 = Ilylll if we consider fy on X = co. However, since co C c C m, this is also true for X E {c, m}. 0

Next, on the way to a functional analytic proof of the ToeplitzSilverman theorem 2.3.7I, we draw an immediate corollary from 7.4.1 saying that for any infinite matrix A = (ank) the statement

E Iankl < oo, that is (ank)k E e for each n E N° k

is satisfied if and only if at least one (thus all) of the inclusions m C WA, c C WA and co C WA holds. These equivalences are a part of Corollary 2.3.4 which we will apply below. However, in the next step we replace the classical gliding hump arguments by an application of the uniform boundedness principle (cf. 6.3.35).

370

Topological sequence spaces: K- and FK-spaces

Theorem 7.4.2 (c C mA). For any matrix A = (ank) the following statements are equivalent: (a) in C MA, that is m C WA and A(m) C m. (b) c C mA, that is c C WA and A(c) C in. (c) co C MA, that is co C WA and A(co) C m. (d) IIAII := supn Ek lankI < oo. In such a case, the matrix map A : (X, 11 III) --+ (m, it Iloo), x --a Ax, where X E {co, c, m}, is continuous, and the operator norm IIAIlx,m of A satisfies II All x,m = IIAII

Proof . The implications (a) = (b) = (c) are obviously true. (a) : If IIAII < oo, then we obtain m C WA from 2.3.4 and (d) (x = (xk) E m, n E N°).

<- ilxll. IIAII < oo

(7.4:1)

Thus A(m) C m as required and the map A : (X, 11 11,,.) --i (m, Il

Iloo),

where X E {co, c, m}, is continuous with IlAllx,m <- IIAII. (d) : Let A(co) C m. We prove IIAII < oo and IIAIIco,m >- IIAII (c)

(which also implies IIAIlx,m >- IIAII for X E {c,m} and, therefore, that IIAIIx,m = IIAII for X E {co, c, m} ).

We note (ank)k E cod = P (n E N°). In particular, by 7.4.1, the `row functionals'

A : X --* K, (xk) - E ankxk

(n E N° ),

k

where X E {co, c, m}, are well-defined, linear and continuous with operator

norms 11AII = Ek lankl. Since co C mA they are obviously pointwise bounded, and thus norm bounded by the uniform boundedness principle 6.3.35, that is supn iifnli = 11AII < co To prove that IlAllco,,n > IIAII holds, for an arbitrarily given c > 0 we

choose an r E N with llfrli ? 11AII - 2 and then an x E co such that IIxII. < 1 and l f,.(x)l > ilf,.Ii - 2. This gives ilAxlloo >-

I fr(x)i >- ilfril -

2

> IIAII - 2 - 2 = IIAII - e.

Hence, IlAilco,m = IIAII

Before we proceed to a functional analytic proof of the ToeplitzSilverman theorem, we give a further functional analytic proof of the main part of the last theorem. This proof is based on the fact that matrix maps between FK-spaces are continuous (cf. 7.3.7). .

Proof of 7.4.2 based on 7.3.7. We prove here only the non-trivial part, (d)'. Concerning notation, we refer to the namely the implication `(c) foregoing proof of 7.4.2.

Functional analytic proofs of some Toeplitz-Silverman-type theorems

371

Let A(co) C m. Then A : co -+ m, x -+ Ax is a matrix map between the BK-spaces (co, II 11,,.) and (m, II 11,,) and hence continuous by 7.3.7. Therefore, IIAIIco,m < oo and (cf. 6.3.14 and 6.3.19)

I f.(x)I = E ankxk <_ IIAxII. < IIAIIco,m 11xIIoo

(x = (xk) E Co).

k

Hence, IIAIIco,m >_ IIfn11= Ek la.kl (n E N°); thus IIAII S IIAlkco,m < oo. Moreover, from (7.4:1) we get II All x,m <_ I1Ai1, where X E {co, c, m}. Thus, 0 IIAII = IIAIIx,m < oo for any X E {co,c,m}.

Applying the last theorem and the (general version of the) BanachSteinhaus theorem we now characterize those matrices which map all null sequences to convergent sequences.

Theorem 7.4.3 (conservative for null sequences). For any infinite matrix A = (ank) the following statements are equivalent: (a) A is conservative for null sequences, that is Co C CA. (b) Co C WA and A(c°) C c.

(c) A satisfies (Zn) and (Sp), that is IIAII < oo and Aek = (ank). E c for each k E No . LIMIT FORMULA: If A is conservative for null sequences, then

(ak) E t and

limA x = E akxk

(x = (xk) E co),

k

where ak := limn ank (k E NO).

Proof. We prove here only that (c) implies (a) and the statements concerning the limit formula. For the other parts we refer to the proof of 2.3.6, since they are done by standard arguments applying 7.4.2 (which is identical with 2.3.5). So we assume that (c) holds. Then, since IIAII < oo, we have that the sequence (f,,) of all (continuous) row functionals of A, which we considered in the proof of 7.4.2, satisfies 11fnll 5 IIAII (n E N1°). That is, (fn) is norm bounded, and thus pointwise bounded. Moreover, since (Sp) holds, (fn(ek))n converges for all k E N°. Therefore, (fn) is pointwise convergent on gyp, which is, because (co,11 Iloo) is an AK-space (cf. 7.2.15(c)), a dense subset of co with respect to 11 By the general version of the BanachSteinhaus theorem 6.3.39, (fn) is pointwise convergent on co and the limit

functional, that is limA (restricted to co ), is continuous on co. Further, by the continuity of limA and the AK-property of (coi11 lloo), it follows that r limA x = 1 Tm limA xlrl = firm E akxk = k=°

akxk

(x = (xk) E Co)

k

which is the desired limit formula. In particular, (ak) E coR = 2.

0

372

Topological sequence spaces: K- and FK-spaces

Now, the Toeplitz-Silverman theorem 2.3.7, which characterizes the conservative matrices, is an immediate corollary of the foregoing theorem since c = co ® (e). So we have now given a functional analytic proof of the Toeplitz-Silverman theorem.

Using the same methods as in the proof of the Toeplitz-Silverman theorem, we now characterize those matrices which sum all absolutely summable sequences. The proofs closely follow the scheme of the foregoing proofs.

Theorem 7.4.4. Q = m, and for each y = (Yk) E m the (well-defined) linear functional

fy : t --+ K, (xk) -* E ykxk k

is continuous on (t, II Iloo) with operator-norm Ilfyll = IIyIIao

Proof. First, let y = (yk) E m and x = (xk) E t be given. Then n

L.1Iykxkl

(n E N°),

IIxII.IIYII1 < 00

k=0

which shows yx E t C cs; thus y E PO and therefore m c P'. Further it shows that fy is well-defined, linear and continuous with Ilfyll < Ilylloo (cf. 6.3.14 and 6.3.19). Ilylloo we apply To prove the non-trivial parts PO C m and IIfylI the Banach-Steinhaus theorem 6.3.38. We consider any y = (yk) E P and define for each n E N° the linear functional n

fn : f -+ K, (xk) -

Ykxk k=0

Each fn is linear and continuous with operator norm Ilfnll = supkENO Iykl. Moreover, (f,,) is pointwise convergent on f to f y : e -> K. Therefore, by the Banach-Steinhaus theorem, fy is continuous with Ilfyll <_ supra llfnll = supkENO lykl < oo, that is y E m. So it remains to prove Ilfyll ? llylloo, which is trivial. Indeed, for each k E N we have

Ileklli = 1 and lfy(ek)l = Iykl < Ilfyll Ileklli = Ilfyll;

0 thus Ilylloo : Ilfyll and so Ilfyll = Ilylloo. Now, on the basis of 7.4.5, it is easy to characterize the matrices which are `conservative for absolutely summable sequences'.

Theorem 7.4.5 (P C MA)- For any matrix A = (ank) the following statements are equivalent: (a) t C MA, that is t C WA and A(P) C m. (b) IIAII°° := Supra,k Iankl < 00.

Functional analytic proofs of some Toeplitz-Silverman-type theorems

373

In such a case, the matrix map A : (P, 11 111) -+ (m, 11 11"), x -+ Ax is continuous and the operator norm IIAIIt,m of A satisfies IIAIIk,m = IIAII°°

Proof. (b) = (a) : If IIAIIO° < oo, then (ank)k E 0 (n E N°) by 7.4.4, and we obtain 2 C WA from

E ankxk

<- 11x1{, IIA1100 < oo

(x = (xk) E e, n E N°).

(7.4:2)

k

Thus A(f) C m as required and the map A :

(f, II

II00) -> (m, 11 11.) is

continuous with (IAIIt,m cfAII°°.

(a) = (b) : Let f C mA. We prove IIAII°° < oo and IIAIIe,m >- IIAII-, which gives IIAIlt,m = IIAII°° Obviously, (ank)k E 113 (n E N°). In particular, by 7.4.4, the `row

functionals'

fn : e

K, (xk) -+ 1: ankxk

(n E N°)

k

are well-defined, linear and continuous with norm IIfnII = supk Iankl. Since

t C mA, they are pointwise bounded and thus norm bounded by the uniform boundedness principle 6.3.35, that is IIA1100 = supra IIfnII < 00. To prove that IIAIIt,m >- IIAII°° holds, we consider an arbitrarily given

£ > 0 and choose r E N with IIfrII ? IIAII°° - 2. Then we choose x E f such that IIxhii < 1 and I fr(x)I ? IIfrII - L. Hence, we obtain IIAxii00

- Ifr(x)I - IIfrII->-IIAII°°-2-2=IIAII°°-£. o

Thus, IiAIlt,m = IIAII°°

Applying the last theorem and the (general version of the) Banach-

Steinhaus theorem we now characterize those matrices which map all absolute summable sequences into convergent sequences. We will apply this theorem in Section 10.2 to prove a functional analytic statement concerning the so-called two-norm convergence.

Theorem 7.4.6 (1 C CA). Let A = (ank) be an infinite matrix. The following statements are equivalent:

(a) f C CA, that is t C WA and A(t) C c. (b) IIAII°° < oo and A satisfies (Sp). LIMIT FORMULA: If I C CA, then

(ak) E m and limA x = Lr akXk (x = (xk) E 1), k

where ak := limn ank (k E NO).

374

Topological sequence spaces: K- and FK-spaces

Proof. We prove here only that (b) implies (a) and the statements concerning the limit formula. The proof of '(a) = (b)' can be done by standard arguments applying 7.4.5. We assume that (b) holds. Then, since IIAII°° < oo, we have that the sequence (fn) of all (continuous) row functionals of A, which we considered in the proof of 7.4.5, satisfies llfnll < IIAII°° (n E NO), that is (fn) is norm bounded and thus pointwise bounded. Moreover, since (Sp) holds, (f, (ek)) converges for all k E No. Therefore, (fn) is pointwise convergent on cp, which is a dense subset of t with respect to 1111, because (1, 11111) is an AK-space (cf. 7.2.15(b)). By the general version of the Banach-Steinhaus theorem 6.3.39, (fn) is pointwise convergent on t and the limit functional, that is limA (restricted to 1), is continuous on 2. Further, by the continuity of limA and the AK-property of (1, 11 111), it follows that limA x = lira limA xl''l = i m E akxk = k=o

(x = (xk) E P)

akxk k

which is the desired limit formula. In particular, (ak) E PQ = m.

0

It is easy to characterize those matrices A which sum all sequences with bounded partial sums, that is bs C CA (cf. Exercise 2.4.19). Using the fact that bs is dense in (fo,ll II00), see Exercise 2.9.11, and applying the (general version) of the Banach-Steinhaus theorem (cf. 6.3.39) we get a `soft' proof of the following extended version of Theorem 2.4.9.

Theorem 7.4.7. For any conservative matrix A = (ank) with column limits ak (k E N°) the following statements are equivalent: (a) A is strongly conservative. (b) bs C CA.

(c) lim sup,, Ek lank - an,k+1 - ak + ak+1 I = 0Proof. The conditions (b) and (c) are equivalent by Exercise 2.4.19. Further, we omit the proof of the implication '(a) . (c)' which is the trivial part in the proof of 2.4.9. (b) (a): Since IIAII < oo, the row functionals

fn : to -> K, (Xk) -} 1: ankxk

(n E

)

k

of A are well-defined and continuous on the BK-space (fo, II II00). More-

over, for the same reason, we have to C M C MA; therefore, the row functionals are pointwise bounded. Since bs C CA, they are pointwise con-

vergent on bs, which is dense in the BK-space (fo,11 I1.). Thus, by the (general version of the) Banach-Steinhaus theorem, (fn) is pointwise convergent on fo which is equivalent to to C CA. Now, f C CA, that is (a), 0 follows because f = to ® (e).

The dual of FK-spaces

375

Remark 7.4.8. Instead of applying the Banach-Steinhaus theorem, we can show that (b) implies fo C CA, by the use of bs0 III = fo and by

standard estimates similar to those in the proof of 2.9.3. A As we mentioned in Section 2.4, there is no purely functional analytic proof of Schur's theorem. However, by combining (classical) analytic and functional analytic arguments we will give a relatively short proof of (an extended version of) Schur's theorem in Section 11.3.

Exercise 7.4.9. Prove cco C e by applying the fact that matrix maps between FK-spaces are continuous (cf. 7.3.7), to a certain triangular matrix defined by an arbitrarily given y E C6 .

Exercise 7.4.10. Prove `(a)

(b)' in Theorem 7.4.5 by applying the fact that matrix maps between FK-spaces are continuous (cf. 7.3.7).

Exercise 7.4.11. Let A = (ank) be an infinite matrix, and let 1 < p, q < oo with n + 9 = 1 be given. I. Prove the equivalence of the following statements: (a) ep C mA, that is ep C WA and A(ep) C m. (b) (ank)k E eq for each n E No and sup,, 11(ank)kllq < 00. II. Let Pp C MA be satisfied. Verify that the matrix map A : (eP, 1111p) ---j (m, 11 ll.) , x -+ Ax is continuous with operator norm sup,, ll(ank)kllq < 00111. Show that ep C CA if and only if ep C mA and A satisfies (Sp). Prove, further, that in such a case (ak) E eq

and

lima X _

akxk

(x E ep),

k

where ak denotes the limit of the kth column of A. Bibliography: [254]; [267], [159]

7.5 The dual of FK-spaces First we complete the statements in 7.2.11 and 7.2.14 on the sequential dual of K-spaces for the case of FK-spaces.

Theorem 7.5.1. (a) Let (X, T) be an FK-space. Then for every y = (yk) E Xa the linear functional fy defined by fy//\x) :=

YkXk

(x = (xk) E X)

k

is continuous, that is fy E X' for each y E X's. In particular, XR C Xf if cp C X.

(b) If (X, T) is an FK-FAK-space, then Xf = XR.

376

Topological sequence spaces: K- and FK-spaces

Proof. (a) For given y = (yk) E XQ and n E No we define fn E X' by n

fn(x) E ykxk

(x = (xk) E X).

k= G

Then fn is continuous since (X, rr) is a K-space. Obviously, the sequence (fn) converges pointwise to fy which implies f E X' by the Banach-Steinhaus theorem (cf. 6.7.16) because (X, T) is an F-space. (b) This statement is an immediate consequence of part (a) and the last statement in 7.2.14.

Theorem 7.5.1 enables us to augment the statements in 7.2.15 on distinguished subsets and sequential duals of special K-spaces (d also 7.3.2). For convenience we repeat the statements in 7.2.15.

Examples 7.5.2. (a) (w, r) is an FK-AK-space which particularly implies w = S,,, = W. = F,,, = A, and wf = cp. (b) For each p E [1, oo[ the space (tP,11 f lp) is a BK-AK-space. In particular, Sep = Wep = Fep = Bep = 2p, 2f = m and (tv)f = 1q for each p > 1 where v + q = 1. [Apply 7.5.1 and 7.1.11(c).]

(c) (co,11{1.) is a BK-AK-space and c/ = 1. [Cf. 7.5.1(b) and 7.1.11(d).] (d) In the case of (m, II lk,,,) we have Sm = Wm = co, Fm = Bm = m and mf = t. Therefore, (m, 11 Qom) is a BK-FAK-space. [Apply 7.5.1(b) and 7.1.11(c).]

(e) If co C X < m, then for (X, I(

we have SX = Wx = co,

FX = BX = X and Xf = 1. In particular, (X,11 fl,) is an FAK-space. [Note,

(f)

= co in (X, II I1,o), part (c) and 7.2.11(a).]

(bvo, 11 1lbv)

is a BK-AK-space with bvof = bs. [See 7.5.1(b) and

7.1.12(b).] (g) (bv, 11 I1bv) is a BK-AB-space, Sbv = Wbv = Fbv = bvo and bvf = bs.

Proof. To prove the statements in (a)-(f) use the hints in square brackets. (g) First we show bvf = bs. Let f E bv' be given. Then f lbvo E bvo' and

because baof = bs (cf. part (f)) we get (f (en)) E bs, that is bvf c bs. Conversely, by part (f) for every y E bs there exists a g E bvo' with (g(en)) = y. Now, the Hahn-Banach extension principle gives us a G E bv' with G1bvo = g. Since (G(en)) = (g(en)) = y we also get bs C W. To prove Sbv = Wbv = Fbv = bvo it is sufficient to verify Fbv C bvo since bvo = Sbv C Wbv C Fbv holds by 7.2.15(g) and (7.2:9). So, let x = (xk) E Fbv be arbitrarily given. Then (xk f (ek)) E cs for all f E bv' which is by bvf = bs equivalent to x E bs' = bvo (cf. 7.1.12(b)).

The following example shows that Xf = XQ does not hold in general for SAK-spaces (cf. Theorem 7.2.14).

The dual of FK-spaces 377

Example 7.5.3. Let X := cp and let r be the K-topology on cp generated by 1111oo. Then (X,-r) is an AK-space (because Sx D w) and Xf = F (by 7.2.11(a), since W is dense in (co, JIB) and because co = F (cf. 7.5.2(c))

p

and X16 = w (cf. 7.1.11(c)).

The statements on the dual of an SAK-space obtained in 7.2.14 (cf. 7.5.1

also) enable us to determine the dual space of particular FK-AK-spaces discussed in the first part of this section. From this we can deduce the dual space of the BK-spaces c and by. In the second part of this section we will determine the dual spaces of FK-spaces, in the sense of 7.3.8-7.3.14, in terms of the dual spacesof the generating FK-spaces. By 7.2.11, 7.2.14 and 7.5.1 and by the fact that w, FP (1 < p < oo), co and bvo are FK-AK-spaces (cf. 7.5.2) we immediately obtain the following theorem.

Theorem 7.5.4 (dual spaces of w, i'P, co, bvo ). If X is an FKSAK-space, then

X' = Xf = XR by virtue of f ---+ (f (ek)), f(x) = E xkf(ek)

(x = (xk) E X and f EX')

k

and

fu E X'

(y = (yk) E X')

where

fy : X -* K, x = (xk) - EYkxk.

(7.5:1)

k

The following statements also hold: (a) w' = cp and (f (ek)) E cp for each f E w'. (b) (FP)' = F9 and (f (ek)) E Fq for every f E (FP)' where 1 < p < oo and

v+Q =1. (c) F' = m and (f (ek)) E m for each f E F'. (d) co' = t and (f (ek)) E F for every f E C0'. (e) bvo' = bs and (f (ek)) E bs for each f E bvo'.

Additional statements: (i) Let X E {FP, F, Co} where 1 < p < oo be given and endowed with the natural norm. Then the isomorphism

T:X'-+XQ,f-4(f(ek)) is an isometry when X' and XR are endowed with the operator norm and natural norm, respectively.

378

Topological sequence spaces: K- and FK-spaces

(ii) If by is given the norm

Ixlbv:bvx=(xk)-+

EIxn-xn+ll+llnmlxn) n

which is equivalent to II Ilk, (cf. 6.3.18), then the isomorphism ((bvo, I Ib )', 11 11) --+ (bs, II

IIba) , f -} (f (ek))

in (e) is also an isometry.

Proof. The main statements of this theorem are contained in 7.2.11, 7.2.14 and 7.5.1 and in 7.5.2. It remains to verify the additional statements.

(i) Let y E XQ be given and let fy be defined as in (7.5:1). Then T is proved to be an isometry if II f y 1j equals the norm of y in V. If X = co or X = f, this has already been shown in 7.4.1 and 7.4.4, respectively.

X := 1P(1 < p < oo) : Let q be a real number with 1+ q = 1. We Ilyllq in the proof of (P1)Q in 7.1.11(c) where fy is just the map T. To prove IIfyII > Ilyllq we consider, as in the proof of 71.11(c), the sequence X(n) = (xkn})k defined by already showed IIfyII

xkn)

f lyklq

10

ifyk#0andk
(kENO).

otherwise

Replacing fn by fy in the proof of 7.1.11(c) we obtain thus Ilyllq = supra Ily'n'IIq

IIY"n'IIq

IIfyII;

IIfyII

0

(ii) This is left to the reader in Exercise 7.5.11(a).

Because c = co ® e and by = bvo ® e we can reduce the determination of the dual space of c and by to that of co and bvo, respectively. We carry out this idea in the case of c and leave it to the reader in the case of by (cf. 7.5.11(b)).

In the case of the BK-

Theorem 7.5.5 (dual space of (c, II

space (c, 11 11.) and its dual c' the following statements hold: (a) aye+Ek_o(xk-ay)ek Tt:L(O x holds in (c, 11 II0) for each x =(xk) E c where ay := limk xk. In particular, the BK-space c is separable.

(b) t1 f Ec' Vx=(xk)Ec: (i)

(f(ek)) E £,

(ii) f (x) = ay (1(e)

(iii)

11f 11 = If (e)

- k=a Ew f (ek))

- 1 f(ek)I 00

+

0.0

+

k=O

k-o

xkf (ek),

If(ek)I

The dual of FK-spaces

(c) (c', II

II) = (e, I {

379

111); more precisely, the map

T : (c`,1111) - (e,

11111), f -+ (tk),

where

to := x(f) := f (e) - Eco f (ek) and tk+1 := f (ek) for k E No, k=o

is an isometric isomorphism.

Proof. (a) For every x = jxk) E c we prove n

x(n) - axe + E(xk - ax) ek n:!q x

in

(c, II 11.),

(7.5:2)

k=0

n->qo

which is obviously equivalent to JIx - x(n)II 0. From this, using standard arguments, we get that the BK-space c is separable by 6.3.29(a) and by considering the countable set n

{cie +

(xk - a)ek

l

I

a, xk E Q* , n E No )

k=0

with

JJ

ifK=R 1Q+iQ ifK=C. JQ

Now, let x = (xk) E c be given. Then x - axe = (xk - ax)k E co and we have n

(x - axe)fn} = E(xk - ax) ek

n- x - axe in (co,11 1100).

k=O

This implies Il x - x(n) II00 = II (x - axe) - (x - axe) (n]II. n- 0.

(b) Let f E c' be given. Because f 1,0 E co we obtain (f (ek)) E P by 7.5.4(d); thus (i) is proved.

Since f is linear and continuous, we have for each x = (Xk) E c the following relation where x(n) is defined as in (a)

f ( n-aoo lim x(n)) = lira f (x(n)) n-+oo

f (x)

[f is continuous, (7.5:2)]

n

=

lim f axe +

(xk - ax)ek) k=0 n

axf (e) + Jim E(xk - ax)f (ek) k=0

[f

is linear]

380

Topological sequence spaces: K- and FK-spaces

(cf. (i)].

axX(f) + E0,0xkf (ek) k=0

This proves (ii). Moreover, 00

If (x)I

00 If (ek)I} (IxwI + E

Ia.I IX(f)I + E Ixkl if (ek)I <-

<-

k=0

IIXIIo,

k=0

that is (cf. 6.3.19),

llf11 <- IX(f)I+EIf(ek)I = Iltlll with t=(x(f),f(e0),f(el),... 00 k=0

To prove also IIf II >- IItII1 we consider for any n E No the sequence z(n) = (z(n)) defined by (n)

zk

ak if k < n

(k, n E Rl° )

if k > n

On

with

ak

sgn f (ek)

and

F+n

sgn (1(e)

-

f (e

k))

k=0

Trivially, Z(n) E c with limk z(kn) = f3 and 11Z(n)II. < 1 for each n E N°.

Further, we have f (e)

e

n

- k=0 f (ek)

n

+ E If (ek)I k=0

On t f (e) k=0

f (ek)) +

n

akf (ek) k=0 o0

akf(ek)+ E /3nf(ek)

fnX(f)+ k=0

k=n+1

00

e

NnX(f)+Ezkn)f(ek)j = If(z(n))I

[cf. (ii)]

k=0

<

IIf II Ilz(n) II. <- IIf II

[cf. 6.3.19(c)];

thus, as n tends to oo, the desired inequality IX(f )I +>k if(ek)I < HA I is obtained.

(c) That the well-defined and linear map T is an isometric map follows

The dual of FK-spaces

381

from (iii) in part (b). In particular, T is injective. Finally, T is surjective. For any y = (Yk) E e the linear functional 00

f : c -4 K, x = (xk) -* yo l km xk +

2kyk+1 k=o

is well-defined, since cO = 1, and linear. Moreover, f is continuous, since it has the representation f = y°lim+flyk+,i, where f(yk+,) is defined as in 7.5.1(a), as the sum of two continuous linear functionals. Obviously, 0 f (ek) = Yk+1 for each k E No.

In the next remark we sketch another method, using the dual map of an isometric isomorphism, to determine the dual of the BK-space by and to gain a suitable representation of a continuous linear functional on by.

Remark 7.5.6. The map E-1 : (e, 11 111) ---4 (by, II Ilbv)

:

(xk) -} (xk - xk-1) where x-1 := 0

is-as is well known-an isometric isomorphism. Then the dual map :

(by', 11

11) - * (t', 11

11)

is also an isometric isomorphism. More-

over, from 7.5.4(c) we know that T : (t', 11 11) -* (m, 11 11.)

: f -+ (f (ek))

is an isometric isomorphism too. Hence (by', it ) = (m, 1111.) via the isoI metric isomorphism To (E-') . Now, using the representation of continuous linear functionals on t we may obtain a representation of continuous linear functionals on by. 0

As we said earlier, we will determine the dual space of FK-spaces in the sense of 7.3.8-7.3.14 in terms of the dual spaces of the generating FKspaces. In the case of (closed) subspaces the solution of this problem is an easy consequence of the Hahn-Banach extension principle (cf. 6.5.4). Theorem 7.5.7. Let (X, r) be an FK-space and Y be a closed subspace of (X, r). Then (Y, rl y) is an FK-space by 7.3.8 and Y' = {fly I f E X') .

The mapping T : X' - Y', f - fly is an (algebraic) isomorphism if and only if X = Y.

Proof. The identity Y' _ {fly I f E X' j holds since on the one hand each continuous linear functional on (X, r) is also linear and continuous on (Y, rly) and on the other hand, by the Hahn-Banach extension principle (cf. 6.5.4), for every g E Y' there exists an f E X' with f {Y = g. To prove the additional statement we assume first that T is injective,

that is T (f) = fly = 0 implies f = 0 for each f E X'. Accordingly, using 6.5.5 we get immediately X = Y. Conversely, if X = Y, then T is obviously an isomorphism.

0

382

Topological sequence spaces: K- and FK-spaces

Corollary 6.5.7 enables us to determine the dual of the intersection of at most countably many FK-spaces.

Theorem 7.5.8. Suppose that X (n = 1, 2, ...) are FK-spaces with dual spaces X. Then the dual X' of the FK-space X := ()n=1,2,... X is determined by

X'=

{fIx I f EXn'},

L.:

n=1,2,...

that is

V f E X' 3 j1,...,jk E {1,2,...}Vi E Nk 3 fi E XX.' : k

f(x) _ E fi(x) for each x E X i=1

and, conversely, each linear functional k

(fi E Xi,', ji E {1,2,.. .} and i E Nk)

f := E fi l x

(7.5:3)

i=1

is continuous on the FK-space X. Proof. That each linear functional of the kind (7.5:3) is continuous on the FK-space X follows immediately from the monotonicity of FK-topologies (cf. 7.3.4(a)).

To prove that each f E X' can be represented according to (7.5:3), we assume that Pn is a family of semi-norms on Xn which generates the FK-topology of X n (n = 1, 2, ...). The FK-topology of X is generated by

P:= U {plx IpEPn} n=1,2,...

(cf. 7.3.9). Now, let f E X' be given. Then there exist

M > O and j1i ... , jk E {1, 2, ...} and pi E P (i E Nk ) such that

k

lf(x)I < MEpi(x)

(x E X).

i=1

Applying Corollary 6.5.7 we may choose gi E X' (i E Nk) such that k

f=

gi

and

jgi (x) l < M pi (x)

(x E X and i E NO.

i=1

By the Hahn-Banach extension principle and by the choice of pi there 0 exist fi E Xj,' with fix = gi (i E Nk) with the desired property.

The dual of FK-spaces

383

The determination of the dual of the finite sum of FK-spaces (as FKspace) is left to the reader in Exercise 7.5.13. It now remains to determine the dual of FK-spaces generated according to 7.3.14 by continuous linear maps between FK-spaces.

Theorem 7.5.9. Let X and Y be FK-spaces and let T : X -- w be linear and continuous. Then for the dual of the FK-space YT :=T (Y) (cf. 7.3.14) the following statements hold:

(a) YT'={gJyT I gEX'}+{hoTlyT I hEY'}, that is b' f E YT' 3 g E X', h e Y' : f (x) = g(x) + h(T(x))

(x E YT). (7.5:4)

(b)If ?:YT -+Y,x-+ T(x) isbijective,then YT'={hoT I hEY'}; that is, for each f E YT' there exists a representation according to (7.5:4)

off with g=O. Proof. (a) Because FK-topologies are monotone and since T : YT --i Y is continuous (cf. 7.3.14(b)) we obviously get

(gEX' and hEY').

f:=glyT+hoTEYT'

To prove that each f E YT' has a representation of such a kind, we assume that P and Q are families of semi-norms which generate the FK-topology of X and Y, respectively. Then, by Theorem 7.3.14(a), the FK-topology

-I

of YT :=T (Y) is generated by

P':={pIYT I pEPI U {gogEQ}. Now, if f E YT' is given, then we can choose pi,...,pn E P and q,,...,qk E Q and an M > 0

such that

If(x)E < M

n

k

j=1

s=1

(P(X) + Ei(T(x))

for each xEYT.

Applying Corollary 6.5.7 to f and the semi-norms n

p. EPj j=1

k _ and q := > (qt o 1')

i=1

(being continuous relative to P') we obtain linear functionals

and h : YT - K with f = g + h such that

YT - K

384

Topological sequence spaces: K- and FK-spaces

n

lg(x)I

(x E YT)

<_ M E po(x)

(7.5:5)

and k

Ih(x)I < M

q?(T(x))

(7.5:6)

(x E YT).

.i=1

By means of the Hahn-Banach extension principle we can extend g to a g E X' with glyy, = g (which satisfies (7.5:5) for each x E X). Now, to determine an h E Y' with

f =g+hoT=glyy+hoT, we define a linear functional h on T (YT) by h(y) := h(x) for each y E Y with y = T(x) and x E YT.

It is well-defined since if T(x) = T(x*) (x, x* E YT) we get h(x) = h(x*) because k

Ih(x) - h(x*)I _ Ih(x - x*)I < M E q?(T(x - x*))

[cf. (7.5:6)]

k

= MEq,(O) = 0. ;_1

Further, for all y = T (X) E T (YT) we obtain k

k

Ih(y)t = 1h(x)I < M E qq (T (x)) = M E qj(y). j=1

j=1

Therefore, using the Hahn-Banach extension principle, we can extend h

from T(YT) to the FK-space Y; that is, there exists an h E Y' with hIT(yr) = h. In particular, by the definition of h, we have

(h oT)(x) = h(T(x)) = h(x) for every x E YT. The chosen g E X' and h E Y' give us the desired representation f (x) _ g(x) + h(T(x)) (x E YT) of f. (b) If f is bijective, then by 7.3.14(c) the FK-topology of Yr is generated by (q o I q E Q) . Therefore, in the proof of (a) we can choose g = 0, and

thus g = 0. Further, the statement { h o T I h E Y' j C YT' is contained 0 in the proof of part (a).

Distinguished subspaces of FK-spaces

385

In Section 8.1 we will make use of Theorem 7.5.9 in connection with domains of matrix methods. In Exercise 7.5.12 one may find an application of that theorem, and we draw the reader's attention to its solution. Exercise 7.5.10. Show that (bs, II Ilbs) and (cs, II fibs) is a BK-AB-space and BK-AK-space, respectively, and that

Sbs=Wbs=F'bs=ip=cs and bsf =csf=bv hold (cf. 7.2.17).

Exercise 7.5.11. (a) Prove part (ii) of the additional statements in 7.5.4. (b) Verify, in the case of the BK-space by, the following statements. Each f E bv' has the representation f (x) = f (e) lim x + J(xk - lim X) f (ek)

(x = (xk) E by).

k

Conversely, gy E bv'

for each y = (yk) E bs,

where gy is defined by g3 '(X) := y0 lim x + E yk+l (xk - lim x) for every x = (xk) E by. k

Moreover, (bv, I Ibv)' - (bs, 11 16). More precisely, T : (bv, I Ibv)' -+ (bs, 11 16)

, f -9 (f (e), f (e°), f (e')....

Exercise 7.5.12. Let A be a row-finite matrix. Prove that to :=A (L) is an FK-space and determine a family of semi-norms which generates its FK-topology. Furthermore, give a representation of the members of LA in terms of the elements of t' (cf. 7.5.4).

Exercise 7.5.13. Let X = E= 1 Xi be the sum of FK-spaces Xi (i E Na). Determine the dual space of the FK-space X in terms of the duals Xi' (i E Nn). Bibliography: [254], [250], [261], [267]

7.6

Distinguished subspaces of FK-spaces

In Section 7.2 (cf. also Section 7.5) we considered distinguished subsets of K-spaces and the corresponding properties AK, SAK, FAK and AB. The distinguished subsets get more structure and therefore more importance if we deal more specially with FK-spaces. A comprehensive presentation of

386

Topological sequence spaces: K- and FK-spaces

the distinguished subsets in this special case is given in Wilansky's book [254, Chapters 10-15] whereas we will restrict our interest in this connection to results which will be useful in what follows.

First we prove that in the case of an FK-space X the distinguished subspaces SX, Wx, Fx and BX are also FK-spaces. For this it is useful to introduce two further distinguished spaces.

Definition and Remark 7.6.1. For any K -space (X, T) with p C X and dual X' we put {x E w I {xtni , n E N} is bounded in (X,-r) }

BX and

FX

{x = (xk) E w I b f E X' :

xk f (ek) converges

.

k

Analogously to 7.2.10 we obviously get FX C BX , Bx = BX n X and

Fx = FX n X. Examples 7.6.2. (a) In the case of (ca,1(j F,a = B,Q = co

) we have

and Fro+ =_ Bh+ p -_T1L

as one can easily check with 7.5.4(d), see also 7.2.15(c). (b) In contrast to (a) we get for (e, 11111) the identities

Bt=Ft=Fi+=Be+=e by 7.2.15 since (xk) E e if and only if supnENO Ek_fl kxkI < oo.

Theorem 7.6.3 (distinguished subspaces as FK-spaces). If X is an FK-space with W C X, then BX , FX , Bx, Fx, Wx and Sx are also FK-spaces. Moreover, Fx, Wx and Sx are closed subspaces of the

FK-space B. In particular, if the FK-topology of X is generated by the family P = (p?

( j E I) of semi-norms and if for each j E I the semi-norm pJ is

defined on BX by p? (x) := supp,(xt"t) n

(x E BX ),

(7.6:1)

then BX , Fx , Wx and Sx, endowed with Q := (Pi i j E I), and also Bx and Fx, endowed with P U Q, are FK-spaces9. 9 Note, in the notation we do not distinguish between semi-norms and their restrictions to subspaces.

Distinguished subspaces of FK-spaces

387

Proof. Because Bx = BX fl X and Fx = FX fl X the last statement follows from the preceding statement, since the intersection of two FKspaces is also an FK-space (cf. 7.3.9).

First we verify that (BX , Q) is a metrizable K-space where Q = (pj I j E I) and pj (j E I) is defined in (7.6:1). Obviously, pj (j E I) are

semi-norms on BX . The K-property follows from that of (X, P). Namely, for each k E NO there exists an M > 0 and a finite subset J C I with

M

xkl

pj(x)

(x = (xj) E X);

jEJ

so we get for every y = (yj) E BX the inequalities lykI

M Epj(ylkl) < M EPj(y) jEJ

jEJ

because ylkl E cp C X. To prove the metrizability of (BX , Q) we choose

an at most countable family R = (ri, r2i ...) of semi-norms which generates the FK-topology of X. Then, we define a corresponding family R = (Ti,T2.... ) on BX by (i = 1, 2, ... , and x E BX )

Ti (x) := sup rj (xll) n

and show that Q and R generate the same topology on BX . (Then, since

R is at most countable, (BX , Q) is a metrizable K-space.) Let j E I be given. Since pj is continuous on (X, R) there exists an M > 0 and

ii,...,i E {1,2,...} such that pj (x) < M

rik (x)

(x E X).

k=1

Consequently, for each x E BX and n E No we obtain pj (Xlnl)

rik (xlnl) < M

M

ii,, (x). k=1

k=1

Therefore

pj (x) < M

rik (x) k=1

which proves that Fj is continuous on (BX , R). Analogously, one may show that each member in k is continuous on (Bx , Q). Hence, Q and R generate the same (metrizable) locally convex topology on B, . In the next step we show that (By l, Q) is complete. (Then (BX , Q) is proved to be an FK-space.)

388

Topological sequence spaces: K- and FK-spaces

Let (x(r)) be a Cauchy sequence in (BX , Q). In particular, by the definition of pj, for every n E NO the sequence (x(r)[n]) r of the nth section

of x(r) is a Cauchy sequence in (X, P). Thus, x(r)[n]

x[n]

in (X, P)

(7.6:2)

for a suitably chosen x E w. (Note, convergence in (X, P) implies coordinatewise convergence but that x is not necessarily in X.) In particular, (7.6:2) implies

Pj((x(r) - x)[")) 0 for each j E I and n E No . We now prove

(r -+ oo)

(7.6:3)

x E BX and x(r) --- x in (BX , Q).

Let j E I and e > 0 be given. Since (x(r)) is a Cauchy sequence in (BX , Q) we can choose a kj such that pj

(x(V)

_ x(µ)) = snppj (x(.) _

(v,p

kj).

(7.6:4)

For a fixed n E No we choose, according to (7.6:3), an r° > kj with pj{(x(r°) -x)[n))

<2

For r > kj this gives


Pj((x(r)-x)[n))

<

2+2=e.

Therefore we get

V j E I b e> 0 3 kj E N° b n E R)° `d r> kj : pj ((x(r) -x)[-]) <e. Thus, on the one hand x E BX , since for j E I and a := 1 we can choose a k j E NO and obtain suppj(x[n))

<

suppj ((x(kj) - r)[n)) + suppj (x(ki)[n))

<

l+ suppj(x(kj)[n]) < 00

n

n

because x(ki) E BX ; while on the other hand we have V j E I V £ > 0 3 k j E N O V r > k j : pj (x(r) - x) :5e,

that is x(r) --i x in (B,, Q). This proves that (B, , Q) is complete. In the next step of the proof we show that (Fx+, Q) is an FK-space by

verifying that Fx is closed in (B, , Q) (cf. 7.3.8). For this let x = (xk) E

Distinguished subspaces of FK-spaces

389

Fx+ be given. Since (BX , Q) is metrizable we can choose a sequence (x(r))

in FX with x(*) -4 x in (BX : Q).

(7.6:5)

We have to prove x E FX , that is E xk f (ek) converges for each f E X'. k

Let f E X' be arbitrarily given. On account of the continuity of f we can

choose ji,...,j,, E I and an M > 0 such that P

IAY)1:5mEpjj(y) for every y E X.

(7.6:6)

i=1

Now, if e > 0 is also arbitrarily given, we can, according to (7.6:5), choose

foriENP an rEN° with Pi. (x('

- x) < 4pM

(7.6:7)

Then, because x(r) _ (xkrl)k E Fx+ we can find for r E No a v E N such that v+µ

<

xkrl f (ek)

9

(p E N°).

(7.6:8)

k=v

Thus, for v and each p E NO, we obtain v+µ

E xkf (e k) k=v

Iv+µ

v+µ

<

E(xk-xkr))f(ek) + kv

<

If ((x _

<

4 - + 2 = s 2pM

Therefore,

(r))Iv+Pl)

_

xkr)f(ek) k-v f ((X _ x(r))Iv-11)1

[cf. (7.6:8)]

+2

[(7.6:6) and (7.6:7), definition of p`J;

P

by

Cauchy's

convergence

criterion

the

convergence

of

Ek ak f (ek) follows so that x E FX . We now prove that (WX,Q) is an FK-space. We show first that

(PEP and x E WX)

(7.6:9)

(Wx, Q) is a subspace of the FK-space (Fx, P U Q).

(7.6:10)

p(x) < supp(x1*1) holds and therefore

390

Topological sequence spaces: K- and FK-spaces

For a given p E P, we consider the quotient space (cf. 6.3.6)

Y:= WX/Kernp = {x + Kernp I x E Wx } endowed with the norm

P : Y --+ R, x + Kernp ---i p(x).

(7.6:11)

The dual Y' of (Y, f) together with the operator norm I I I I y' : Y' -4 R is a Banach space. If II II denotes the operator norm on (X, p)', then, applying the Hahn-Banach extension principle, we may check

Y' _{ f E Hom(Y, Ilk 13 f E (X, p)' C X' b x E Wx f (x + Kernp) = f (x) and IIf II = IIf IIy, }

.

(7.6:12)

As is known in functional analysis (cf. 6.3.30), we can isometrically embed (Y,p) into the second dual Y" (endowed with the operator norm II IIY" } by virtue of the map

y-*F, with Fy:Y'--*K, g-+FF(9):=9(y). This shows that for each x E Wx, we have

p(x) = p (x + Kernp) [cf (7.6:11)] = I I Fa+Kern P I I y" [isometrical embedding of (Y, p) into (Y", lily")] sup

I f(x +Kernp) I

11f Ily' <1

sup

If (x) I

[choose f associated with f as in (7.6:12)]

IIf Ily,<1

<

sup 11f1iy,<1

sup I If (x(nl) I n

< supp(xlnl)

[because x E Wx and f E X']

[since 11f 11 = Elf DDY' < 1]

n

and consequently (7.6:9), and thus (7.6:10). If we show in addition that WX is closed in (FX, PUQ), then (WX, Q)

is proved to be an FK-space. Thus, let x = (xk) E Fx and let (x(')) be a (x(r))k and sequence in W.v with x(') = X(r)

-> x in (Fx, P U Q), so in particular, in (X, P) and in (Fx, Q).

We have to show x E WX, that is

f (x) = E xk f (ek) for each f E X' k

where the convergence of the series Ek xk f (ek) is already known because x E FX. Now, let f E X' be given. Then for each r E No we have

f (x) - 57 xkf (ek) k

Distinguished subspaces of FK-spaces

=

xkr)f (ek) r

f (x) - f (x(r)) + 1 (x(r)) -

xk) f (ek)

k

k

-x)(ni) r (f(x) - f(x(r))) +lim n f((x(r)

=

391

(7.6:13)

0

[note x(r) E Wx and x(r) - x E Fx ]

where the first term in (7.6:13) converges to zero since f E X' and x(r) converges to x in (X, P), and the convergence to zero of the second term can be proved using the convergence of x(r) to x in (Fx, Q) as follows. Since f E X' we can choose jl,... , jP E I and an M > 0 with P

I f (y)1 < M

pji (y)

for each y E X

(7.6:14)

s=1

and get

lim f [ (x(r) - x) fn,! I n P

<

x)[n]\

sup J f ((x(r) -

I

/f

pj; ((x(r) -

< Msnp

x)[nj

4=1

P

< M E pjj (x(r) - x) r::* 0

[since x(') --- x in (Fx, Q) J.

i=1

Thus, we have proved f (x) = Ek xk f (ek), that is x E W. The proof that SX is closed in (Wx, Q), that is (Sx, Q) is an FKspace, is left to the reader in Exercise 7.6.9; in this proof the inequality (7.6:9) is also an essential tool.

0

On account of the importance of (7.6:9) we emphasize it in the following remark.

Remark 7.6.4. If X is an FK-space and p is any continuous semi-norm on X, then p(x) < supp(x(' l) for each x E Wx. n

(Note, in 7.6.3 as a family P generating the FK-topology one may consider the set of all continuous semi-norms on X.) 0 The following corollary is an immediate consequence of 7.6.3.

Corollary 7.6.5. If X is an FK-space with cp C X, then BX , FX, Bx, Fx, WX and Sx are FK-AB-spaces. As a further consequence of 7.6.3 we get the following corollary.

Corollary 7.6.6. Suppose that X is an FK-space with cP C X and that Bx is closed in X. Then Fx, Wx and Sx are closed in X too.

392

Topological sequence spaces: K- and FK-spaces

Proof. If Bx is closed in the FK-space (X, P), then (Bx, P) is an FKspace (cf. 7.3.8). From 7.6.3 we know that Sx, Wx and Fx are closed in the FK-space Bx. So, on account of the uniqueness of the FK-topology, 0 we get that Sx, Wx and Fx are closed in X. Using 7.6.6 we can easily characterize FK-spaces which are AK-spaces.

Theorem 7.6.7 (FK-AK-spaces). If X is an FK-space with V C X and dual X', then the following statements are equivalent: (a) X is an AK-space. (b) X is an SAK-space. (c) X is both an AD- and an FAK-space. (d) X is both an AD- and an AB-space. (f (ek)). (e) X' = XO by virtue of f

Proof. (a)

(b) : This implication is trivial (cf. 7.2.12). (b) #* (c) : Each SAK-space is an FAK-space (cf. 7.2.12) and, on account of (7.2:9), also an AD-space. (d) : In particular, FAK-spaces are also AB-spaces. (c)

(a) : If X is an AB-space, then Bx = X. Thus, by 7.6.6, Sx is (d) closed in X since Bx is. Because X is also assumed to be an AD-space we get Sx = iT = X (cf. (7.2:9)). (e) : This is the first statement in 7.5.4. (b) (e) . (b) : This implication follows from 7.5.1(a) noting that there yk = 0 fy(ek) (k E N°) holds. Theorem 7.6.7 enables us to characterize jP as an AK-space. Thereby we also complete Corollary 7.6.6.

Theorem 7.6.8 (gyp as an FK-AK-space). Let X be an FK-space (endowed with the induced topology of the FKspace X) is an FK-space, and the following statements are equivalent: (a) 7 is an AK-space. (b) Sx = iT.

with W C X. Then

(c) Sx is closed in X.

(d) Wx =7(e) Wx is closed in X.

(f) ipCFx. (g) ip C Bx. Proof. As a closed subspace 7 of an FK-space it is an FK-space itself. (a) . (b) : If the subspace iP of X is an AK-space, then 7 C Sx and therefore Sx = ;P since Sx C 7 holds always (cf. (7.2:9)). (b) (c) is trivially satisfied. (d) : If Sx is closed in X, then cp c Sx C Wx C 7 C Sx which (c)

Notes on Chapter 7 393 implies Wx =
(e) = (f) : If Wx is closed in X, then ip C Fx because W C Wx c 7 C

WxCFx. (f) (g) : Note Fx C B. (g) = (a) : If iT C Bx is satisfied, then the FK-space 7 is both an ABand an AD-space, and thus an AK-space because of 7.6.7.

0

We close this section with the remark that we will come back to the results obtained here in subsequent sections, in particular in Section 8.2 where we will deal with distinguished subsets of matrix domains.

Exercise 7.6.9. Let X be an FK-space with cp C X, and let the FKtopology of X be generated by a family P = (pj I j E I) of semi-norms. Prove that Sx is closed in (Wx, Q) where Q = (pj I i E I) is defined as in 7.6.3. (Consequently, (Sx, Q) is an FK-space as is stated in 7.6.3.)

Exercise 7.6.10. Show that, if X is an FK-space with sp C X, then the FK-space Sx is even an AK-space.

Exercise 7.6.11. Let X be an FK-space with cp C X and let Fx be the closure of Fx in X. Verify that Fx C Bx if and only if Fx is closed in the FK-space X. Bibliography: [2541, [260], [249]

7.7

Notes on Chapter 7

Essentially, we restrict the notes on Chapter 7 to those investigations and developments in FK-space theory which took place in the period after the publication of Wilansky's book [254] and to those which play an important role in the context of this book. Regarding the countable union of FK-spaces (LFK-spaces) we refer to Remark 7.3.17. For notes on the so-called Wilansky property we refer the reader to the notes on Chapters 9 and 11 in Section 11.6. In generalizing `sections properties' of FK-spaces G. Meyers (cf. [174]) and M. Buntinas (cf. [55]) introduced Toeplitz sections. For a (fixed) regular row-finite matrix T, also called a Toeplitz matrix, sectional convergence, weak sectional convergence and sectional boundedness for FK-spaces are defined in terms of T. Important special cases of Toeplitz sections are the `original sections' for T := I introduced by K. Zeller and the Cesaro

sections for T := Cl introduced by M. Buntinas in [54]. For further investigations and applications of Toeplitz section properties of FK-spaces we refer, for example, to the papers of G. Goes [94], T. Leiger [144], D. J.

394

Topological sequence spaces: K- and FK-spaces

Fleming [82], D. Noll [186], W. H. Ruckle and S. A. Saxon [212], and to K.-G. GroBe-Erdmann [102].

If E, F are sequence spaces then M(E,F) :_ {(yk) E w I (ykxk) E F for all (xk) E E} is called the multiplier space (of E and F) and the elements of M(E, F) are called multipliers. Obviously, we may iden-

tify the multipliers with the matrix maps between E and F which are given by a diagonal matrix (diagonal transformation). Further, note that

E° = M(E, £), E13 = M(E, cs) and E' = M(E, bs). In general, it is interesting to determine M(E, F). For an overview of the extensive literature see, for example, MathSciNet with `Classification: 46A45' and `Anywhere: multipliers'. GroBe-Erdmann's Lecture Notes [104] are a further good source for references. Regarding the role of multiplier spaces in FK-space theory, the following observations lead to a series of problems:

If E and F are FK-spaces, then M(E, F) is not necessarily an FKspace as the trivial example E := w and M(E, cs) = E1 shows. If E is a BK-space, then E13 is also a BK-space where the converse is not true in general. If E is an FK-space, then E16a is an FK-space too. We list some recent papers within the scope of these observations: D. J. Fleming and J. C. Magee [83], K: G. GroBe-Erdmann [97], M. Benholz [19], J. C. Magee [162], M. Buntinas and G. Goes [58].

In the last cited paper M. Buntinas and G. Goes (see also Buntinas' paper [57]) introduced and investigated the product of FK-spaces (FKproduct). In a private communication, G. Goes gave a very elegant (unpublished) proof, using properties of FK-products, of the fact that EE0 is an FK-space if E is. This proof and the results in the papers mentioned indicate that there are still more applications of FK-products and that the FK-products are still interesting research subjects in FK-space theory. We understand by Toeplitz-Silverman-type theorems, also called mapping theorems, characterizations of matrix maps A from a sequence space E into a sequence space F in terms of the defining matrix A. In Chapter 3 we proved in the case of special spaces E and F a series of ToeplitzSilverman-type theorems by using gliding hump arguments. In Section 7.4 we proved some of them by applying functional analytical tools such as the uniform boundedness principle, different versions of the Banach-Steinhaus theorem and the fact that matrix maps between FK-spaces are continuous, which is based on the closed graph theorem. We present another access to Toeplitz-Silverman-type theorems in Section 11.3 via sectional convergence and gliding hump properties. In all the cases considered the domain and range spaces of the matrix maps under consideration have in common that they are special BK-spaces. For more general treatments of the characterization of matrix maps between

FK-spaces, which are partially recognizable in our proofs, we refer the

Notes on Chapter 7 395

interested reader to Wilansky's book [254, Chapter 8]. It is still an unsolved problem to give a (general) characterization of

the matrix maps from PP into t', 0 < p, r < oo, in terms of the matrix entries. There are solutions for special classes of matrices: for example, in [23] G. Bennett gives a complete description of mapping properties, from PP into Pr, 0 < p, r < oo, of a class of lower triangular matrices A = M(a, b), defined by ank = anbk if k < n and ank = 0 if k > n, where a = (an) and b = (bn) are two given sequences of non-negative numbers. Further interesting results concerning necessary and sufficient conditions for a matrix to map & into f', 0 < p, r < oo, are obtained, for example, by D. Borwein and X. Gao in [49]. For an overview of the literature see, for example, MathSciNet with 'Classification:(26 OR 40 OR 46A45)' and `Anywhere: (matrix map* OR matrix op* OR Matrixabb* OR mean op*)'. In Chapter 4 we have become acquainted with several kind of Tauberian theorems. All of them have in common that they are valid for special summability methods or, more generally, for special classes of summability

methods and that the proofs are based on classical methods. By contrast there is a series of papers dealing with Tauberian theorems for FK-spaces, in particular for parts of summability domains (having, for example, the AK-

property). This research topic is connected, for example, with the names A. Jakimovski, W. Meyer-Konig and K. Zeller (cf. [119], [120], [173]), G. Goes (cf. [94], [95]), M. Buntinas (cf. [56]), J. Connor and A. K. Snyder (cf. [62], [66]), and H. Tietz and K. Zeller (cf. [241], [242], [243]).

8

Matrix methods: structure of the domains In the first part of this chapter (8.1-8.5) we continue the investigation of the structure of domains which we started in Chapter 2. Whereas in Chapter 2 we made use exclusively of classical methods we now apply functional analytic methods, which we discussed in Chapters 6 and 7. However, we should remark that within the scope of this book it is not possible to present

all the details of the structure of domains. For more details we refer the interested reader to Wilansky's book [2541.

First we deal in 8.1 with the domains of matrix methods by way of introduction. The FK-space structure of domains will prove to be very useful for the investigation of the structure of matrix domains. In particular, we give representations of the continuous linear functionals on the domain -1

EA :=A (E) of any matrix A relative to any FK-space E and apply the results to the special case CA.

Based on the statements about distinguished subsets of FK-spaces E in Section 7.6, we consider, in Section 8.2, in the special case CA, further distinguished subsets which are determined in some cases alone by the FKtopology of CA and in other cases directly by the matrix A. Since precise knowledge of the distinguished subsets lets the topological structure appear more transparent, we deal essentially in this section with the determination of the closure and comparison of distinguished subsets. In this connection we introduce the notion of `replaceable matrices'. Since replaceable matrices play an essential role in summability we consider them by way of introduction and the notion of `µ-uniqueness' of matrices which is connected with the representation of the continuous linear functionals on CA.

In Section 8.4 we apply the results in 8.2 and 8.3 to special examples of matrices. Using the classical methods we obtained after much effort in 2.5.8 that each conservative matrix, which sums a bounded divergent sequence, also

sums an unbounded sequence. By comparison with that, in Section 8.5, we easily get this result from the results in Section 8.2 which we have

Domains of matrix methods as FK-spaces

397

obtained by applying functional analytic methods. Similarly we find that any conservative matrix does not sum a bounded divergent sequence if c is closed in the FK-space CA.

In the second part of Chapter 8 we apply results on particular distinguished subsets. Namely, in Section 8.6, we continue the investigation of the consistency of matrix methods. For this we introduce the notion of `P-perfectness' and that of 'perfectness' of matrices. The last section of this chapter, namely 8.7, contains a more detailed study of replaceable matrices and-closely connected with replaceabilitythe study of invariant objects of matrix domains. Here `invariant' means that the object, under consideration, depends exclusively on the domain. 8.1 Domains of matrix methods as FK-spaces Background and motivation for the present (and also for subsequent) sections are first the knowledge that matrix maps between FK-spaces are (linear and) continuous, cf. 7.3.7, and second that the inverse image of an FK-space under a linear and continuous map between FK-spaces is again an FK-space, cf. 7.3.14. More precisely, if we have identified the application domain of a given matrix A as an FK-space, then A : WA -> W is continu_1

ous (if WA and w carry their FK-topology) and, for example, CA (=A (c)) is an FK-space. First we consider more generally an FK-space E (instead

of c) and then study the particular case E := c. We begin by extending the notion of the domain of a matrix.

Definition 8.1.1 (domain). If A is an infinite matrix and E is a se{x E WA I Ax E E} is called the domain of A

quence space, then EA

(relative to E). Before proving that WA is an FK-space we show that the domain CA is an FK-space if A is row-finite. Note, in this special case, we have WA = W and w is an FK-space (cf. 7.3.2).

Theorem 8.1.2 (domains of row-finite matrices). If A = (ank) is row-finite, then CA endowed with the family

{1111.-A}U{q; ! jEN°} of semi-norms is an FK-space. (Recall, qj (x) := I xj I for x = (xk) E W and (111100 o A)(x) = SUP- IEk ankxkl for x = (xk) E CA.)

Proof. Since wA = w (as noted above) is an FK-space, the matrix map

T:w-4w, x ---+ Ax is well-defined, linear and continuous as a matrix map between FK-spaces (cf. 7.3.7). Applying 7.3.14(a) in the case

(X,P) := (w, (q? I j E N°)) and (Y,Q) we get the desired result.

(c, (11 11.)),

0

398

Matrix methods: structure of the domains

Theorem 8.1.3 (application domain as an FK-space).

For each infinite matrix A = (ank) the application domain wA is an FK-space and its FK-topology is generated by the family (pn I n E N°) of semi-norms where pn is defined by

(nEN°,x=(xk)EWA)-

P. W := IX- 1 + sup E ankxk k=0

Proof. For every n E NO let the matrix An = (aik))i,k be defined by j ank if 0 < k < i (i, k E N°). aik otherwise t0 Applying 8.1.2 to the row-finite matrix An we get that (n)

CA,. _ {(zk) E w 11: ankxk converges k

is an FK-space and its FK-topology is generated by the family {qj I j E No } U {pn}

of semi-norms, where qj is defined as in 8.1.2 and pn is given by (x = (xk) E CA,.)'

fin(x) := sup

i

k=0

Obviously, wA = fln CA.. Therefore, wA as an at most countable intersection of FK-spaces is itself an FK-space (cf. 7.3.9) and its FK-topology is generated by the family {qj I j E No } U {pn I n E NO } of semi-norms. Because pn = qn+pn (n E N°) the FK-topology of WA can also be generated by the family (pn I n E NO), see 6.4.12.

Using 8.1.3 and 7.3.14 we are able to verify that EA is an FK-space if E is an FK-space.

Theorem 8.1.4 (domains as FK-spaces). Let A be an infinite matrix, (E, Q) be an FK-space and let qj (j E NO) and pn (n E N°) be defined as in 8.1.2 and in 8.1.3, respectively. Then the following statements hold:

(a) EA is an FK-space and IN I n E No} U {q o A I q E Q} generates its FK-topology. (b) If A is row-finite, then {qj I j E No } U {q o A I q E Q} generates the FK-topology of EA too.

(c) If the matrix map A : EA -a E is bijective, then the FK-topology of EA is generated by (q o A I q E Q) alone. In particular, if E is a BKspace, then EA is also a BK-space. (Note, for example, A is bijective if the matrix A is a triangle.)

Domains of matrix methods as FK-spaces

399

Proof. (a) By 8.1.3 WA is an FK-space and its FK-topology is generated by (pn ( n E N°) . Since the matrix map A : WA -a w is linear and continuous (cf. 7.3.7) we get statement (a) by 7.3.14(a). (b) If A is row-finite, then (q? I j E N°) generates the FK-topology of WA = W (cf. 7.3.2(a)). Thus, applying 7.3.14(a), as in part (a), we get that {qj I j E NO} U {q o A I q E Q} generates the FK-topology of EA. (c) If the map A : EA --3 E is bijective, then (q o A ( q E Q) generates the FK-topology of EA by 7.3.14(c).

We emphasize particularly the special cases E = co, c, m and E = t.

Corollary 8.1.5. Let A be an infinite matrix. (a) Then COA, CA and MA together with {pn I n E No } U {(( X100 o A} are FK-spaces and, if in addition A is row-finite, then {q? I j E No } U {I( Iloo o A} generates their FK-topologies. Furthermore, COA, CA and mA o A, if the corresponding matrix map A is are BK-spaces with norm bijective (cf. 8.1.4(c)). (b) eA together with {pn I n E N) U {ll f 1 o A} is an FK-space, and it o A if A : eA -i a is bijective. (See also is a BK-space with norm 7.5.12.)

Proof. With reference to 7.3.2(b) and (c) these statements are obviously special cases of 8.1.4.

On account of the dependence of the FK-topology of EA on the FKtopology of E we expect that some of the topological properties of the FK-space E are carried to the FK-space EA. For example, this proves to be the case for the property of being separable, as we now state.

Theorem 8.1.6. If A is any matrix and E is a separable FK-space, then EA is separable too. In particular, the FK-space CA is separable since (c, II

Iloo) is.

Proof. For a proof of the general statement we refer to Wilansky's book [2541. The particular statement follows from the general one since (c, 1111"") is separable (cf. 7.5.5(a)).

We seek a representation of the dual of the FK-space EA in terms of the duals of E and WA. This applies in particular, of course, to CA and LA. We begin by obtaining a representation of the dual of the FK-space WA. Analogously to our earlier determination of the duals of the FK-SAKspaces b oi e, ca and w, we have that WA is an FK-AK-space. Thus we may identify wA with w0 .

Theorem 8.1.7 (dual of WA). If A = (ank) is an arbitrarily given matrix, then for the FK-space WA the following statements hold: (a) WA is an FK-AK-space.

400

Matrix methods: structure of the domains

(b) wA = wA by virtue of f --* (f (ek )) . In particular,

f (X) _ E xkf (ek)

(f E wA and x = (xk) E WA)-

k

(c) If A is row finite, then wA = V and, in particular, (f (ek)) E V for each f EwA.

Proof. (a) Let pn be defined as in 8.1.3. Then pn (x - x1il) = sup

j>i+1

ankxk

-i0 (i-+oo)

k=i+1

holds for every x E WA and n E NO since Et ankxk converges. (b) Apply 7.5.4 to X := WA.

(c) If A is row-finite, then WA = w. Thus (c) comes from (b) because 0 `w' = V by virtue of f --+ (f (ek)) , (cf. 7.5.4(a)).

Theorem 8.1.8 (dual of EA ). If A = (ank) is an arbitrary matrix and E is an FK-space, then the duals of EA, CA and IA have the following representations: E

akxk = h(Ax) + ax (x = (xk) E EA).

f (x) = h(Ax) +

(8.1:1)

k

Conversely, if a E wA and h E E' are given, then an f E E ' is defined by (8.1:1). E

f (X) = p limAx +E to E ankxk + E akxk n

k

(8.1:2)

k

= Et limAx + t(Ax) + ax (x = (xk) E CA). Conversely, for all p E 1K, t E e and a E wA an f E CA' is defined by (8.1:2).

(c) V f E eA 3t = (tn) E m 3a = (ak) E wJ :

f (x) = t(Ax) + ax (x = (xk) E LA).

(8.1:3)

Conversely, an f E eA is defined by (8.1:3) for all t E m and a E wA.

Additional statements: (i) If A is row-finite, then in (a)-(c) we may take a E w. (ii) If A : EA --3 E (CA -4 c and eA -+ e) is bijective, then in (a) ((b) and (c), respectively) we may put a := 0.

Domains of matrix methods as FK-spaces

401

(iii) The statement in (a) ((i) and (c)) remains true, if we replace wA by EA (cJ and QA, respectively). (iv) If A is conservative for null sequences, then in (b) we may take a E 1.

Proof . (a) Apply Theorem 7.5.9(a) to the case X := WA, Y := E and T := A and note 8.1.7(b). (b) Statement (b) is the special case E := c of (a) if one also considers the representation of h E c' given in 7.5.5.

(c) We put E := I in (a) and make use of the representation of h E I' given in 7.5.4(c).

To prove the additional statements it is obviously sufficient to verify them in the case of part (a). (i) If A is row-finite, then we have wA = WO and therefore we may take a E cp in the representation of f E EA.

(ii) If A : EA -+ E is bijective and f E EA, then h := f o A-i E E' by the theorem of the inverse operator (cf. 6.7.8). Using again that A is

bijective, we obtain h(Ax) = f (x) for each x E EA; that is, we get a representation (8.1:1) of f with a = 0. (iii) Since wA C EA we may obviously choose for each f E EA an a E EA and an h E E' such that (8.1:1) is satisfied. Conversely, using 7.5.4(a) we get for each a E EA a continuous linear functional g if we put

g(x) := ax (x E EA). Thus, we have shown that statement (a) remains true if we replace wA by EA. (iv) If in (b) the matrix is assumed to be conservative for null sequences, 0 that is CO C CA, then a E wA C ca = e holds.

In particular, Theorem 8.1.8(b) contains the statement lima E cB for every matrix B (which follows also from lima = lim oB and the continuity

of the maps B : CB - c and lim : c -i K). Therefore, on account of the monotonicity of FK-topologies (cf. 7.3.4(a)) we also get limn Ix E X' for each FK-space X with X C cB, for example if X := CA and CA C CB. Proceeding from this statement, we now show that each f E CA' gives rise to a matrix B such that CA C cB and limB ICA = f which is a result due to K. Zeller (1951). During the last decade this observation of Zeller's has turned out to be of great importance. Equally important is the class of K-spaces characterized in this way (cf. [44]).

Theorem 8.1.9 (representation as limB). If A =

is any ma.

trix, then the following statements hold: (a) If B is a matrix with CA C cB, then lima IAA E c' A'

(b) For each f E cA there exists a matrix B with CA C cB and lima IAA = f. Moreover, if there exists a representation (8.1:2) of f with ,u ¢ 0, where a E wA or a E CA , then we may choose B so that CB = CA holds.

402

Matrix methods: structure of the domains

Proof. (a) We have already verified this statement in the preamble to the present theorem. (b) Let f E CA' be arbitrarily given. Then f (X) = It limAx + t(Ax) + ax

(x E CA)

(8.1:4)

where y E 1K, t = (tn) E e and a = (ak) E wA C CA are suitably chosen

(cf. 8.1.8(b)). First we define the matrix D = (d,,,) by

p

ifv=n

10

ifn>v

d,,n :=

to ifn
(v,nEN°).

Obviously, D is conservative since t E e (cf. also 2.3.71), to is the limit of the nth column and X(D) =,a holds. Thus, using the limit formula (cf. 2.3.71) we get

CC CD and limp x = p lira x + E tnxn

(x = (xk) E c).

n

Now, if we put C = (cnk) := DA, then CA C cc and

limAx = limDAx = plimAx+E to n

ankxk

(x = (xk) E CA) (8.1:5)

k

on account of 2.6.1(a) and 2.2.5, since D is a lower triangular matrix. We now show that B = (bnk) defined by

ak + cn-l,k

bnk

if n > 0

(n, k E N° )

has the desired properties. Because a = (ak) E CA and CA C cc we obviously obtain CA C WB,

[Bx]o = ax and [Bx]n = ax + [Cx]n-1

(n > 0)

for each x E CA; thus CA C CB and (cf. (8.1:5))

limAx = limAx + ax = p limA x + t(Ax) + ax = f (x) for every x E CA. Now, we assume that there exists a representation (8.1:2) of f with p $ 0. Then D is a triangle. In particular, there exists a uniquely

determined (bi-)inverse of D, that is I = D-1D = DD-1 and D-1 is also a triangle (cf. 2.2.9).

If we can prove that D-1 is conservative, then we get A = D-1C and cC C CA (cf. 2.6.1(a)) and so CA = CC. Moreover, we obtain CB C CC = CA, and thus CA = CB C.

To see this, note that x E cB C wB implies the existence of [Bx]o = ax and therefore the existence of jCx]n_1 = [Bx]n - ax (n > 0); we now get x E cc because x E cB and since ax exists.

Domains of matrix methods as FK-spaces

403

To prove that D-i is conservative we first verify m C MD-1 which is obviously equivalent to VYEW:

(y V m' Dy

m).

(8.1:6)

Thus, let y = (Yk) f m. Then we may choose an index sequence

(vj) with

(0 < v < vj).

I yv I < I yv; I

(8.1:7)

Obviously, Iyv, I -+ oo (j -; oo). Further, since p # 0 and t E e, we may choose an 00

N E N with E It-1 < Ial.

(8.1:8)

n=N

Hence, for each j E NO with vj > N, we obtain vJ-I

vJ

I[Dy]v,I

=

4j.yn

Pyv;+

n=0

vi-i >-

tnyn

n=0

N-i

-v-

ItIl Iyv, I - E Itni IYnl - 57, Itnl lYnl n=0

n=N

=:M

1i

4 Iyv; I - Iy,, 157, Itn I- M

[cf. (8.1:7)]

n=N CO

>-

(Ijil

- n=N Itnl)

Iyv; I - M.

Now Dy 0 m since the last term diverges to oo as j -+ oo on account of (8.1:8) and Iyv; I -> oo. In the next step we prove c C CD- I which is equivalent to

VyEw: (DyEc=yEc). For this let y = (yk) E w with Dy E c be given. Since y E m on account of (8.1:6) and because t E e the series En tnyn converges. Applying the definition of D we obtain

v-I F`yv = [Dy]v - E tnyn n=0

thus y E c because Dy E c and p# 0.

(uEN);

404

Matrix methods: structure of the domains

Remark 8.1.10. As an application of 8.1.9(a) we get the limit formula

limA x = X(A) lim x + E akxk

(x = (xk) E c)

0

for conservative matrices A which we stated already in 2.3.7. Obviously c = ct C CA, and thus 8.1.9(a) implies f :=1imA I, E c'. Therefore by 7.5.5 we obtain limA x = AX) = A f lim x + E xk f (ek )

(x = (xk) E C)

k

with

= limA ek = ak

f (e

(k E N°)

and

f (ek) = hnA e -

µf = f (e) -

limA ek = X(A)

k

k

Remark 8.1.11. Now we ask whether the representation (8.1:2) of f E cA in 8.1.8(b) is unique. The answer is negative. In general, p and t in (8.1:2) are not uniquely determined as we can easily verify in the case A = 0. Moreover, if A is any matrix with A $ 0, then t and a are not uniquely determined, as we now show. If A = (ank) 0 0, then there exist no, ko E N° such that anoko # 0. Now, let f E cA be arbitrarily given and let it be represented according to (8.1:4) with p E K, t = (tn) E e and

a

(ak) w. If we put

t'

=

ftno-1 ifn=no

if n#no

to

(n E N°)

and

ak + anok

ak

(k E NO),

then F:= (tn) E e, a :_ (ak) E w q and

P X) = Jl limA x + E anokxk + E Fn L ankxk + E akxk k

= iclirAX + t(Ax) + ax

n

k

k

(x = (xk) E CA).

Thus, p, t and & yield a representation of f with To t and a # a. A In Section 8.3 we will deal more precisely with the uniqueness of µ and show among other things that p is uniquely determined for all g E CA, if it is uniquely determined for at least one f E cA. Further, we will prove in Section 8.7 that uniqueness of p in the case of a fixed matrix A implies uniqueness of p in the case of all matrices B with CA = CB; therefore, in this case we speak about the invariance of µ-uniqueness.

Domains of matrix methods as FK-spaces

405

As we have stated in 8.1.6 the domain cA of any matrix A is a separable FK-space. In particular, this is true for the discrete Abel and Borel methods. The following theorems tell us that the corresponding statement also holds in the case of the domain of the Abel method, and that the domain of the Borel method is an FK-space.

Theorem 8.1.12. The domain of the Abel method is a separable FKspace. In detail, we have the following statements: (a) (CA,, {p} U {pj I j E N}) is an FK-space where the semi-norms pj and p are defined by

pj(x) :=supIxk1(

-j)k

and p(x) := sup (1 - t) o
k

E xktk

(x E CA,)-

k

(b) t-,limEalnm((xk -xk_1)tk)lnl = (x

for each x = (xk) E CA,

in the sense of the FK-topology of CA, .

Proof. We only sketch the proof. Statement (b) is an immediate consequence of (a). Further, applying (b) to (xk) E CA, with xk E Q (k E N°) we get that the FK-space CA, is separable by using standard arguments. (a) First we consider the `application domain' of the Abel method:

III =

{xi)

xktk converges for Itl < 1

Ew1 k

which is an FK-space with the semi-norms pj (j E N) by Example 7.3.16(a). Then we consider the linear map

T : III -* f f :]0,1[---+ R} , (xk) -i f where f (t) := (1 - t) E xktk. k

Endowing the range space with the (locally convex Hausdorff) `topology of pointwise convergence' and III with its FK-topology, we get that T is continuous. Obviously,

X:= f E C[0,1[ +

I

o

m _ f(t)

endowed with the supremum norm

exists } ,

is a Banach space. Now, CA, _

T (X) and, applying Theorem 7.3.14 (with Remark 7.3.15), we get that (CA, , {p} U { pj ! j E N}) is an FK-space. Theorem 8.1.13. The domain of the Bore] method is an FK-space. In detail, (CB {p} U { pj I j E N}) is an FK-space where the semi-norms pj and p are defined by pj (x) := sup Lx-k1 jk and p(x) := sup

k'

S

e` I ! k

k!

tk

(x E CB, ). I

406

Matrix methods: structure of the domains

Proof. The proof is quite similar to that of 8.1.12(a).

0

Now, applying Theorems 8.1.12 and 8.1.13 it is easy to complete the proof of Theorem 3.6.11. We repeat the statements in question. Theorem 8.1.14. U CC. C CAI and a>o

U cEa C CBI. °
Proof. It remains to prove that the inclusions are strict. If they are not = CB1 which contradict the strict, then U°O_1 cCk = CAI and U' 1 CE1 fact that CA, and CB, are FK-spaces since both (cc,) and (cE are strictly increasing sequences of FK-spaces (cf. Theorem 7.3.12).

0

Exercise 8.1.15. Let A be a matrix, E be an FK-space and M C EA. Prove the equivalence of the following statements:

(a) M is bounded in the FK-space EA. (b) M is bounded in the FK-space WA, and A(M) is bounded in E. (c) Va E wA : SUPZEM I axI < 00 and V f E E: sup.EM If (Ax)I < oo.

Exercise 8.1.16. Let A be a matrix and TA : CA -+ c, x -- Ax be bijective. Show that the following statements hold: (a) There exists a uniquely determined right inverse of A.

(b) There exists a matrix B with Ek Ibnkl < oo (n E N°) and a v E w such that TA 1(y) = (lim y)v + By for every y E c. (c) Every matrix B according to (b) is the right inverse of A. (d) The matrix D (formally) defined by r

if r = 2n

E x2k r +

x2k

if r = 2n + 1

k

has the application domain WD = { (xk) E w I (x2k) E cs } and yields a bijective matrix map TD : CD -Y c. Give a representation of TD 1(y) (y E c) in accordance with (b). Bibliography: [254]; [250], [261], [267], [44]

8.2

Distinguished subspaces of domains

In Section 7.6 we dealt with `distinguished subspaces' of FK-spaces. Now, it would be natural to consider `distinguished subspaces' of domains EA

where E is an FK-space. We do not do this and study just the particular case of `distinguished subspaces' of domains CA. For that we first decide on some conventions.

Distinguished subspaces of domains 407

Conventions 8.2.1. (a) If it is not otherwise stated, let A = (a,,k) be a matrix with p C CA. This is equivalent to the existence of the column limits ak := lim,, (k E N°). (b) In the case of the subsets SCA, WW,,, FCA and BcA of the FK-space CA,

introduced in 7.2.10, we write simply $A, WA, FA and BA.

A

The subspaces SA, WA, FA and BA are determined just by the sequence space CA (and its FK-topology) and not by the particular matrix A, that is if A and B are matrices and CA = CB, then SA = SB etc. In addition, we also consider `distinguished subsets' of CA whose definition is given in terms of the matrii A. However, in the case of some of them we will see that they are already determined just by the FK-space CA.

Remarks and Notation 8.2.2. Analogously to 2.6.6 we define

(inset),

IA := {x = (xk) E CAI (akxk) E cs}

AA : IA -- * K, x = (xk) -> limA x - E akxk k

and

AA := KernAA = {x E IA I AA(x) = 01

.

Furthermore, we put LA

{x E CA I Vt E I : (tA)x exists (in the sense of 2.2.2)} (associative part of CA, see 8.2.4(b))

and

PA := {x E CA I Vt E TA : (tA)x = t(Ax)}, where

TA := {tEtIVYEcA: (tA)y exists}. Obviously, WA C AA and FA C IA because of limA E cA.

IL

First we verify that section boundedness of each x E CA in the FK-space CA is equivalent to the boundedness of the set of its sections relative to the semi-norm 11fl , o A and that LA = BA. The latter result is surprising

since BA is only determined by the FK-topology of CA which depends only on the sequence space CA, whereas the matrix A enters essentially into the definition of LA. In the proof of this statement we also obtain the important fact that we can `calculate associatively' on LA, which justifies LA having the name `associative part' of CA. We begin with a remark. Remark 8.2.3. For every matrix A we obviously have cp C WA, and WA 0 is an FK-AK-space (cf. 8.1.7(a)), and thus an FK-AB-space.

408

Matrix methods: structure of the domains

Theorem 8.2.4. The following statements hold: (a) LA = BA = {(xk) E CA I supn,v IEk=O ankxkl < 00} .

(b) VtEf VxELA : t(Ax)=(tA)x. (c) BA = LACPA. (d) VxECA : xESA

IIA(x-x1''')!1. ---*0 (r-+ oo).

Proof. (a) On account of 8.1.3, 8.1.5(a) and 8.2.3, for each x = (xk) E CA the equivalence x E BA

sup (II oo o A)

(X[-'])

= sup

< oo,

(8.2:1)

n,v k=0

v

is valid, that is the second equality in (a) is satisfied. In the next step of the proof we show BA C LA. If x = arbitrarily given, then

k) E BA is

v

oo, where unv

sup n,v

E ankxk, k=0

follows from (8.2:1). This implies, for each t = (tn) E f, that

E tnunv converges uniformly for v E No .

(8.2:2)

n

Since t(Ax) exists because Ax E c and t E t, we get t(Ax)

=

EtoEankxk = Etolim unv v n

(definition of unv ]

n

k

lim E tnunv v

[cf. (8.2:2)]

n

=

lim

1: xk k=0

tnank

(tA exists since V C CAI

n

(tA)x,

which implies the existence of (tA)x. Thus, x E LA, and in addition t(Ax) = (tA)x. (Note, the latter observation together with LA = BA proves statement (b).) Therefore, we have shown BA C LATo prove LA C BA, let x = (xk) E LA be given. Then we define for each v E N° the linear functional

t -l K, t = (tn)

xk 1: thank, k=0

n

which is well-defined since cp C CA. Further, fv E (P, 7.5.4,

because of

Distinguished subspaces of domains 409 v

E xk 1: thank = 1: to E ankxk k=0

n

n

(t = (tn) E £)

k=0

and

(tankxk) k=0

Em

[since c P C C A I .

n

By the additional statement in 7.5.4(i), for every v E NO, we obtain

= sup

dnkxk

IIf,41 =

k=0

n 00

ankxk

n

k=0

where IIffII denotes the operator norm of f,,. Since X E LA, that is (tA)x satisfies exists for each t E t, the sequence

-i (tA)x

(v -+ oo, t E 2),

that is it converges pointwise. Therefore, (f I v E N°) is a pointwise bounded family of continuous linear functionals, which implies

sup IIffII = sup v

n,v

< 00 k=0

by the Banach-Steinhaus theorem, that is x E BA. (b) See the remark in the part `BA C LA' of the proof of (a). (c) This statement is an immediate consequence of (a), (b) and the definition of PA. (d) Let pn (n E N°) and II II00 o A be the semi-norms on CA according to 8.1.5(a). Then for each x E CA we have

xESA

limJlA(x-xf1)11.=0 and lirapn(x-xt''])=0 (nEN°) lim II A(x - x[r)) II0 = 0

[WA is an FK-AK-space, c£ 8.1.7(a)],

which proves the statement in (d). In the next theorem we give the inclusions between the `distinguished subsets' which we have so far established.

Theorem 8.2.5. The following statements hold: (a) FA = LA fl IA = BA fl IA.

(b) WA = LAfAA = FAfAA . (c) FA = WA or FA = WA ®(u) withanyuEFA\WA = FA \AA. (d) If e E FA and x(A) j4 0, then FA = WA G (e). In the case of conservative matrices A we have e E FA; therefore FA = WA E) (e), if A is coregular.

410

Matrix methods: structure of the domains

Proof. (a) Because FA C BA = LA and FA C IA it is obviously sufficient to prove LA n IA C FA. For this let y = (yk) E LA n IA and f E CA' be given. By 8.1.8(b) we may choose for f a p E 1K, t = (tn) E P and an

a = (ak) E wA such that

7 a,,,kxk + Y akxk

AX) = p limA X + n

k

(8.2:3)

k

= p limA x + t(Ax) + ax

(x = (xk) E CA).

In particular,

f(ek) = pak+[tA]k+ak and therefore, for the given y, the identity

Eykf(ek) =

p

E akyk+(tA)y+ay

k

(8.2:4)

k

holds. We should note that on the right hand side of the last equality the first, second and third terms exist because of y E IA, y E LA and a E wA C CA, respectively. Thus, we have proved y E FA. (b) Obviously, WA C BA = LA and also WA CAA . On account of part

(a) it remains to prove LA n AA C WA. Let y = (yk) E LA n AA C LA n IA = FA and f E CA' be given. If we represent f in the same way as in the proof of (a), then we get by (8.2:3) and (8.2:4)

f (y) - E ykf (e

k)

=p

(limA y

-

akyk) + (t(Ay) - (tA)y) (8.2:5)

= PAA(y) + (t(Ay) - (tA)y)

=0

[because y E AA and 8.2.4(b)].

This verifies y E WA.

(c) Since AA is a linear functional on IA we have codim pA (Kern AA I FA) < 1

[note FA = LA n IA C IA],

hence the desired statement holds since W A = FA n AA = Kern AA IFA (d) e V WA since limA E cA and X(A) 0 0. Thus, FA = WA E) (e) by (c). If A is conservative, then e E FA since e E IA because (ak) E 2 (cf. 2.3.6), and e E BA because of 8.2.4(a) and sup,,,,, IE'_o ankl < IIAII < oo. If A 0 is coregular, then A is conservative and X(A) $ 0.

In the first equality in (8.2:5) we have only made use of y E LA n IA = FA and a E CA (instead of a E wj). This observation gives us a representation of f E cA on FA which we make explicit in the following theorem.

Distinguished subspaces of domains 411

Theorem and Notation 8.2.6. Let f E CA' be given and let p E K be suitably chosen in accordance with the representation of f in 8.1.8(b) (cf. also the additional statement (iii)). Then we have following results: k) (x (a) f (x) = p (limA X - Ek akxk) + Ek xkf (e = (xk) E FA).

(b) If A is conservative, then x(f) := f (e) - Ek f (ek) = pX(A). In particular, p is uniquely determined if A is coregular. (c) If FA # WA, then p is uniquely determined. Proof. We verified (a) in the discussion prior to the theorem.

(b) If A is conservative, then for each f E cA and p E K chosen in accordance with 8.1.8(b) we get X(f) = f (e) - Ek f (ek) = p X(A), since, by 8.2.6(a),

f(e)=p(limAe->ak)+Ef(ek)=pX(A)+Ff(ek) k

f/

k

k

holds because e E FA (cf. 8.2.5(d)). If A is coregular, then x(A) # 0 and consequently p = -TA-) x(f ). In particular, p is uniquely determined. (c) Now, we assume FA # WA. By 8.2.5(c) there exists a u E FA \ A,q ,

that is a u = (uk) E FA with ) := limA u - Fk akuk # 0. Therefore, by (a), we get p = a f (u) - Ek Uk f (ek). This proves that p is uniquely

0

determined.

We draw some corollaries from 8.2.6 in the case of conservative matrices.

Corollary 8.2.7. If A is a conservative matrix, then the following statements hold:

(a) ccCSA andcCmncACFA=LAnIA. (b) WA C iP = ca in the FK-space CA. (c) A is coregular if and only if e WA. Proof. (a) We have co C SA since (co, 11 11.) is an AK-space (cf. 7.2.15(c)) and because FK-topologies are monotone (cf. 7.3.4(a)). Furthermore, c C

m n cA holds since A is assumed to be conservative, and FA = LA n IA is valid by 8.2.5(a). If x E m n CA, then x E IA because (ak) E P, and x E LA since sup.,v I Ek=o ankxkl < IIAII IIxiloo < oo (cf. 2.3.71 and 8.2.4(a)). (b) The inclusion WA C (7.2:9). Now, we prove

(in the FK-space CA) is already contained in

bf Ec,q : (cpCKernf Then we get co C

coCKernf).

by 6.5.21 and thus i7 = ea.

412

Matrix methods: structure of the domains

So let f E CA with W C Kern f, in particular with f (ek) = 0 (k E N°), be given. Then for every x = (xk) E co we get, by 8.2.6(a),

f (X) = p (limA x- E akxkl J= p X(A) lim x = 0 k

where p E K is suitably chosen. Here, we made use of the limit formula in the case of conservative matrices (cf. 2.3.71 or 8.1.10). (c) If A is coregular, then e f WA on account of limA E cA and X(A) y6 0. Conversely, if e 0 WA, then there exists an f E cA with X(f) A 0 which implies X(A) # 0 by 8.2.6(b). It is useful to summarize some of the results in 8.2.4 and 8.2.5 as follows.

Theorem 8.2.8. The relations

cp C SACWA = LAfAA C WA®(u) = FA = LAflIACLA = BACPA hold, where u = 0 or u E FA \ WA = FA \ AA is arbitrarily chosen. In Section 8.4 we will show by way of examples that the inclusions in 8.2.8 are in general strict. In 7.6.6 we stated that in every FK-space X containing V the subspaces

Fx, Wx and Sx are closed if Bx is closed. Next we prove that in the particular case of X := CA, the subspaces BA, FA, WA and SA are closed in the FK-space CA if and only if at least one of them is closed. For this we look for results on the closure of those subspaces. For simplification of the proofs we introduce the notion of replaceability of matrices.

Definition 8.2.9 (replaceability). A matrix A with O C CA is called replaceable if there exists a matrix D such that CD = CA and dk

limDek=0 (kEN°). A necessary condition for replaceability results from the uniqueness of FK-topologies.

Theorem 8.2.10. If A is replaceable, then FA = LA = BA. Proof. If A is replaceable, there exists a matrix D satisfying CD = CA and dk = 0 (k E NO). The latter implies ID = CD; thus FD = LD = BD. Thus, we get FA = LA = BA since FD = FA and BD = BA hold by the definition of those sets and by the uniqueness of FK-topologies.

We get a sufficient condition for the replaceability of a matrix A from the representations of the continuous linear functionals on CA.

Theorem 8.2.11. If there exists an f E CA with , C Kern f and p 96 0 for at least one representation of f with a E cA (according to 8.1.8(b) and the additional statement (iii)), then A is replaceable.

Distinguished subspaces of domains 413

Proof. Using 8.1.9 we may choose for f, with the assumed properties, a matrix D such that CD = CA and limD = f- Further, since cp C Kern f holds we obviously get dk =hmD ek = f (ek) = 0 (k E NO) -

In Section 8.3 we will enquire more precisely into replaceability. Now, we deal with the closure of `distinguished subsets'.

Theorem 8.2.12. The following statements hold, where we take in all cases the closure in the FK-space CA (and therefore relative to each dual topology of the dual pair (CA, CA), see 6.6.13): (a) cp C SA C WA C tp.

(b) cp = SA = WA and FA = BA = LA.

(c) FA = WA ® (u) _ i7 ® (u) where u = 0 if FA = WA and U E FA \ WA otherwise.

(d) If A is coregular, then LA = FA = Z. Proof. (a) These inclusions are already established in (7.2:9). (b) and (c): By (a) we immediately get 7 = TA = WA. To prove FA = BA = LA we distinguish two cases. Case I: Let A be not replaceable. Then we prove LA C WA, thus WA = LA, which implies FA = BA = LA because of WA C FA C BA = LA. For that (cf. 6.5.21) let f E CA' with cp C WA C Kern f be given. We prove LA C Kern f. By 8.2.11 we get p = 0 (for each representation of f according to 8.1.8(b)); that is, if we choose a suitable t E t and a E wA , then f has the representation

f (x) = t(Ax) + ax

(x E CA).

Because t(Ax) = (tA)x for each x E LA we get

f (x) = (tA)x + ax = yx

(x E LA),

where y = (yk) := to + a. Since p C WA C Kern f we now obtain 0 = f (ek) = yk (k E N°). Thus, LA C Kern f. Now, by 6.5.21 we get LACWA-

Case II: Let A be replaceable. Then FA = LA = BA by 8.2.10 and therefore FA = LA = BA. Since in addition we know FA = WA or FA = WA ® (u) where u E FA `WA = FA \ AA is arbitrarily chosen (cf. 8.2.5(c)), we get FA = WA or, if u V WA, FFAA = WA e (u) = WA (u), where we apply 6.5.9(b) in taking the closure of the sum. Thus, we proved (b) and (c). (d) This follows from 8.2.5(d), 6.5.9(b), 8.2.12(b) and 8.2.7(b): FA = WA ® (e) = WA + (e) = 'P + (e) = c° + (e) = CO + (e) = 'c. Remark 8.2.13. In 7.5.2(g) we verified that (bv, 11 Ilb") is a BK-AB-space

satisfying Sbv = Wbv = Fb, = bv° $ by = Bb,,. Since bv° is closed in

414

Matrix methods: structure of the domains

(bv, 11

this example proves that the second statement in 8.2.12(b) does

not hold in general in the case of FK-spaces X with cp C X. However, because cp C Wx C (cf. (7.2:9)), the statement ; = Sx = Wx is true for each K-space X with p C X. 0 The next theorem completes 7.6.6 in the particular case X := CA.

Theorem 8.2.14. In the FK-space CA the subspaces SA, WA, FA, BA and LA are closed if at least one of them is dosed. In such a case we have

either ip = SA = WA = FA = BA = LA or ip = SA = WA and 7 ® (u) = FA = BA = LA, where u E FA \ WA is arbitrarily chosen. Proof. Because SA C WA C FA C BA = LA we get by 8.2.12(a)-(c) the following chain of implications: SA is closed WA is closed = FA BA = LA is closed. Moreover, applying 7.6.6 to the special is closed, SA is closed' case X := CA we also get the implication `BA is closed which completes the proof of the first statement. The second statement is 0 an immediate consequence of 8.2.12(b) and (c).

Next we prove that PA is always closed in the FK-space CA in contrast to the subspaces SA, WA, FA and BA. Furthermore, we will compare PA with ip. For this it is useful to have the notion of test functions and some properties of them at hand.

Definition and Theorem 8.2.15 (test function). Let f E CA. Then f is called a test function (of A), if cp C Kern f and if p = 0 in at least one representation of f with a E cA according to 8.1.8(b) and its additional statement (iii). (a) PA C Kern f holds for each test function f E cA.

(b) If WA # LA and if f E cA satisfies LA C Kern f, then f is a test function.

(c) Let f E cA be given. Then f is a test function of A if and only if there exists a t E TA (cf. 8.2.2) with

f (x) = t(Ax) - (tA)x

(x E cA).

(8.2:6)

In addition, for each t E TA a test function ft is defined by (8.2:6). (d) PA = ntETA Kern ft = n {Kern f I f is a test function of A } .

Proof. (a) If f is a test function, then there exist t E P and a E cA such that f (x) = t(Ax) + ax (x E CA). Because a = (ak) E cA and 0 = f (ek) = [tA]k + ak (k E N), for each x E CA the existence of (tA)x follows and, moreover, we get (tA)x = -ax. Thus t E TA, and

f(x) = t(Ax) - (tA)x (x E CA).

Since t E TA we obtain t(Ax) = (tA)x, therefore f (x) = 0 (x E PA), which proves PA C Kern f.

Distinguished subspaces of domains 415

(b) Let f E CA with LA C Kern f be given. If f is not a test function, then we may assume u = 1 and choose t E P and a E CA satisfying

f (x) = lifA x + t(Ax) + as

(x E CA).

For every x E LA we obtain

P X) = limA x + (tA)x + ax = limA x + ryx with -y = (-yk) := to + a. From this and Thus,

C LA C Kern f we get 0 = f (ek) = ak + 1k (k E NO) . akxk

11mA x =

(x E LA)-

k

Consequently LA C AA , and thus WA = LA (cf. 8.2.5(b)).

(c) If t E TA and ft : CA -+ K, x -1 t(Ax) - (tA)x, then ft E CA' by 8.1.8(b) and its additional statement (iii) since to E CA 3. Therefore, ft is a test function because tp C Kern ft holds on account of 8.2.4(b) and V C LA. That each test function has a representation (8.2:6) has already been shown in the proof of (a). (d) This follows from the definition of PA, (a) and (c).

Theorem 8.2.16. (a) PA is closed in the FK-space CA. or PA = P ® (u), where (b) codimpA iT < 1; that is, either PA = u E PA `ip is arbitrarily chosen.

(c) If LA 5WA, then FA=LA=PA. (d) If A is coregular, then FA = LA = c = PA.

Proof. (a) As in 8.2.15(c) we consider for each t E TA the functional

ft E cA defined by ft(x) := t(Ax) - (tA)x (x E CA). By 8.2.15(d), PA = ntETA Kern ft. Furthermore, Kern ft is closed in the FK-space CA since ft is continuous (cf. 6.5.3); thus PA is closed as an intersection of closed sets.

(b) Because of ip C PA and the fact that PA is closed we have i7 C PA. So it remains to consider the case 7 C PA. Let us choose an arbitrary u E PA `gyp and show PA = i7 ® (u). For that we choose in accordance with

6.5.5 for u an f E cA with cp C Kern! and f (u) = 1 and for f a p E K, t E 2 and a E c0 such that AX) = p limAx + t(Ax) + ax (x E CA). By 8.2.15(a) we have u # 0. If

For this 9 there would exist

® (u) C Kern g and g(v) = 1. E K, T E t and & E cA such that

g(x) = µ limAx + t(Ax) + &x

(x E CA),

416

Matrix methods: structure of the domains

where again ji # 0 would follow from 8.2.15(a). Thus we would obtain for h := p f - lag the statements h E CA', cp C Kern h and

h(x) = (µµ - p) limAx + (µt - pt)(Ax) + (tca - p&)x for each x E CA. Therefore, h would be a test function since ip - E.cµ = 0,

which would contradict 8.2.15(a) because u E PA and h(u) = A f (u) -

µg(u) = µf (u) = µ 0. (c) Let LA # WA. Then we have to prove

AA = PA (cf. 8.2.12(b)).

For that (cf. 6.5.21) we choose an f E cA with LA C Kern f and verify PA C Kern f. However, because f is a test function, by 8.2.15(b), the latter follows immediately from 8.2.15(a). Consequently, we get PA C LA and thus LA = PA (cf. part (a) and 8.2.4(c)). (d) If A is coregular, then FA = WA E3 (e) (cf. 8.2.5(4)); thus LA # WA. 0 Hence, the statement in (d) follows from those in (c) and 8.2.12(d).

At the end of the section we consider the subspace IA which is essentially determined by the column limits of A. It is obvious to enquire into

the invariance of IA, that is whether IA is determined just by the sequence space CA. For this we consider first the intersection of all ID where D runs through all matrices with CA = CD. Later, we show with Example

8.4.6 that it differs from IA in general, that is IA is not invariant.

Definition and Theorem 8.2.17 (internal inset). Let A = (ank) be any matrix (cf. 8.2.1). Then IAt :_ ID is called internal inset, and FA = Iqt. Proof. It is sufficient to prove II t C LA since FA C IA t and FA = LA n IA; hence, x 0 FA implies x 0 IA D IIt or x 0 LA D IAt. Let x E IAt be given. We have to prove the existence of (tA)x for every t E t. We define, for an arbitrarily given t = (tn) E f, the matrix D = (dvn) by dvn :=

1

ifv=n

t,

if n < v

00

ifn>v

(v, n E IJ° ),

which we considered earlier in the proof of 8.1.9(b) in the case p = 1. The matrices D and D-1 are conservative triangles. Applying 2.6.1(a) to this situation we obtain cc = CA where C := DA. By the definition of and D we have C=

cvk = E tnank + avk

(v, k E N° and v> 1);

n_0

thus Ck = limv En tnank+ak for each k E NO. From this we obtain the existence of (tA)x = k xk En thank since x = (xk) E IIt C IAnID.

0

Distinguished subspaces of domains 417

Analogously to the more general case of K-spaces, the case when SA, WA, FA or BA (= LA) is maximal, that is equal to CA, is of interest here too. In this connection we define:

Definition 8.2.18. Let A be a matrix. A has AK, SAK, FAK or AB, if the FK-space CA has.

A is associative :4=i LA = CA. IA = CA. A has maximal inset A has PMI (propagation of maximal internal inset) :4--* IA t = CA. Corresponding to the characterization of the AK-property of FK-spaces we now give a characterization of the AB-property of CA (and A).

Theorem 8.2.19 (AB-matrices). The following statements are equivalent:

(a) A has AB. (b) A is associative. (c) A has FAK. (d) A has PMI. (e) A has SAK, or CA = WA ® (u) holds with an arbitrary u E FA \ WA. (f) A has AK, or CA = SA®(u) is satisfied with an arbitrary u E FA WA. In (e) the case SAK occurs if and only if in (f) the case AK does. Proof. For each matrix A with W C CA we have CA = BA A has AB A is associative CA = LA [cf. 8.2.4(a)] A has FAK CA = FA [cf. 8.2.14] CA = IAt

A has PMI

[cf. 8.2.17]

and by 8.2.14 also

CA = WA or CA = WA ®(u) with an arbitrary u E FA \ WA

CA = BA

CA = SA

or CA = SA ® (u) with an arbitrary u E FA \ WA,

where CA = WA occurs if and only if CA = SA (cf. 7.6.7).

0

In closing this section we aim a (functional analytic) proof of Remark 2.8.10. First, we-specify in the case of domains CA the fact that BA is an FK-space (cf. 7.6.3).

Proposition 8.2.20. For any matrix A = (ask) with to C CA the space (BA, P := {qj ' j E NO } U {q}) is an FK-space where the semi-norm q is defined by

q(x) := supIlAxihI II. = sup n

n,r

ankXk k=0

(x = (xk) E CA).

Matrix methods: structure of the domains

418

If, in addition, A is a triangle, then q is a norm and generates the FKtopology of BA.

Proof. Noting sup r

E dnkxk <_ q(x) (n E N°) and IIAxII. <_ q(x) (x = (xk) E BA) k=0

we get that the topology Tp generated by P is finer than the topology TAIGA induced by the FK-topology TA of CA. Thus, (BA, P) is an FKspace by 7.6.3. If, in addition, A is a triangle, then (CA, II Iloo o A) is a BK-space and, applying 7.6.3 as before, q is a norm and Tp is even 0 generated by q.

Now, applying the last proposition, we may give a short proof of the statement in Remark 2.8.10. For this we reformulate it in a language appropriate to the notions of this chapter.

Theorem 8.2.21. Let A be a regular triangle with p C CA. Then A has AB if and only if A satisfies the mean value property.

Proof. By definition, A = (ank) satisfies the mean value property if there exists a K > 0 such that (note WA = W) r

00

E ankxk

< K sup

avkxk

(0 < r
0
k=O

If applicable, we have

r

sup E avkxk < K IIAxII. ((xk) E E dnkxk < K 0
A

k=0

k=O

for all n, r E NO with 0 < r < n. Since A is a triangle the inequalities are trivially satisfied in the case n < r and, consequently. we have r T. ankxk

< K IIAxII. < oo ((xk) E CA and n,r E N°),

k=O

that is A has AB by 8.2.4(a). On the other hand, let A have AB. Then, by the second statement in Proposition 8.2.20, (BA, q) is a BK-space and, since CA = BA, the norms q and 1111 oo o A are equivalent. Thus, there exists a K > 0 such that

q(x) = sup JE ankxk n,r

k=O

KIIAxII.

(x E CA).

(8.2:7)

Replaceability and p-uniqueness of matrices

419

Now, let r E I6, x = (xk) E w and Ax =: y be given. Obviously, y[r] E c, i = (xk) := A-lylrl E CA, and xk = xk (k E N°). Therefore, by (8.2:7), we have r

co I7,

k=O

ankxk

KII Aal{0, = K 0sup


l

(n > r).

Thus A satisfies MK(A).

C]

Exercise 8.2.22. Let A be a matrix with O C CA. Verify that WA = (lcD=CA AD is satisfied.

Exercise 8.2.23. Show that for each coregular matrix A the inclusion BA C x holds where the closure is taken in the FK-space CA.

Exercise 8.2.24. Prove for any matrix A = (ank) with cp C CA (a) Y x = (xk) E CA : sup S l E ankxk

X E BA

n, v, 1A E No :

v>

k=µ

(b) AAC BA

< oo.

WA = SA. Hint: The application of 2.5.3 may be useful.

Bibliography: [254]; [260], [249], [20], [267], [16], [256], [248]

8.3

Replaceability and ts-uniqueness of matrices

Referring to 8.2.9-8.2.11 we deal further in this short section with replaceability of matrices. First let us say something about the motivation of the notion of replaceability. If a conservative, non-regular matrix A is given,

then it is obvious to ask for a regular matrix B with the same domain, that is CA = cB. Because CA = CB, the statements

X(B) 0 0

e 0 WB = WA

X(A) 54 0

(cf. 8.2.7(c)) hold, a necessary condition for a positive answer is that A is coregular. (As Example 8.4.4 will show, coregularity of A is not sufficient for the existence of a regular matrix B which is equivalent to A.) Therefore, generalizing the notion of regular matrices, it is useful to consider so-called multiplicative matrices which lead to the notion of replaceability, if we ask for the existence of multiplicative matrices B with CA = ca.

Definitions and Remarks 8.3.1 (multiplicative). Let D = (dnk) be a conservative matrix. Then D is called multiplicative if there exists a t E K 'such that hmD x = t lim x holds for each x E c. In such a case, D is more precisely called t-multiplicative. (a) D is multiplicative if and only if dk = 0 for each k E N, where dk denotes the limit of the k`h column. If D is t-multiplicative then t = X(D).

420

Matrix methods: structure of the domains

(b) If D is a t-multiplicative matrix, then there exists a regular matrix B which is equivalent to D if and only if t 54 0.

(c) A conservative matrix A is replaceable if and only if there exists a t E K such that A is equivalent to a t-multiplicative matrix. In such a case, t = 0 if and only if A is conull. A

Proof. (a) If D is t-multiplicative, then 4 = limD ek = t lim ek = 0 for all k E NO. Conversely, if dk = 0 (k E NO), then we get limb x = X(D) lim x for each x E c by the limit formula in 2.3.71. In particular, D is necessarily X(D)-multiplicative. (b) If D is t-multiplicative and t # 0, then B := i D is obviously equivalent to D and regular since limB x = i limD x = lim x for every x E c. Conversely, if B is regular and equivalent to D, then e V WB = WD C FD

(cf. 8.2.7(c)); thus limb e 96 Ek limb ek since limD E cD, that is t = X(D) j4 0 by part (a). (c) The first equivalence follows immediately from part (a). Now, let us assume that A is a replaceable conservative matrix. Then we have A is conull

e E WA e E WD

D is conull X(D)=0

[cf. 8.2.7(c)] [because WA = WD] [cf. 8.2.7(c)]

t=0.

This proves the second equivalence. Applying 8.2.11 we now give further necessary and sufficient conditions for replaceability which are mainly due to A. Wilansky (1964) and which allow a characterization of replaceability in the case of coregular matrices.

Theorem 8.3.2 (replaceable coregular matrices). (a) If P C CA and !p 96 PA, then A is replaceable.

(b) For every coregular matrix A the following statements hold: A is replaceable 4--* e f ip ip 54 PA. Proof. (a) Let itp 34 PA. Then we choose a u E PA \;5 and then a functional f E CA such that cp C Kern f and f (u) = 1. By 8.2.15(a) we get JA 34 0 for

each representation of f (with a E Cf) according to 8.1.8(b). Applying 8.2.11 we obtain that A is replaceable. (b) Let A be coregular. If A is replaceable, then there exist a t # 0 and a t-multiplicative matrix D with CD = CA (cf. 8.3.1(c)). Thus, e V ip since limp E c1, cp C Kern (limD) and 0 0 t = lirD a (cf. 6.5.21). Moreover, we obtain PA 0 ip from e ip since e E = PA 8.2.16(d)). Finally, part (a) tells us that PA # implies the replaceability of A. In the proof of (b) we made use of e V ilp D WA, and thus implicitly of FA = WA ® (e); that is, of FA 36 WA. Therefore one can expect that in the case of matrices A with FA 0 WA a characterization of replaceability

Replaceability and la-uniqueness of matrices

421

corresponding to that in (b) works. Let us remark here that conservative matrices with FA # WA are called almost coregular.

Theorem 8.3.3. If O C CA and WA i4 FA, then A is replaceable if and

only ifip0PA. Proof . We leave this to the reader, since it runs completely parallel to that of 8.3.2(b) if one replaces e by a suitable u E FA \ WA. Now, we give further sufficient conditions for replaceability where the first two conditions are deduced from 8.3.2(a).

Theorem 8.3.4. Let cp C trA. Then each of the following statements is sufficient for replaceability of A : (a) FA jA WA.

(b) FA # WA and AA is closed (in CA). (c) A has maximal inset (that is, IA = CA ).

Proof. (a) This statement holds on account of 8.3.2(a) and 8.2.16(c) in connection with 8.2.12(b).

(b) By FA # WA and AA = AA we get FA 96 WA since otherwise (cf. 8.2.5(b)) FA = WA = FA n AA c FA n AA ; therefore FA C AA and so WA = FA n AA = FA which would contradict the hypothesis FA ,-F WA. Thus A is replaceable by (a). (c) If IA = CA, then f := AA E CA (cf. 8.1.8(b) and its additional statement (iii)). Now, A is replaceable by 8.2.11 since u = 1 and cP C Kernf = AA .

By 8.2.11 a matrix A with cp C CA is replaceable if there exists an

f E cA such that W C Kern f and that p 0 0 holds for at least one representation of f according to 8.1.8(b) with a E CA 6. In this connection we introduce the notion of p-uniqueness and clarify its connection with replaceability. We will discuss this in more detail in Section 8.7.

Preliminary Remark 8.3.5. If A is any matrix and f E CA is represented in accordance with 8.1.8(b) by

AX) = p limA x + t(Ax) + ax

(x E CA),

(8.3:1)

then, in connection with the p-uniqueness, which will be defined next, we

also have la E K and t E 1, and we always assume a E cA which is justified by the additional statement (iii) in 8.1.8.

A

Definition and Theorem 8.3.6 (p-uniqueness). A matrix A is called p-unique, if for every f E cA each representation according to (8.3:1) has the same value for p. Each matrix A satisfies the following statements: (a) If

422

Matrix methods: structure of the domains

(b) A is p-unique if and only if in (8.3:1) the value u is uniquely determined in the case of at least one f E CA. (c) If cp C CA and A is not p-unique, then A is replaceable. Proof. (a) is an immediate consequence of 8.2.6(c). (b) Obviously, it is sufficient to prove the implication `#-' which we will prove indirectly. Thus, let g E cA and let 1, u E K with l 36 A, and t, t E l and a, a E c be given such that

g(x) = p lima x + t(Ax) + ax = p lima x + t(Ax) + ax (x E CA). Then we get 0 = (lc - u) lima x + (t -

(Ax) + (a - a)x

(x E CA)

which gives us a representation according to (8.3:1) of the null functional

with p := µ - ji # 0. Consequently, since f = f + 0 the value p is not uniquely determined for each f E c'A' (c) This is an obvious corollary of 8.2.11 since by (b) the null functional has a representation with p i4 0.

In Section 8.4 we will see some examples of replaceable and nonreplaceable matrices. Furthermore, by generalizing 8.3.3 we will give in Section 8.7 a characterization of replaceability of matrices.

Exercise 8.3.7. Let A be a matrix with

C CA and column limits ak (k E NO). Show that each of the following conditions is sufficient for replaceability of A : (a) FA 0 WA and IA is closed in CA. (b) FA 91 WA and (ak) E (IA)7. (c) FA 36 WA and IA C LA.

Exercise 8.3.8. Let A be a conservative matrix. Prove the equivalence of the following statements: (a) A is replaceable.

(b) There exists a matrix B with cB = CA and an s E cA such that the (infinite) system of equations yB = -b - s where b := (limn ek)k has a solution in P.

(c) There exists a matrix D with CD = CA and an f E cD with Kern f D co and to # 0 in a suitable representation of f relative to D and according to (8.3:1).

Bibliography: [254]

Examples 423

8.4

Examples

In the following series of examples we demonstrate that certain inclusions (for example, those in 8.2.8) between distinguished subspaces of matrix domains may be strict. Moreover, we will show that some of the cases, which we considered in 8.2, can in fact occur. For example, we will verify that there exist both examples of matrices A with PA = and examples of matrices A with PA = ED (u). Throughout this section we consider the class of conservative matrices subdivided by three properties, namely: - coregular matrices (that is, e 0 WA), - almost coregular matrices (that is, FA $ WA), - very conull matrices (that is, FA = WA). First we turn to particular examples of coregular matrices.

Example 8.4.1 (identity matrix). If A := I, then CA = c, COA = Co and X(A) = 1. Thus, A is regular, therefore 1-multiplicative and replaceable. Furthermore,

'P=CO=SA=WA=COA and

5®(e)=c=FA=LA=BA=IA=PA=CA. In particular, SA, WA, FA, LA and BA are closed, e % WA $ FA and

tl

A satisfies the property FAK.

Example 8.4.2 (Ceshro matrix). Let A := C1. Then A is regular, thus 1-multiplicative and replaceable. Further we have the identities

Y=CO=SA=WA=COA and

(8.4:1)

=BA =LA =FA DmnCAZ C.

CA

(8.4:2)

In particular, SA, WA, FA, LA and BA are closed in CA, the null domain COA is an FK-SAK-space and A has the property FAK. A

Proof. In 2.4.8(c) and 2.5.8 we proved c q m fl cc, C cc,. By means of 8.2.19 the identities in (8.4:1) and (8.4:2) are verified, after we have shown BA = CA. Namely, the statement WA = BA fl AA = Kern limA = COA is satisfied. However, BA = CA follows from 8.2.4(a) since for each x = (xk) E CA and n, v E No we have that

ifn
k=O

is satisfied.

1

1

v+1

ifv
< 00

424

Matrix methods: structure of the domains

Example 8.4.3 (Zweier matrix). Let A := Zi be the so-called Zweier matrix (cf. 1.2.13(c)). Then the following statements hold: (a) A is regular (thus 1-multiplicative and replaceable).

(b) CCmf1CA=FA=BA=LACCA. (c) co=SACmfCOA=WA9COA (d) CA = PA = c = Z (e) and ip = COA In particular, SA 0 WA and thus the subspaces SA, WA, FA and LA are not closed in CA by 8.2.14.

Proof. (a) The matrix A is obviously regular. (b) Because ((-1)k) E m fl COA we get co ¢ m fl COA and c

m n cA. Therefore A also sums an unbounded sequence (without loss of generality) to zero by 2.5.8, that is m l.COA C COA and m n cA C CA. Moreover, FA = LA = BA follows from IA = CA and FA = LAfIA. Since mf1CA CFA (cf. 8.2.7(a)) it remains to prove BA C m fl CA. However, this is a consequence of 8.2.4(a) since, for each x = (xk) E BA and v E No, the following holds: v

> sup Flankxk n

00 > sup

n,r

k=0

k=O

V

E k=0

Thus X E M. (c) We have already verified co C m fl COA C COA (note, A is regular).

Further, we get m fl COA C WA by the representation of f E CA on FA (cf. 8.2.6(a) and 8.2.7(a)) and therefore m fl COA = WA because m fl cA = (mflc°A)®(e) noting that mf1CA =FA =WA ED (e). Because co C SA (cf. 8.2.7(a)) it remains to show SA C Co. However, this inclusion is valid since, for each x = (xk) E SA and r E NO, we have IIA(x - xlr}) II

. = sup

00

00

ankxk k=r+l

E ar+l,kxk

k=r+1

and IIA(x - xfrl) Iloo -+ 0 (r -+ oo) is satisfied (cf. 8.2.4(d)). (d) We now prove COA C itp. Then C0A = i7 and c = PA = p ® (e) = 'CA follows from limA e = 196 0, codimCACOA < 1 and 8.2.16(b),(d).

So let f E CA with W C Kern f be given. Since A is a triangle, there exist p E ]K and t = (tn) E t with f (x) = p limA x + t(Ax)

(x E CA).

In particular, we get 0 = f (ek) = to 0 + (tk + tk+1), thus tk = -tk+1

for each k E N°, which implies tk = 0 (k E N°), that is t = 0, since t E t. This gives f(x) = A1inAx = 0 (x E c0A). Thus, COA C Kernf, which proves COA C 7 (cf. 6.5.21).

13

Examples 425

Next we give an example of a coregular non-replaceable matrix which is due to Zeller and proves (together with the preceding examples) that for coregular matrices A both the case PA = ip and the case PA 9E ip occur (cf. 8.2.16(b)).

Example 8.4.4 (coregular, non-replaceable).

Let (at) E e with

a2k = 0 and a2k+1 0 0 (k E lkl°) be given. For the matrix

0 1

a3

0

a3

1

the following statements hold:

(a) A is coregular with X(A) = 1. (b) 7 = cfl = c = PA = CA, in particular A is non-replaceable.

(c) ccmf1CACFA=IA={x=(xj)ECA I (x2k)EC}CLACPA. (d) SA=WA={x=(x,)E CA I (x2k)Eco} and FA = WA ED (e). In particular, FA # LA, and none of the sets SA, WA, FA and LA is closed in CA.

Proof. (a) is obviously true. (b) We now verify 7 = CA. (Then the remaining statements follow immediately since W C co c c, c = PA (cf. 8.2.16(d)) and in view of the equivalence `ip = PA e=> A is not replaceable' (cf. 8.3.2(b)).) Now, let f E CA with W C Kern f be given. Since A is a triangle, we may choose p E lK and t = (tn) E f such that f (x) = p limA x + t(Ax)

(x E CA).

In particular, for every k E N° we obtain if k is even

tk + tk+1 00

thank =

0 = f (ek) = laak +

pak + ak

E to

n=k

if k is odd.

From this we get for each odd k, because ak 76 0, the identity 0

=

1 f (ek) ak

=

1

f (e k+2)

ak+2

00

00

n=k

n=k+2

!l-A+Eto- E to

= tk+tk+l

Therefore, tk = -tk+1 (k E N°) so t = 0 since t E 1. Thus,

(8.4:3)

426

Matrix methods: structure of the domains

0=

00

ifW)

+ n=1

follows from (8.4:3). Hence, f = 0, that is Kern f = CA, which proves i7 = CA (cf. 6.5.21).

(c) Because (1, 0, 1, 0, ...) E CA we have c c m n CA. Furthermore, since FA = LA n IA, the identity FA = IA follows from IA C LA, which is true since

x E IA

4=

and

x = (xk) E CA

akxk converges (X2k) k

.=*

x = (x?) E CA

and

E c

E akxk converges

(8.4:4)

and (x2k) E c

k

holds (cf. 8.2.4(a)). By (8.4:4) we also get IA = {x = (x,) E CA I (x2k) E c} . Now, we prove IA Cs LA which involves FA 54 LA. For that we construct a sequence with k-1

X2k = -

a2v+1x2v+1 , (x2k) E m \ c and (X2k - X2k+2) E Co (k E NO) v=0

Consider the divergent series

2Y+1}1 and choose a sequence

with

s E {-1, 1} (v E N°) such that (Fkv_0 2+ ) E m c is satisfied. Then we put Sk

12k+1 =

(k E N°) (2k + 1)a2k+1

and k-1

xo := 0 and X2k = - E a2v+1x2,+1

(k E N).

v=0

This gives Sk (x2k - x2k+2) = (a2k ix2k1) =

2k+1)

ECo

and k-1

(x2k) =

Consequently, x

3v

- v-0 2v+ 1

Em

IA but x E CA because

n0 ifn=2v ifn=2v+1 E akxk= sv

k=O

2v + 1

(n,vEN°).

Examples 427

To prove x E LA, that is sup7,,,, IEk=o ankxk) < oo by 8.2.4(a), we determine for arbitrarily given v, n E No numbers v*, n* E No satisfying

2v*-1 2n*'.

If v < 2n*, then v'-1

v

ankxk = k=O

v*-1

a2µ+1 x2µ+i = .K=O

s µ=0 2µ + 1'

if v = 2n*, then E ankxk = E a2µ+1 i2µ+1 + x2n* = 0, µ=O

k=O

and if v > 2n*, then ankxk

(0

k=0

ifn=2n*

2n'+1

ifn = 2n* + 1.

This shows that x E LA and so LA 34 IA.

On account of 8.2.14 none of the subspaces SA, WA, FA or LA is closed in CA, and therefore we have LA C PA since LA C PA is true in any case (cf. 8.2.4(c)) and because PA is closed in general (cf. 8.2.16(a)). Now, we prove m fl CA 9 IA where we note that m fl CA C IA is true since A is conservative. To see this, consider the matrix B = (bnk) defined by bnk := a2k+1 (n, k E No ). It is obviously conull, and therefore its domain a2k+1xk converges

CB = {ixk) E w ! .k

contains by 2.5.7 unbounded sequences. Because

IA = { (xk) E w I (x2k) E c and E a2k+1 x2k+1 converges k the inset IA includes unbounded sequences, that is m fl CA C IA. (d) FA = WA ® (e) is satisfied since A is coregular (cf. 8.2.5(d)). Furthermore, with IA C LA we get WA = LA fl AA = AA . Therefore, for every x = (xk) E CA we have

x E WA 4

limA x- E akxk = 0 t-* (x2k) E Co. k

428

Matrix methods: structure of the domains

Thus, the second identity in (c) is verified. Now, if x = (xk) E WA, then x E SA (thus SA = WA) follows from n

IIA(x-xlr1)II0-sup

E akxk

+ SUP 1x2k I 2k>r

k=r+1

--4 0 (r -+ oo)

since x E IA and (x2k) E co, that is x1r1 -+ x in CA (cf. 8.2.4(d)).

0

In the next group of examples we consider almost coregular matrices, that is matrices A satisfying WA 96 FA.

Example 8.4.5 (e E WA # FA, M C SA). For A := diag

the following statements hold: (a) A is coercive and 0-multiplicative, in particular conull and replaceable.

) E c} = BA = LA = FA = PA. In particular, (b) CA = {(xk) E w I ( SA, WA, FA and LA are closed in CA.

(C) m9ciA=I(xk)EW

{

I

(f+A

) Eco =SA=WA=iT =CC PA=CA.

(d) FA = WA e (u) with u := (k + 1) E FA `WA. In particular, FA i4 WA, that is A is almost coregular.

Proof. The proofs of (a) and (b) are trivial and are left to the reader.

(c) and (d): Obviously, COA = S (xk) E W I (er) E co} , and therefore in C CoA. Further, since ( k -+1 ) E COA \ m, we have m C COA. Moreover, COA = AA since A satisfies (Spo), and COA = SA = WA = ip on account of (b), 8.2.5(b), 8.2.14. Because u := (k + 1) E CA \ COA and C C COA = P we get altogether (cf. also 8.2.5(c))

mcCoA=SA=WA =iP=CCFA =WA ®(u)=PA =CAIn particular, A is almost coregular.

O

Concerning the statements about distinguished subspaces, the matrix which we now study, while very similar to the Zweier matrix Zj. (cf. 8.4.3), is, nevertheless, conull.

Example 8.4.6 (coregular, almost coregular). The matrix F1 1

A :_

-1 -1

0

1

1 1

-1

has the following properties: (a) A is conull and 0-multiplicative, thus replaceable.

Examples 429 (b} C C m fl CA = FA = LA = BA C CA-

(c) co = SAC c g m fl COA = WA C COA. In particular, none of the sets SA, WA, FA and LA is closed in CA. (d) FA = WA ®(u) with u := (1, 0, 1, 0, ...), that is A is almost coregular.

(e) ip=C=COACCA =PA =7®(u) with u:_ (1,0,1,0,...). (f) CA = IA O PAt = FA, that is IA is not invariant.

Proof. Statement (a) is obvious. The proofs of (b)-(e) are quite similar to those for the corresponding parts of 8.4.3 and are left to the reader in Exercise 8.4.11. One should- note that u = (1, 0, 1, 0, ...) now takes the place of e in the proof of 8.4.3. Statement (f) is an immediate consequence 0 of (b), Theorem 8.2.17 and the preceding discussion.

Examples 8.4.5 and 8.4.6 have the common property that 7 = c C PA is satisfied. Thus, ip =E since the matrices are conull (cf. 8.2.7(b) and (c)), and ip C PA is satisfied on account of replaceability and because FA # WA (cf. 8.3.3). Now, we give an example of a non-replaceable, almost coregular, conull matrix.

Example 8.4.7 (almost coregular, conull, non-replaceable). put ak := 4-k (k E N°) and a0

ao ao ao ao

A :_

4-0 4-0 4-0

ai

N

0 a2 a2 a2

as as

ao

ai

a2

as

a0 a0

ai

a2 a2

as

d0

ai

a2

as

ai

We

as

4-i 4-i 4-1 a4 a4

as as as

a6 a6

4-2

a6

4-2

as

as

Then the following statements hold: (a) A is conull. (b) c C m fl cA C SA = WA C FA = WA ®(u) C LA where u = (uk) is defined by 14" if k = 3v+ 1 Uk (k, V E NO). 0 otherwise

In particular, A is almost coregular, and none of the subspaces SA, WA, FA and LA is closed in CA.

(C) CA =C=iP=PA =SA=WA=FA=LA(d) A is not replaceable, and, in particular, IA 0 CA.

Ll

430

Matrix methods: structure of the domains

Proof. (a) and (b): The matrix A is obviously conservative and for each x E m n CA we have

AA(x) = limA x - E akxk = 0

[since (4-vx3v+1) E co],

k

which implies m n CA C AA n LA = WA. In particular, A is conull and therefore c c m n CA (cf. 2.5.7). To prove m n CA # WA we consider v = (vk) defined by vk :=

f 2' if k = 3v + 1 0

(k, v E 1``l0 ).

otherwise

It has the desired properties: obviously v 0 m. Further (4-'v3,+,) E CO and (akvk) E cs. Thus, v E IA n AA = WA. We next prove FA = WA E) (u) where u = (Uk) is defined as above. To

prove u E LA let v, n E N be arbitrarily given and let p E No be chosen such that 3p + 1 < n < 3p + 3. Then µ-1

3µ+3

E a3µ+3,kuk = E a3k+1u3k+1 + 4-µ4u k=0

k=0

k=0

µ-1

1:4 -2k-1 +, 1 < 3 k=o

and thus u E BA = LA by 8.2.4(a). Further, since (akuk) E cs, we get u E IA; thus u E IA n LA = FA. However, u V Al- because (n-1

3n+1

E a3n+l,kuk = lim E

limA u =

4-2v-1 + 1

v=0

k=0

=

L r akuk + 1. k

Now, using 8.2.5(c) we have FA = WA ® (u). To prove SA = WA, by 8.2.4(d), it is sufficient to show

lim IIA(x - x[°')) II. = 0

(x E WA).

For that, let x = (xk) E WA and e > 0 be given. Then (4-ux3u+1) E co follows from 3µ

akxk

a3µ+l,kxk -

4-ux3µ+1 =

v

0

[since x E WA CAA

k=0

k

For each n E N, we denote by p(n) the uniquely determined k E NO with an,3k+i = 4-k. Then there exists an no E N such that

b n > no

ankxk k

akxk! + k

14-A(n)x3c,(n)+1)

11

<2

Examples

431

and

br>no : Hence we get n

1: ankxk

E ankxk - E akxk

k=r

+

+

I4-ti(n)x3µ(n)+1I < E

k

k

for all r > no and n > r which implies just II A(x - x(r1) III < e for each r > no. Therefore x E SA. The inclusion FA C LA is in general true. To verify FA ¢ LA we consider y = (Yk) defined by - k+1

ifk = 3v

k+1

ifk = 3v + 2

yk

0

(v E N°).

ifk=3v+1

Then (akyk) E cs\L. Modifying the sign of yk suitably we obtain a sequence z = (zk) which satisfies

)zkl =

yk 0

ifk543v+1

(v EN°f) otherwise and (akxk) E bs \ cs. (Draw a figure!) Now, we define x = (xk) by

xk:=zk if k#3v+1 (vEN°) and inductively by 3v

xk :_ -4° E akxk if k = 3v + 1 (v E N°). k=0

Then

ifn=3v+1

0 1

E ankxk k

a3v+2y3v+2 = 3v+2 3v+2+2

+ 3v+3

if n = 3v + 2 if n = 3v + 3

(n,v E N°);

thus x E COA. Further, x V IA follows since (akxk) 0 cs and

4-3v <

I:ta3v+1x3,+1I < Sup k V

Moreover, x E LA because for all n, r E No the inequality ankxk k=0

< 4 sup E akxk k=0

holds (cf. 8.2.4(a)). Hence, we have proved x E LA `FA.

00.

432

Matrix methods: structure of the domains

(c) We now show 7 = CA. (Then, the remaining identities in (c) are simple consequences of cp C C C CA, the inclusions in 8.2.8, and the closedness of

PA in CA.) For this let f E cA with

(x E CA)-

Because

0 =

= ua3,, + a3v

tn.

n=3v Therefore

0 = f(e3v) - f(e3v+3) = t3v + t3v+1 + t3v+2

(8.4:5)

for every v E N°, and thus

A = - E to = - E03v + t3v+1 + t3v+2) n

= 0-

v

Noting p = 0, we analogously obtain CO

3v+2)

0 = f(e

=a3v+2

tn n=3v+2

-

Thus,

00

0 = t3v+2 +

L

(v E N°)

to = t3v+2

(8.4:6)

n=3(v+1)

and, by (8.4:5), also (8.4:7)

(v E N°).

t3v = -t3v+1

Moreover, for each v E N° the statement 0

=

f(e3r,+1)

00

E

4-v(t3v+1 + t3v+2 + t3(v+l)) + a3v+1

to

n=3(v+1)+1

e 4-v(t3v+1 + t3(v+l)) - a3v+lt3(v+l) 4-vt3v+1 + 4-v(1 -

[cf. (8.4:6) and (8.4:7))

4-2v-1)t3(v+l)

holds. Applying (8.4:7) we get It3&I =

(1-4-2v-1)It3(v+1)I

(t3(v+l)I1

which implies t3v = 0. Therefore, by !(8.4:7), also t3v+l = 0 for each v E No

because t E C. Hence, p = 0 and t = 0 force f = 0.

Examples 433

(d) The matrix A is non-replaceable by 8.3.3 since FA WA and 7 = PA. Furthermore, we have IA ?6 CA because otherwise A would be replaceable (cf. 8.3.4(c)). Note, we have already verified IA 0 CA in part (b). Finally, we consider examples of very conull matrices. In the last decades

it has been found that these matrices present many problems when one attempts to answer certain questions; for example, in trying to characterize replaceability. Naturally, `most conull' is the zero matrix which is contained in the following example as a special case.

Example 8.4.8 (very conull, replaceable, ip = PA). For any given (ak) E 8 we define

0 A '_

Then A is coercive, and thus conull. Furthermore,

m C

t (Xk) E w

akxk converges } k

CA=IA=SA=WA=FA=LA=

BA

PA=gyp.

Also A is very conull and replaceable, namely by a1

a2

0

0

Proof. Obviously, CA = IA j m since a E P, and A is replaceable by the matrix D since CA = WD = CD (cf. also 8.3.4(c)). Further, in # CA because

each coercive matrix is conull and each conull matrix sums unbounded sequences by 2.5.7. The remaining statements are trivially satisfied if SA = CA holds. However, the latter is true since for each x = (xk) E CA we have n

I!A(x - xlrl) III =sup

E

akxk

--+ 0

(r -} oo)

n>r k=r+1

(cf. 8.2.4(d)).

By modifying the last example slightly we get a very conull but nonreplaceable matrix.

434

Matrix methods: structure of the domains

Example 8.4.9 (very conull, non-replaceable). For a given (ak) E t with ak # 0 (k E N°) let H = (hnk) be defined by hnk :=

Iak ak 0

ifk
if k = n = 2v otherwise

that is ao

H _

ao

0

ao ao ao

ai

0

al

a2 a2

0

aF

a2

a3

a4

Then H has the following properties: (a) H is coercive, and thus conull. (b) SH = WH = FH = IH C LH. In particular, H is very conull. (c) H is not replaceable and T = PH.

(d) None of the sets SH, WH, FH and LH is closed in CH, therefore LHCPH. A

Proof. (a) and (b): If A denotes the matrix in 8.4.8, then obviously m (; CA = Iii C CH. In particular, H is coercive. Because FH C IH and since the FK-topology of cH induces on CA a weaker topology than the FK-topology of CA, we get CA = SA = SH = WH = FH = IH from 8.4.8. The inclusion FH C LH is in general true. To prove IH y-f LH, therefore ((-1)k k ). We get x V IH because IH C LH, we consider x = (Xk) n

Eakxk k=O

- {0

1

if n is even if n is odd

(n E N°),

x E CH since Ek+ankxk = 1 (n E N°), and x E BH = LH because supn,r JEk=° ankxkl = 1. (c) Obviously, H is not replaceable since FH 0 LH (cf. 8.2.10). In particular, ip = PH by 8.3.2(a). (d) FH LH implies by 8.2.14 that none of the sets SH, WH, FH and 0 LH is closed. Hence, PH # LH since PH is in general closed.

As the last example we consider a matrix A that we already know, which is very conull and (trivially) replaceable and which satisfies c C PA = iT ® (u) with u V LA (cf. 8.2.16(b)).

Examples 435

Example 8.4.10 (very conull, PA = p ® (u) with u 0 LA).

The

matrix

A:= .1

has the following properties: (a) A is 0-multiplicative, in particular it is conull and replaceable.

(b) CcmnCA=mnC0ACC0A (c) SA=CO and mnCA=WA=FA =LA=BA. /(d) None of the sets SA, WA, FA and LA is closed in CA.

(e) u:=(n)fEN0 ECA`LA and CA=PA=7ED (u).

(f) 'P=c=LACPAProof . (a) is obviously true. (b) We have c C mncA since A is conull (cf. 2.5.7). To prove mncA C CoA

we assume K := lit (Then we may reduce the case K := C to this one by considering the real and imaginary parts separately.)

Let x = (xk) E CA with a := limA x # 0 be given. Then we may assume a > 0 without loss of generality (otherwise we consider y := -x)

and choose an no E N with xn - xn_F > z (n > no). Thus, for each n > no we obtain xn - xn_1 + xn_1 - xn_2 + ... + xno+l - xno + xno >

xno + (n - no)

a 2

which implies x V m because a > 0. Hence, m n CA = m n COA C CoA is proved. The latter inclusion is strict since x = (xk) E COA where xk k _v=0 7+1 (k E NO). (c) This part of the proof is similar to the corresponding part of 8.4.3 and is left to the reader (cf. Exercise 8.4.12). (d) is true because SA $ WA (cf. 8.2.14). (e) Obviously, u = (n) E CA and limA u = 1. Moreover, u 0 LA since 1

r n-1 sup L ankuk > sup ankUk n,r k=0 n k=0

= sup(n - 1) = oo.

n

We now prove CA = PA. (Then PA = iP ® (u) since u W C Kern IimA and limA u = 1.) For this, let

t=(tn)ETA={yEe'VxECA: (yA)x

on account of exists}

436

Matrix methods: structure of the domains

be arbitrarily given. We have to verify (tA)x = t(Ax) for each x E CA. Applying Abel's summation formula, we get, for every n E No and x (xk) E CA, the identities n

n

E tv F avkxk v=o

e

E tv(xv - xv_1)

[with x_i:= 01

v=0

k

n

E(tk - tk+1)xk - to+lxn k=0 n

tvavk - to+lxn-

xk

(8.4:8)

k=0

Si nc e

t E TA we get from this for each x = (xk) E CA.

(tn+lxn) E c

(8.4:9)

If we put y=(yk):=Ax (xECA) and note n

to+lxn = to+l E yv

(x = (Xk) E CA),

v=0

then (8.4:9) tells us that the matrix

0 t3

is conservative since A is a triangle and since therefore y = Ax runs through c completely if x runs through CA. Thus e E CT, hence (ntn) E C.

Moreover, since t E t, we have (ntn) E co because otherwise we could choose an e > 0 with'ntnt > e (n > no, no suitably chosen) which would imply 00

n=no

00

I to I >- E

Ln

n=no

contradicting t E 1. However, since (ntn) E co implies that T is 0multiplicative we have

(tn+lxn) = Ty E Co

(x = (xk) E CA and y = Ax).

Therefore (cf. (8.4:8)) the desired identity t(Ax) = (tA)x (x E CA) holds. (f) follows from i7 = WA (cf. 8.2.12(b)) in connection with WA = LA (cf. 13 (c)), PA = ape (u) (cf. (e)) and by the preliminary remark to 8.4.7.

Examples 437 example

property regular coregular conull

almost coregular very conull replaceable co = SA

SA=WA WA =7

c=mncA FA=mncA FA=LA BA=PA PA =7p WA

TA

8.4.1 8.4.2 8.4.3 8.4.4 8.4.5 8.4.6

+ +

-

+

+

-

-

-

+

+

+

+ +

- - - - - - + +

+ +

+ +

+

+

- - + + + + - + + - + + + - + + - - + - + + + + + - + - - - + - - - + -

8.4.71 8-4.8 8.4.91 8.4.101

+

+

+

+ +

- -

- - + + +

+ +

+

- - -

-

- + + + - + + + - - + + -

+ +

+ +

-

+

+ +

+ + + +

-

+ +

-

+

Fig. 8.4.1

We now leave matters at these examples for Sections 8.2 and 8.3 and close this section with an overview of the matrices in tabular form (cf. Figure 8.4.1). The signs, `+' and `-' mean that the matrix under consideration respectively has, or does not have, the property.

Exercise 8.4.11. Prove parts (b)-(e) of 8.4.6. Exercise 8.4.12. Verify part (c) of 8.4.10. Exercise 8.4.13. Prove that the matrix 1

-2 A:=

0

1

-2

1

-2

1

is (-1)-multiplicative, thus coregular and replaceable. Moreover, show

SA=WA=C0=mnCOA and FA=LA=BA=PA=C=mnCACCA, and that each of the sets SA, WA, FA and LA is closed in CA. Hint: As we will prove in 8.5.4, c is closed in CA if m n cA = c holds.

Bibliography: [254]; [20], [30], [249]

438

8.5

Matrix methods: structure of the domains

Bounded divergent sequences in the domain

In Section 2.5 we proved that each conservative matrix, which sums a bounded divergent sequence, also sums an unbounded sequence (cf. 2.5.8)

and that each conull matrix sums both a bounded divergent and an unbounded sequence (cf. 2.5.7). With the help of the results in Section 8.2 we easily obtain the first result and the second part of the second result. For this we make some preliminary observations.

Theorem 8.5.1. For any conservative matrix A we consider the following statements: (1) CA C m. (2) co is closed in CA. (3) co is closed in CA, and x(A) $ 0. (4) c is closed in CA. (5) PA = C-

(6) mflcA=c. Then we have:

(2) : If CA C m, then the FK-topology TA of CA is stronger (1) than the topology generated by 11 1I,,,, since (m, 11 fly) is an FK-space and because FK-topologies are monotone. Consequently, co is closed in (CA, TA) since it is closed in (m, 11 (jco). (2) . (3) : If co is closed in (CA, TA), then WA C co = co (cf. 8.2.7(b)) and, in particular, e 0 WA to is equivalent to X(A) # 0 (cf. 8.2.7(c)). (3) . (4) : By the closedness of co in (CA, TA) we get c = co ® (e) co ® (e) = co ® (e) = c (cf. 6.5.9(b)). is an FK-space and (4) . (2) : If c is closed in (CA,TA), then on account of the therefore TAj,, equals the topology generated by 1(I uniqueness of FK-topologies. Consequently, co is closed in (c,TAI,), and thus in (CA, TA).

(3)' is already (4) . (5) : Let c be closed in (CA,TA). Since `(4) proved, the matrix A is coregular. That implies PA = c by 8.2.16(d). The converse conclusion is obvious since PA is closed in the FK-space CA by 8.2.16(a). (5) . (6) : This implication holds since c c m fl CA C FA C PA (cf. 0 8.2.7(a) and 8.2.8). As an immediate corollary of `(1) (6)' and `(1) = (2)' in 8.5.1 we get statements which we proved in Part I by applying classical methods.

Corollary 8.5.2 (cf. 2.5.8 and 2.5.7). Let A be a conservative matrix. (a) If A sums a bounded divergent sequence, then it also sums an unbounded sequence.

439

Bounded divergent sequences in the domain

(b) If co is not closed in CA, then A sums at least one unbounded sequence. (Note, in this situation A is conull by `(2) (3)' in 8.5.1.) 0

Generalizing 8.5.2(b) we now prove that each conservative matrix A, for which co (therefore c too) is not closed in CA, sums not only unbounded but also bounded divergent sequences. This result goes back to Wilansky and Zeller [255] (1955) and it was proved in the general version, which we now give, by Meyer-Kong and Zeller [172] (1962) and later on by Bennett [21] (1972).

Theorem 8.5.3 (bounded divergent sequences).

If E is an FK-

space and if co nE is not closed in E, then E contains bounded divergent sequences, that is (m n E) \ c 0 0. We prove the theorem by means of the gliding hump method where we now apply the gliding hump method to a countable family of semi-norms which generates the FK-topology, whereas, up till now, we applied it only to infinite matrices. A further proof will be given in Section 10.2 as an application of Theorem 10.2.7.

Proof. Let (pj I j E N°) be a countable family of semi-norms generating the FK-topology r of E. Without loss of generality we may assume lxj 1 < pj (x) < pj+i (x)

(x E E and j E N° ).

(Otherwise we consider (pj I j E N°) with pj(x) := Ev_0 (Ix"I + p, (x)) for each x = (Xk) E E and j E N°.) First we verify

Ve>0 VjENO 3XEconE, Ilxllm=1 : pj(x)<e.

(8.5:1)

If (8.5:1) failed, then we could choose an c > 0 and a j E No such that

VxEconE, llxllo=1 : pj(x)>e, that is V

so that

IIxII. <

x#0 : pj

1'e,

pj(x) for every x E co n E.

Consequently, the FK-topology of co n E would equal rlconE (cf. 7.3.9 and 6.4.13) and, as a complete subspace of a metric space (cf. 6.2.21(b)), co n E would be closed in the FK-space E which would contradict the assumptions. Next we construct, with the aid of the gliding hump method and (8.5:1),

a sequence (x(")) in co n E such that the series x := E" x(v) converges in the FK-space E and x E (m n E) \ c holds. For that we determine inductively an index sequence (j") and a sequence (x(")) in co n E in the following way:

440

Matrix methods: structure of the domains

We put jo := 0 and choose, in accordance with (8.5:1), an x(°) (xko)) E co fl E with and

11x(O) II = 1

pjo (x(°)) < 1.

Now, if v > 1 and if j and x(O) = (xk")) k are already determined for µ E N,0_1, then we choose iv E No with jv > j,,_1 such that

(k > jv and 0 < .a< v)

Ix(") I < 2'"

(8.5:2)

which is possible since x('`) E co. In conclusion we choose for jv, in accordance with (8.5:1), an x(") = (xk"))k E co fl E with

and pjv (x(v)) < 2-v.

II x(v) II. = 1

(8.5:3)

It is justified to speak in this connection about a gliding hump method since the coordinates of x(") have a hump which slides to infinity as v tends to infinity; namely, x(") has the properties (k E No with k > jv+1)

Jxkv) I < 2-" (cf. (8.5:2)) and

Ixk") I < pk(x(")) < pjy (x(")) < 2-"

(k E No with k < jv)

(8.5:4)

(cf. (8.5:3)). For each v E No the latter ensures the existence of a

kENo with jv
(8.5:5)

n y(n)

E x(v)

(n E NO).

v=°

The sequence (Y (n)) is a Cauchy sequence in E, since for each (fixed)

jENO and forallr>n>j we have pj(y(r)

_ y(n))

e r

E p,Y(x(v))

[jv > j if v> j]

v=n+1

r

<

E 2-"-*0 (n -* oo and r > n). v=n+1

Since E is complete there exists an x = (xk) E E with

Bounded divergent sequences in the domain

yi"i -> x, that is x =

441

x(I) in the FK-space E.

Moreover, E is a K-space; thus the convergence is coordinatewise, that is xka)

xk =

for each k E NO.

Thus, (m fl E) `c # 0 is proved, if we also show x E m \ c. By the choice

of (jr,) and (x(")) we obtain for each (fixed) k E NO with j. < k < j"+1 (cf. (8.5:2)-(8.5:4)) Ixk") I

<

2-" ifµ
if p = v

2-JI

if14>v

(µ E NO)

and thus

v-1

Lr

Ixk

p=0

00

I xk ) I +

I

xkµ)I

xkv) I+ EI µ=V+1

V-1

00

57 2-" + 1+ E 2-9 Fz=v+l

/z=O

v 2-v + 1 + 2-" < 2, which implies 0x110 < oo and lira sup Ixk 1 < 1. Therefore, x E m. k

that

Now, we show x f c. Consider first for 00

Ixj,l µ=0

jz=v+1

Ixi")I

v2-"+2-"+2-" <

(v+2)2-" -+0(v-4 oo).

Then we choose an index sequence (kv) in accordance with (8.5:5) such

that

j"
and

Ix(")I=1 (vEN°)

and get, for (xk ), the statement v-1

Ixk' i

Ixk' I

Ixk, I

00

Et=O

>

1- v2-" - 2-"

v

µ=v+l

Ixk'

I

1 = lim sup Ixk k

Therefore x does not converge since there exist two subsequences of x with 0 different limits.

442

Matrix methods: structure of the domains

We now draw some corollaries from 8.5.1-8.5.3.

Corollary 8.5.4. For every conservative matrix A the following statements are equivalent: (a) m n cA = c.

(b) co is closed in CA. (c) c is closed in CA.

(d) PA = C. Proof. Since co C c C CA, `(a) . (b)' is the case E := CA of 8.5.3 whereas `(b) . (c)', `(c) #- (d)' and `(d) (a)' are contained in 8.5.1.

Corollary 8.5.5 (cf. 2.5.7). Each conull matrix sums both bounded divergent and unbounded sequences.

Proof. Assuming that m n CA = c, then co is closed in CA by 8.5.4 which gives us the contradiction X(A) # 0 by '(2)a (3)' in 8.5.1. Therefore c C m n CA and consequently CA \ m # 0 by 8.5.2(a).

Corollary 8.5.6. If A is conservative and if at least one of the subspaces SA, WA, FA, LA and PA of CA contains a divergent sequence, then CA contains a bounded divergent sequence.

Proof. In any case we have PA 34 c (cf. 8.2.8) which forces, by `(4) 4* (5)' in 8.5.1, that c is not closed in CA. Thus, m n CA # c by 8.5.4. We close this section with a consequence of Corollary 8.5.4 concerning the associativity of the matrix product A(BC). We promised in 2.2.3(c) the proof of the existence of infinite matrices A, B and C for which all of the

products AB, BC, (AB)C and A(BC) exist but still (AB)C # A(BC) holds.

Remark 8.5.7. By Theorem 2.5.14 there exist (a lot of) regular triangles B such that m n cB = c C cB. Then we have c = PB C CB (cf. Corollary 8.5.4). Thus, by the definition of PB (see also 8.2.15(d)) there exist an x E cB and a t E TB C f such that (tB)x and t(Bx) exist but (tB)x # t(Bx) holds. (Note, tB and Bx exist since t E P, B is regular and x E CB.) Now, let A be the matrix which has t in each row and C be the matrix which has x in every column; then, obviously, AB, BC, (AB)C and A(BC) exist but still (AB)C 34 A(BC) holds.

Exercise 8.5.8. Let A and B be conservative. (a) Prove that CA n cB n m \ c o 0 if A and B are conull. (b) Apply (a) to. prove 2.5.13.

Exercise 8.5.9. Verify that for each conservative matrix A the following statements hold: (a) If B is a matrix with IIBII < oo and BA = I, then m n CA = C.

Consistency and perfectness

443

(b) If B is a matrix with IIBII < oo and BA = I = AB, then B is conservative.

Exercise 8.5.10. Verify that AA = co if A is a conservative matrix with A,q C m fl CA. Hint: Consider the cases (lima ek)k = 0 and (limA ek) k 0 0. Bibliography: [2541, [172], [21]; [255], [267]

8.6 Consistency and perfectness The material in this sectiorr and in Section 9.1 is connected with the consistency theorem 2.6.11. Recall that Theorem 2.6.11 showed that for matrices A and B which are conservative for null sequences and satisfy cB D m fl cA = (m n AA) ®(u) where u = 0 or u E IA `AA , we can conclude the consistency of A and B on m n CA from their consistency on the smaller subspace WED (u). The significance of that theorem becomes clearer if we notice that, in general, we do not know the bounded domain

m fl CA and that it is very easy to check consistency of A and B on the simple space cp ® (u), since it is equivalent to ak = bk (k E N°)

and

lima u = lima U.

In this section we make use of the facts that PA = ip ® (u), where u = 0 or u is suitably chosen in PA \ gyp, and that PA is closed in the FK-space CA which implies that PA is an FK-space (where it is endowed with the relative topology of the FK-topology of CA). Consequently, if we assume cB D PA, then the linear functional limB IPA is continuous on PA on account of the monotonicity of FK-topologies. Therefore the consistency of

A and B on V ® (u) implies the consistency of A and B on PA, as we now show. We consider the continuous linear functional

f : PA -4 K, x -* AX) := limB x - limA x. Because PA = ® (u) = cp ® (u), the consistency of A and B on cp ® (u) gives us cp ® (u) c Kern f; thus PA C Kern f. Hence A and B are consistent on PA. We thus have the following theorem.

Theorem 8.6.1. Let A and B be matrices with cp C ® (u) = PA C cB where u = 0 or u E PA \ ip. Then consistency of A and B on cp ® (u) implies consistency of A and B on PA. As we stated in 8.1.9, we have lime I IA E c,q for each matrix B with CA C CB and for each f E cA there exists a matrix B with CA C CB and limB IAA = f. Since PA = CA is not necessarily true, we may guess that in the situation of Theorem 8.6.1 we cannot conclude from consistency of A and B on cp ® (u) their consistency on CA, if we consider matrices B with CA C CB.

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Matrix methods: structure of the domains

First we consider matrices A which satisfy PA = CA and (because c C PA in general, and since c = PA for coregular matrices A) those conservative matrices A satisfying c = CA-

Definition and Remarks 8.6.2 (perfectness). A matrix A with cp C CA is called P-perfect if PA = CA. A conservative matrix A is called perfect if c = CA (in the FK-space CA). (a) Because Z! C PA (cf. 8.2.7(a) and 8.2.16(a)) each perfect matrix is also P-perfect. (b) Each coregular matrix A is perfect if and only if it is P-perfect (cf. 8.2.16(d)). (c) For example, the coregular matrices in 8.4.1-8.4.4 (in particular, I, Z1,,2 and C1 ), the almost coregular matrix in 8.4.7 and the very conull matrix in 8.4.8 are perfect. (d) The matrices in 8.4.5 and 8.4.6 and also the very conull matrix in 8.4.10 are P-perfect, but not perfect. (e) One-sequence methods are not P-perfect (cf. 2.5.14 and 8.5.7). We now formalize the remarks prior to 8.6.2 in a theorem.

Theorem 8.6.3 (consistency and P-perfectness). If A is a matrix with W C PA = ® (u) where u = 0 or u EPA \ gyp, then the following statements are equivalent:

(a) A is P-perfect. (b) A is consistent on CA with every stronger matrix B which is consistent with A on W ® (u).

Proof. (a)

(b) follows from 8.6.1 because of PA = CA.

(b) = (a) : Let f E cA with W ® (u) C Kern f be given. Since PA =

Now, if (b) holds, then the consistency of A and B on CA follows. Thus, 0 = limb x = f (X) for each x E cA.

If we consider in 8.6.3 the case of coregular matrices, then we have

PA = c = ilp ® (u) with u = 0 or u = e (cf. 8.2.16(b) and 8.3.2(b)). Further we get that P-perfectness is equivalent to perfectness, and that we can replace in part (b) of 8.6.3 consistency on cp ® (u) by consistency on c (cf. 2.6.3(b)). The corresponding theorem is true for conservative matrices as we now state.

Consistency and perfectness

445

Theorem 8.6.4 (consistency and perfectness). For each conservative matrix A the following statements are equivalent: (a) A is perfect. (b) A is consistent with every stronger matrix B which is consistent with A on V ® (e) (or equivalently on c). Proof. `(a) =*- (b)' is obviously contained in `(a) (b)' of 8.6.3. The proof of the converse implication is analogous to that of the corresponding part O in 8.6.3 and is left to the reader.

To get consistency results analogous to 8.6.3(b) and 8.6.4(b) we seek necessary and sufficient conditions for P-perfectness and perfectness, respectively. The examples 8.4.3, 8.4.4 and 8.4.7 will help us as we try to formulate such conditions. These examples are triangles, and to prove PA = CA in each case we proceeded in the same way. We represented each f E CA' with cp C Kern f by f (x) = p limA x + t(Ax)

(x E cA)

with suitably chosen p E K and t E Z. Then we deduced to = 0 from V C Kern f and finally t = 0. This observation motivates the study of matrices of type M which we introduced in Definition 2.7.3. This definition

tells us that a matrix A is of type M if and only if the zero matrix is the only matrix B which has rows in P and is a left divisor of zero of A. Before we clarify the connection between `perfectness' and `type M' we give more examples of matrices of type M.

Examples 8.6.5. (a) Each matrix in the examples 8.4.1-8.4.7 and 8.4.10 is of type M. (b) Example 8.4.9 is not of type M. (c) The matrix in 8.4.8 is of type M if and only if ak # 0 (k E N°). Proof. (a) That the matrices in 8.4.1 and 8.4.2 are of type M is already proved in 2.7.3. The proof that the matrices in 8.4.3, 8.4.5, 8.4.6 and 8.4.10 are of type M is straightforward and is left to the reader. We give here short proofs for the remaining examples. 8.4.4: Let a = (ak) E P with a2k = 0 and a2k+1 0 0 (k E N°) be given and let A = (ank) be the matrix defined in 8.4.4. To prove that A is of type M let t = (tk) E e with to = 0 be given. Then 0=

tk + tk+1

if k is even

E

if k is odd [because ak 54 0].

00

n=k

t

By the second identity we get for each odd k E No the statement 00

0 = Lr to - E to = tk + tk+1 00

n=k

n=k+2

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Matrix methods: structure of the domains

and from it and the first identity, tk = -tk+1 (k E N). Thus t = 0 since

tE2. 8.4.7: Let ak := 4-k (k E N0) and A = (ank) be the matrix defined in Example 8.4.7. To prove that A is of type M we look at the proof of ip = CA in 8.4.7(c). Let t = (tk) E e with to = 0 be given. This means 00

00

00

0 = a3v E to , thus t3v + t3v+1 + t3v+2 = E to n=3v

57, to = O, n=3v+3

n=3v

(8.6:1)

and 00

00

0 = a3v+2 E tn = a3v+2 n=3v+2

t3v+2

+E

to

,

n=3(v+1)

which implies in combination with (8.6:1) t3v+2 = 0 and

t3v = -t3v+1

(8.6:2)

Furthermore, with to = 0 we get 00

0 = 4_v (t3,+1 + t3v+2 + t3(v+l)) + a3v+1

E

tn,

n=3(v+l)+1

and thus we obtain, analogously to the proof of 8.4.7(c), the identity t3v = 0 and therefore

t3v+1 = 0

[cf. (8.6:2)].

Thus, t = 0. (b) The matrix H defined in 8.4.9 is not of type M since

t:= (1,-1,12,- 2,4,-4,g,-&,...) Et and tH=O. (c) Let (ak) E e be given. The matrix A = (ank) defined in 8.4.8 is given by ank := ak if k < n E N°, and ank = 0 otherwise. If ak 96 0 for each k E No and if t = (tk) E 2 with to = 0 is given, then

0=av E

(vEN°)

n=v

and therefore 00

00

0= E to - E to = tv n=v

(v E N0);

n=v+1

thus t = 0. Conversely, if there exists a k E No with ak = 0, then we define t = (tv) E t by t := e° if k = 0 and, if k>1, by

Consistency and perfectness

ifv=k-1 -1 ifv=k

447

1

(VEN°)

otherwise

10

and we get t E e, t # 0 and to = 0, that is A is not of type M. In the following theorem we establish the connection between `type M' and `P-perfectness' which we sketched earlier in the remarks preliminary to 8.6.5.

Theorem 8.6.6 (type M and P-perfectness). Let A be a matrix with tP C CA, and let A.: CA -+ c be bijective. Then the following statements hold:

(a) If A is P-perfect, then A is of type M. (Note, A is P-perfect if it is perfect.)

(b) If, in addition, A is fit-unique, then A is P-perfect if and only if A is of type M.

Proof. (a) Let A be P-perfect, that is PA = CA, and let t = (tk) E e with to = 0 be given. Obviously (tA)x exists for each x E CA; thus t E TA (cf. 8.2.2). By the definition of PA we get from this t(Ax) = (tA)x = 0

(x E PA = CA).

Since A : CA --+ c is assumed to be bijective, there exists for each k E NO a Y E CA with Ay = ek, which implies tk = t(Ay) = 0, and thus t = 0. (b) On account of part (a) it is sufficient to verify that the property `type M' implies P-perfectness. Let A be a p-unique matrix of type M. To prove P-perfectness, that is PA = CA, we take a t E TA and verify that the test function ft defined by

ft(x) := t(Ax) - (tA)x

(x E CA)

(8.6:3)

(cf. 8.2.15(c)) is zero on CA. Since A : CA -+ c is bijective, we may choose

by 8.1.8(b) and the additional statement (ii) a ft E K and a T= (tn) E e such that ft(x) = p limA x + t(Ax) (x E CA). We get necessarily p = 0 on account of (8.6:3) and of the p-uniqueness of A. Therefore we obtain t'A = 0 from (cf. 8.2.15(a)) 0 = ft(ek)

thank = [tA]k

(k E N°).

n

Thus T= 0, because A is of type M. Hence, A is P-perfect. In the case of coregular matrices the statement in 8.6.6(b) involves the following statement about perfectness and therefore about consistency.

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Matrix methods: structure of the domains

Corollary 8.6.7 (perfectness and type M).

If A is coregular and A : CA -3 c is bijective, then A is perfect if and only if A is of type M. Proof. The statement is an immediate consequence of Theorem 8.6.6(b) since coregular matrices are p-unique (cf. 8.2.5(d) and 8.3.6(a)) and since in the case of these matrices perfectness and P-perfectness are equivalent

0

(cf. 8.6.2(b)).

It is unknown whether in 8.6.6(b) the condition `A is 11-unique' is necessary. Moreover, let us remark that non-p-unique matrices are replaceable

(cf. 8.3.6(c)) and that in the case of conservative matrices A such that cp C COA and A : CA -+ c is bijective, we have by 8.6.8 that A is of type M if and only if codimCA 'P = 1. However, it is unknown whether non-µ-uniqueness implies P-perfectness.

Exercise 8.6.8. Let A be a multiplicative matrix and A : CA --* c be bijective. Prove that A is of type M if and only if codimCA = 1. Bibliography: [2541; [2671

8.7 Replaceability and invariance In Section 8.3 we characterized the replaceability of matrices A under the assumption FA 54 WA. In Section 8.7 we will provide the basis for a full characterization of replaceability. That consists essentially of two theorems which also imply the `invariance of p-uniqueness' and the `invariance of

PA'. Furthermore, the results of this section make clear the connection between replaceability, p-uniqueness (cf. 8.7.9) and PA. The following theorem is fundamental for the current section, but it is also of independent interest.

Theorem 8.7.1. Let A be a p-unique matrix with W C CA and B be a matrix with CB D CA. Then there exists a quotient representation

B=CA+D where C and D are matrices with 11Ch1 < oo

and t(Dx) _ (tD)x for all t E f and x E WA.

(8.7:1)

Moreover, (CA)x exists and C(Ax) = (CA)x holds for each x E CA. Proof. We consider the continuous linear functionals

fn : CA -+ K, x = (xk) -* 1: bnkxk (n E NO) k

defined on the FK-space CA by the matrix B = (bnk). Since CB D CA, the sequence (fn) is pointwise convergent, thus pointwise bounded, and

Replaceability and invariance

449

therefore equicontinuous (cf. 6.7.15). Now, if pn (n E N°) are the seminorms defined in 8.1.3, then, applying 6.5.16, we may choose (independently

of n E N°) numbers K > 0, L > 0 and r E NO such that pi(x)

fn(x)l < LllAxIIoo + K

(x E CA, n E N°).

Further, by the proof of 7.5.9(a) we may choose hn E c' and gn E wA with

fn (x) = hn(Ax) +gn(x), 1hn(y)l

LIIylloc

(y E c),

(8.7:2)

r Ign(x)I < K E p.(x)

(8.7:3)

(x E WA)-

i=1

Since WA is an FK-AK-space, we obtain

gn(x) _ E xkgn(ek) = [Dx]n

(x = (xk) E WA)

k

where D = (dnk) is defined by dnk := gn(ek) (n, k E N°). Further, for n E NO and hn we may choose a tun E K and t(n) = (tkn)) E

hn(y) = tln lim y +

tk I/k

with

(y = (yk) E e).

k Therefore,

hn(Ax) = An lifA x + t(n) (Ax)

(x E CA).

Since fn is representable on the one hand by fn(X) = An limA x + t(n) (Ax) + (gn(ek)) k x

(x E CA)

and on the other hand by fn(x) = E bnkxk = (bnk)kx

(x E CA),

k

we get Mn = 0 by the tu-uniqueness of A. This gives

Bx = (fn(x)) = C(Ax) + Dx

(x E CA)

where we put C = (Cnk) := (t(kn)). By the substitution x := ek (k E NO ) we get the desired identity

B=CA+D.

450

Matrix methods: structure of the domains

Now, we show that C and D satisfy (8.7:1). We have IICII < co since

(nEN°)

IIt1n'111=IIhnII
holds on account of (8.7:2) and p = 0. If t = (tn) E 1, then r I:Itn[Dh]nl < K E Pi(X) IItIIi < oo

(x E WA)

i=1

n

on account of (8.7:3), and a continuous linear functional ft on the FK-space WA is defined by

ft(x) := t(Dx)

(x E WA)

(cf. 6.5.3). Thus, since WA is an FK-AK-space, we get (cf. also 8.1.7(b))

xkft(ek) = (tD)x

t(Dx) = ft(x) =

(x = (xk) E WA)

k

which proves (8.7:1) for the chosen matrices C and D. It remains to verify the additional statements. We have already proved

B= CA + D and Bx = C(Ax) + Dx

(x = (Xk) E CA).

(8.7:4)

Thus it follows for n E No and x = (xk) E CA that cnva.k + dnk)

xk

E bnkxk

(8.7:5)

k

E xk E cnudvk + k

V

dnkxk

[Bx and Dx exist]

k

and therefore (including the existence of (CA)x) the desired identity

C(Ax) = Bx - Dx = (CA)x

(x E CA)

holds because of (8.7:4) and (8.7:5).

Our next objective is to see under what conditions the A-uniqueness

of A implies that of B where CA C cB. We start by trying to find a connection between the value of p in a representation of f E cB relative to cB and the value of p in a representation of f I1A relative to CA. Of course, f I1A E cA. We first introduce some notation.

Notation and Remarks 8.7.2. Let A and B be matrices with CB J CA, and let f E cB. Since FK-topologies are monotone we have f I CA E cA

and, in particular,

limB I

CA

E c'

Moreover, applying 8.3.5 to B and f, we get a p E K, t E £ and an a E ca such that

Replaceability and invariance

f (x) = p lima x + t(Bx) + ax

(x E CB).

451

(8.7:6)

Because lima IAA E cA there exist a PA(B) E K, Y E f and a -y E ca with IimB X = PA(B) limA x + y(Ax) + yx

(x E CA).

(8.7:7)

Applying 8.3.5 also to A and f 11A we obtain a 16(f leA) E K, t E e and a & E CA with

f(x) = la(ffeA) hmAx+t(Ax) +ax

(x E CA).

(8.7:8)

The values µA (B) and µMeA) are uniquely determined if A is p-unique.

If A is not p-unique, then we put AA(B) := 0 =: µ(f ICA) which is in accordance with the above discussion, since in such a case for each g E ca there exists a representation with p = 0. A

Thus, our immediate objective is to relate the values p, PA(B) and AM IA) in 8.7.2 with each other. For the purpose of motivation and a better understanding of the notation we have just introduced we make some remarks.

Remarks 8.7.3. (a) If A := I in 8.7.2, then limB X = X(B) lim x + 1: bkxk

(x = (xk) E C),

k

thus pi(B) = x(B), and for f E cB the statement p(fIC,) = p(fIC) = ppi(B) = uX(B), holds as one can easily check using (8.7:6)-(8.7:8). (b) If A is conservative and CB J CA, then X(B) = PA(B) X(A). (c) For each matrix A we have (according to 8.7.2): A is p-unique if and only if AA (A) $ 0 for each representation of limA according to (8.7:7) or equivalently A is not p-unique if and only if MA (A) = 0 for at least one representation of limA according to (8.7:7).

Proof. (b) We assume CB D CAD c and verify X(B) = AA(B) X(A). For this we consider the functional f := limB IAA E cA which has, by (8.7:7), a representation AX) = AA(B) limA x + y(Ax) + yx

(x E CA)

with certain AA (B) E K, y E t, y E CA O. Thus, by 8.2.6(b),

x(B) = limB e - 1: lime ek = X(f) = pA(B) X(A). k

(c) If A is p-unique, then the value pA(A) is determined in all representations of limA according to (8.7:7), therefore pA(A) = 1 (since

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Matrix methods: structure of the domains

limA x = 1 limA x + O(Ax) + Ox (x E CA)); in particular, µA (A) 54 0.

On the other hand, if A is not p-unique, then the null functional has a representation according to (8.7:7) with it i4 0 (cf. the proof of 8.3.6(b));

thus there exist p E K\ {0}, t E I and a E cA such that 0 = p limA x + t(Ax) + ax

(x E CA).

limAx = -µ1 t(Ax) - 1ax

(x E CA),

Thus,

so limA has a representation according to (8.7:7) with PA(A) = 0. The statements in 8.7.3 suggest the following theorem due to Beekmann [14] (1976).

Theorem 8.7.4. Let cB 3 CA 3 gyp, let f E cs, and let p, p(f I cA) and 1AA(B) be chosen in accordance with 8.7.2. Then u(f I CA) = p PA (B).

Proof. There is nothing to prove if A is not p-unique since in such a case AMA) = 0 = pA(B) according to our agreement.

If A is p-unique, then, on account of cB 3 CA and 8.7.1, we may represent B by

B=CA+D

where the matrices C and D satisfy the conditions listed in 8.7.1. Then we obtain for each x E CA, by (8.7:6) and (8.7:7), the statement f (x) = A AA (B) limA x + p y(Ax) + (py + a)x + t(Bx)

+ t(Bx) = ... + t((CA + D)x) [cf. 8.7.1] + t(C(Ax) + Dx) [cf. 8.7.1] + t(C(Ax)) + (tD)x [11C11 < oo, 8.7.1] + (tC)(Ax) + (tD)x p PA (B) limA x + (py + tC)(Ax) + (p-y + a + tD)x.

Since µy, a, tD E cA and µy, tC E e the functional PIA may be represented in accordance with (8.7:8) with p(f kCA) = p (CA(B) ,

t :_ (µy + tC) and a :_ py + a + tD.

This proves the theorem.

By means of Theorem 8.7.4 we are now able to prove the invariance of p-uniqueness and of PA and to characterize replaceability. For that we first make the notion of invariance precise.

Definition and Remarks 8.7.5. Let A be an arbitrary matrix, and let H = H(A) be a statement about A or a set which is fully determined by A. Then H is called invariant (relative to A), if for each matrix B with

Repiaceability and invariance

453

CB = CA the statements H(A) and H(B) are equivalent or, if H is a set, H(A) = H(B). (a) CA and its FK-topology are trivially invariant. So also are SA, WA, FA and BA. Thus LA (since LA = BA) is invariant. In each case the set in question is determined by the FK-topology of CA which is uniquely determined.

(b) IA and AA are not invariant. (c) The property coregular (thus conull) is invariant since e f WA is invariant (cf. 8.2.7(c)). However, both, regularity and t-multiplicativity are

not invariant as A := I and B := 21 prove.

IL

Proof. It remains to verify part (b). For IA see 8.4.6(f). Let us assume that AA is invariant. Then AA -L= fcD=CA AD and, by 8.2.5(b) and 8.2.17,

WA

FA n AA = I t n AA = n IQ n n ACD=CA

CD=CA

n (IQ n AD) = n AD = AA . CD=CA

CD=CA

This is a contradiction to WA C COA = AA in 8.4.6(c), where the last identity follows since ak = 0 (k E N°). Thus AA is not invariant. More surprising than the invariance statement in 8.7.5 is the invariance of p-uniqueness which is also due to Beekmann (14].

Theorem 8.7.6 (p-uniqueness). The p-uniqueness of matrices A with w C CA is invariant.

Proof. Let A be p-unique and let B with CB = CA be given. The peuniqueness of B follows obviously by 8.7.4, if PA(B) # 0. Applying 8.7.4 to f := limA E cB, we obtain ff

PA(A) = p(hmA !CA) = APA(B),

and then the p-uniqueness of A gives pA(A) $ 0 (cf. 8.7.3(c)). Thus, liA(B) 0 0. In the next theorem we show the monotonicity and invariance of PA which was proved by Boos in a joint paper (cf. (15]) with Beekmann and Zeller.

Theorem 8.7.7 (invariance of PA ). If cP C CA C cB, then PA C P8. In particular, P := PA is invariant for each matrix A with W C CA. Proof. By 8.2.15(d) we have

PA = n {Kern f I f is a test function of A} .

(8.7:9)

Now, if V C CA C cB, then, by (8.7:9), PA C P8 is proved, if we show that

each test function of B is also a test function of A. However, this is an obvious consequence of 8.7.4 and the definition of a test function.

454

Matrix methods: structure of the domains

Remark 8.7.8. It results from the proof of 8.7.7 that for each matrix A with cp C CA the set of test functions of A is invariant. This fact is surprising since this set is determined by TA (cf. 8.2.15(c)) which is not invariant as MacPhail and Wilansky [158] proved. In closing this section we give a characterization of replaceability due to Beekmann [14].

Theorem 8.7.9 (replaceability). Let A be a matrix with cp C CA. If A is not p-unique, then A is replaceable. On the other hand, if A is punique, then A is replaceable if and only if there exists an f E ca with cp C Kern f and p 54 0 in at least one representation of f according to 8.3.5.

Proof. The first part was shown in 8.3.6(c). To prove the second, we assume A to be p-unique. Then, the sufficiency follows from 8.2.11. Now, if A is replaceable, we can choose a matrix D with CD = CA and dk

limD ek = 0 (k E N°). We put f := limb E cD = cA and get IA(f) =

ii(fLA) = PpA(D) = uA(D)

from 8.7.4. As we stated in the proof of Theorem 8.7.6, PA(D) 54 0, so

u(f) # 0, since A is p-unique. Because cp C Kern f, the necessity is proved too.

0

Exercise 8.7.10. Examine the examples in 8.4 for p-uniqueness.

Exercise 8.7.11. Let A be a matrix with V C CA and /a := (f E CA' I f has a representation according to 8.3.5 with p ¢ 0 }

.

Prove the following statements:

(a) µA is invariant. (b) V f E cA : (3B with CB = CA and limB = f e=* f E p. ) . Exercise 8.7.12. Show by means of Exercise 8.7.11 that the statement in Exercise 8.3.8 remains true if we replace B and D by A. Bibliography: [254]; [14], [15], [17], [18], [20]

8.8 Notes on Chapter 8 The structure of matrix domains CA and properties as replaceability and p-uniqueness are well investigated in the case of almost coregular matrices

A. This is no longer true in the case of very conull matrices. For example, on the one hand Theorem 8.7.9 gives us a full characterization of the replaceability of a p-unique matrix and, on the other hand, in general, it is not easy to find out whether a special very conull matrix is (or is not) p-unique.

Notes on Chapter 8 455

As we have stated in 8.1.8, each continuous linear functional f on the FK-space CA, that is each f E cA, has a representation f (X) = A f limA x + t(Ax) + ax

(x = (xk) E CA)

E t and a E cA . Further, if p f is uniquely where p f E K, t = determined for at least one f E cA, then for all f E cA; therefore A is p-unique. Thus for each matrix A the map

pA:cA -aK, f -{pf ifAisp-unique otherwise 0

is well-defined. In [253] Wilansky called a matrix A p-continuous if PA : (CA, J3(cA, CA)) -+ K is continuous. (More generally he introduced for

FK-spaces the notion of a p-space which in the case of domains CA is equivalent to the p-continuity.) He conjectured that all matrices are pcontinuous. For example, Boos in [35] and J. C. Magee and W. H. Ruckle in [163] contributed a positive answer to Wilansky's conjecture. Finally, K: G. Grof3e-Erdmann proved Wilansky's conjecture in his paper [98] in which he obtained important information about the structure of EA when E is a distinguished FK-space.

Part III Combining classical and functional analytic methods Part III consists of three chapters with very different topics. However, they have in common that the proof of almost all results requires the combination of classical and modern methods. Following the idea of the bounded consistency theorem, we try in Chapter 9 to answer the question whether two matrix methods, which are consistent on a (suitable) set, are consistent on a given superset Y of it (intersected with the domains under consideration). Since in general the answer is negative, on the one hand we give conditions on Y which guarantee a

positive answer and have their origin in the gliding hump arguments of classical proofs of the bounded consistency theorem. This leads to gliding hump properties. On the other hand, we show that the answer is positive if Y is the set of all p-bounded sequences. The key to this result is the possibility of the comparison of the µ-bounded domains of regular matrix methods (µ-comparison) by quotient representations. Closely related to the ti-comparison is the answer to the question whether, for given reg-

ular b-consistent matrices A and B, there exists a regular matrix which is b-stronger than and b-consistent with both. The negative case leads to different types of singularities. In Chapter 10 we develop a further functional analytic tool, namely two-norm convergence and, more generally, Saks spaces, which is mainly designed for the study of the topological structure of bounded domains of conservative matrix methods. For instance, this tool is suitable for proving characterizations of the comparison of bounded domains in the case of regular matrices via continuity estimations, comparison of the corresponding FK-topologies, and via quotient representations. As a corollary we again get the bounded consistency theorem. Whereas we studied in the last chapters subjects in summability which are influenced by functional analysis, for instance by FK-space theory, we deal in Chapter 11 with subjects in topological sequence spaces, for example with the sequential weak completeness of the #-dual, which can be characterized by summability statements. We close the book with a study of the role of sequences of zeros and ones in a given sequence space.

9

Consistency

of matrix methods In this chapter we continue the investigation of the following question. Given matrices A and B and a subspace (P C Y C CA with Y c cB, if A and B are consistent on the `basic subset' (p ED (u), for a suitable u, are A and B consistent on Y? In Section 9.1 we deal with the above question in a very general context where the basic idea comes from Corollary 2.6.11 which we draw from a theorem of Mazur-Orlicz type (cf. 2.6.8). For any matrix A with P C CA we find a large class of sequence spaces X containing cp such that a theorem

of Mazur-Orlicz type holds for Y := X n WA. Then we can show that under the hypothesis V c X n FA C CB the consistency of matrices A and B on p ® (u) with a certain u E FA implies the consistency on X n FA. We use similar methods to those in Section 2.6, but we combine the (classical) analytic methods from that section with functional analytic methods developed in Chapter 8. In the remainder of this chapter 'p-bounded domains' of matrix methods

play an essential role. For that we consider-by way of introduction-in Section 9.2 a-bounded sequences and domains. Among other results, we show that the set mµ of all p-bounded sequences is a BK-AK-space and we extend the Schur theorem by equivalent statements which are connected

with m,. In general, it is non-trivial to characterize for a given sequence space X and an arbitrary (regular) matrix A those matrices B which are stronger than A relative to X. In Section 9.3 we characterize in the case of regular matrices A and B the compatibility of the p-bounded domains, that is m, n CA C CB for at least one sequence a. We do it by quotient representations of the form B = CA + D where the `quotient' C is regular and the `remainder' is small in some sense, and

by continuity statements about the maps B : p -+ c and B : m -p m, respectively. This characterization of the p-compatibility includes results concerning consistency. Thus, in the case of p-bounded domains of regular matrices, we give a complete solution of the problem formulated at the beginning of

460

Consistency of matrix methods

this introduction. In the. next chapter (Sections 10.3 and 10.4) we'll give a corresponding solution in the case of bounded domains of regular matrices. However, for that we have to develop further functional analytic tools.

In Section 9.4 we deal with a further consistency problem which is related to p-bounded domains: for arbitrarily given regular matrices

AM,-, A(N) the question arises under which assumptions there exists a regular matrix B which is 6-stronger than each of the matrices AM,-, A(N) (or, for a certain p, is stronger than AM,..., A(N) relative to a). In general-as simple examples prove-the answer to this problem is negative and leads immediately to `singularities of matrices'. We restrict ourselves here to motivation and basic results.

9.1

Consistency and theorems of Mazur-Orlicz type

We have so far established two consistency theorems. On the one hand there is the consistency theorem on PA (cf. 8.6.1), which is a simple consequence of the topological structure of PA (in the domain CA) and the monotonic-

ity of FK-topologies; on the other hand there is the bounded consistency theorem (cf. 2.6.11), whose proof required some classical methods. Now, we may obviously ask whether we can prove the bounded consistency theorem

by applying simple methods as in the first case; that is, if a matrix A is given, does there exists a topology on the bounded domain m fl CA such that (analogously to the case PA ) m f1 cA = ip ®(u)

with u=0 or

u E (m f1 cA) \P

holds and for each matrix B, which is consistent with A on cp ® (u), the functional U MB ImncA is continuous on m fl CA? (Then the consistency of

A and B on m fl cA would follow similarly from that as in 8.6.1.) That question has been answered positively by W. Orlicz [1911 (1957) and by G.

Bennett and N. J. Kalton [261 (1972) as well as by others. This leads to the notions of Saks space and of mixed topology, respectively. We will deal with this question in Chapter 10. In aiming to prove more general consistency theorems than the bounded consistency theorem, we will use, in this section, similar classical methods to those in the proof of the bounded consistency theorem (cf. 2.6.11). However, we combine them with functional analytic methods. Roughly speaking, we will do the `coarse' work with modern (functional analytic) tools and then

the `fine' work with classical (analytical) tools. We deduced the bounded consistency theorem (cf. 2.6.11) from Theorem 2.6.8. Now we give, on the basis of the distinguished subsets (cf. Section 8.2), a version of Theorem 2.6.8 which is suitable for generalization. For the sake of simplicity we consider conservative matrices (instead of matrices conservative for null sequences). This is not a restriction, since we will essentially weaken the assumption subsequently .

Consistency and theorems of Mazur-Orlicz type

461

Theorem 9.1.1 (cf. 2.6.8). If A and B are conservative matrices, then m n WA C CB implies m n WA c AB . Proof. Let m n WA C cB. Since m n cA C LA and WA = LA n AA (cf. 8.2.7(a) and 8.2.5(b)) we get m n AA C m n WA C cB, and thus

0

m n AA C AB by 2.6.8. Hence m n WA C AB since WA C AA L.

Now, the question arises whether Theorem 9.1.1 remains true if we replace m by other sequence spaces X. That is, does

XnWACCBPXnWACAB

(9.1:1)

hold for certain sequence spaces X and matrices A and B? That the implication (9.1:1) does not hold in general can be shown by the example where

X := c, A = 0 and B = I. Now, in aiming to find suitable conditions for X, one considers the factor sequences which are constructed in the proof of 2.6.8, ultimately the sequence (f j) in 2.5.6, and tries to obtain a 'general description' of these factor sequences. They are sequences which have infinitely many 0-blocks, and behind every 0-block they increase in steps until one, are equal to one in a further block and finally they decrease in steps to zero. We describe such sequences by the use of subsequences of step 1-block sequences which we now introduce. Later in this connection we introduce `oscillation properties' of sequence spaces.

Definition 9.1.2 (step 1-block sequence). A sequence (0)) in w with y(j) _ (y(?)} is called a block sequence, if there exists an index sequence (-yj) with -yo( = 0 and yk?) = 0 for all k < ryj and all k > ryj+l (j E N°). Therefore, yj) can be represented by Y

_

Yj+1-1

ykj)ek, k=Yj

and the coordinatewise sum y = (yk) := Ej y(2) (that is, the limit of that series in (w, r,,,)) is given by

yk=yk?)if ryj
(a) io = 0 and i3j+1 < i3j+2 (j E N°), (j)

3) yk = ( 7)

(yk

0,

if k < k=3j or k > k=3(j+1)

ifk;3j}1 < k < k=3j+2

is monotone for kz 3j

k < kj3U+1) (j E N°),

(j ENO)

and

(a)

yo

> 01,

1 < k < ki3j+1 and also for ks 3j+2

,

-

1<

462

Consistency of matrix methods

(6) y = (yk) := E? yi?) (coordinatewise sum) is constant for k2 < k < ki+1 (i E N°).

Moreover, (y(i)) is called a step 1-block sequence if there exists a (k;) with k° = 0 such that (y(i)) is a 1-block sequence with respect to (k2). We illustrate this definition in Figure 9.1.1. Ilk

1

L J /-F

ki0_O

I

F-

k,( k,2

!

I

k,3

k,3,

I ``I

kj3.+1

k,1i+2

k13(,41) ki(3.+(}+( kt3( -31+2 IC

Fig. 9.1.1: Illustration of a step 1-block sequence

For this section we use the notation S :_ {a E ]K I Jat = 1} for the boundary of the unit interval and disc in R and C, respectively. Definition 9.1.3 (SIGNED P_OSCP). Let Y be a sequence space with

(p C Y. Then, by definition, Y has the signed pointwise oscillating property (SIGNED P_osc ), if for each x E Y and every index sequence (k;) with k° = 0 there exists a step 1-block sequence (y( j)) with respect to (k=) such that for each subsequence of (y(3)) there exist a subsequence (y(.i-)) and a sequence in S with yx E Y, where y h,,y{i.) (coordinatewise sum). We give no examples at this point since we obtain many examples below using 9.1.5, 9.1.6 and 9.1.16. We state here the following theorem. So as not to interrupt the development of ideas, we postpone the proof until near the end of this section. This theorem answers quite generally the question posed at the beginning of this section and includes Theorem 9.1.1 as a special case.

Theorem 9.1.4 (of Mazur-Orlicz type). If Y is a sequence space which contains W and has the SIGNED P_OSCP, then Y C CB implies Y C AB for each matrix B. In Theorem 11.1.2 we will discover that the statement of this theorem remains true if we replace AB by the smaller set WB. We begin by showing that Y := X fl WB has the SIGNED P_OSCP, if E is an FK-space containing cp and X is a sequence space with a gliding hump property (defined below) closely related to the SIGNED P_OSCP. In

particular, if A is a matrix with W C CA, then Y := X fl WA has the SIGNED P_OSCP.

Consistency and theorems of Mazur-Orlicz type

463

Definition and Examples 9.1.5 (SIGNED P_GHP). Let X beasequence space containing W. Then, by definition, X has the signed pointwise gliding hump property (SIGNED P_GHP), if for each x E X and any block sequence (yii)) in X with sup? IIIJW11bv < oo and for each subsequence in S with of (y(i)) there exist a subsequence (y(iv)) and a sequence (coordinatewise sum). yx E X, where y := EY Every solid sequence space containing cp has the SIGNED P_GHP. (For a proof note that y E m holds independently of the choice of the subsequence in the definition of SIGNED P_GHP. Thus yx E X if X (y(i.)) and of is solid.)

In particular, cp, w, co, m, tP (0 < p < oo), d, d,. (r > 0) and d have the SIGNED P_GHP (cf. also 7.1.11).

Further examples are contained in Exercises 9.1.18 and 9.1.19: bs and fo have the SIGNED P_GHP, but neither is solid as we know from 7.1.12 and 7.1.11(b).

a

Remark 9.1.6. If a sequence space containing cp has the SIGNED P_GHP,

A

then it has the SIGNED P_OSCP too.

Proof. Let X be a sequence space containing W and which has the SIGNED P_GHP. To prove that X also has the SIGNED P_OSCP, we consider a given

x E X and an index sequence (ki) with ko = 0, and define a 1-block sequence y(i) = (y(j)) by iii yk

(0, ifkkj+i 1,

if ki < k < ki+1

(j, k E NO ).

We have y(i) E X since w C X and trivially ,(yli> Ilbv < 2 (j E No ). Moreover, (y(i)) is a step 1-block sequence with respect to (ki), as setting (i1) := (0, 0, 1, 1, 1, 2,2,2,3,3,3 ....) -that is, iii := i3i+1 := j and i3i+2 := j+1 (j E N°)-proves. Because X enjoys the SIGNED P_GHP, we see that for each subsequence of (y(3)) there exist a subsequence (y(i-)) in S with yx E X for y := F_ Thus X has and a sequence the SIGNED P_OSCP.

0

Theorem 9.1.7 (of Mazur-Orlicz type). Let X and E be given sequence spaces with v c X n E, and let E be an FK-space. If X has the SIGNED P_GHP, then Y := X n WE has the SIGNED P_OSCP.

In particular, as a corollary of 9.1.4, we get (under the assumptions made here) the following theorem of Mazur-Orlicz type: X n WE C CB implies X n WE C AB for each matrix B. So as not to loose the thread of ideas we delay this proof to the end of the section. Our object now is to deduce, on the basis of 9.1.7, consistency theo-

rems for matrices A and B with cp C X n FA C CB, where X has the

464

Consistency of matrix methods

SIGNED P_GHP. First we deduce a limit formula from 9.1.7 analogously to the procedure used in 2.6.

Corollary 9.1.8 (limit formula). Let X be a sequence space containing cp and having the SIGNED P_GHP, and let A and B be matrices with

XnFA=(XnWA)®(u) with u=0 or uEXf(FA\WA) (9.1:2) and, if u E IB is satisfied,

lima x = a

(limAx_akxk) + > bkxk k

where

(x=(xk)EXnFA)

k

0,

ifu=0

nAU ,

ifuEXn(FA\WA).

Proof. The first statement is an immediate consequence of FA = WA(D(u), where u = 0 or u E FA \ WA = FA \ AA is suitably chosen (cf. 8.2.5(c)). The proof of the limit formula is analogous to that in 2.6.10 and is left to 0 the reader. Now, we are able to deduce from the limit formula in 9.1.8 a very general consistency theorem.

Corollary 9.1.9 (consistency theorem). Let X be a sequence space containing tp and having the SIGNED P_GHP, let A and B be matrices with o C X n FA C cB, and let u be chosen in accordance with (9.1:2). Then the consistency of A and B on cp ED (u) implies their consistency on

XnFA. Proof. Proceed analogously as in Corollary 2.6.11. We should note that

here u E IB is true on account of u E X n FA C X n IA and of the consistency of A and B on gyp, and that in the case u = 0 the inclusion

XnFA=XnWAcXnAA holds.

0

In the following remarks we give some examples of sequence spaces to which Theorem 9.1.7 (together with its corollaries) is applicable, and note that the consistency theorems, proved in 2.6, are contained in 9.1.9.

Examples and Remarks 9.1.10. (a) If X is any of the sequence spaces

Q' (0 < p < oo), co, f°, m, d, d,., II, (r > 0), b, bs or w, then X has the SIGNED P_GHP (cf. 9.1.5), that is 9.1.7 and 9.1.8 as well as 9.1.9 are applicable to it.

(b) Consider X := m and matrices A and B which are conservative for null sequences; then X n FA = m n FA = m n CA because m n CA C FA

Consistency and theorems of Mazur-Orlicz type

465

(cf. 8.2.7(a)). In that case the theorem of Mazur-Orlicz type, formulated in 9.1.7, corresponds to Theorem 2.6.8 (without limit formula), 9.1.8 corresponds to Theorem 2.6.10 and Corollary 9.1.9 to Corollary 2.6.11. Obviously, the bounded consistency theorem is also included in 9.1.9. (c) The consistency theorem for the case of absolutely A-bounded domains of regular matrices A (cf. Exercise 2.6.16) may be deduced also from 9.1.9, which is the task of Exercise 9.1.20. A We will deal with a further example of an application of 9.1.7, 9.1.8 and 9.1.9 in Section 9.3. Namely, we will consider, among other things, consistency theorems in theease of `p-bounded domains'. Thus, we may state that we have obtained a very general answer to our initial question. Now we will give the outstanding proofs of Theorems 9.1.4 and 9.1.7, which are quite technical and extensive. In the first step we prove Theorem 9.1.4 by deriving a `non-summability theorem' for matrix domains.

Theorem 9.1.11 (non-summability). Let B = (bnk) be a matrix with cp C CB and column limits bk (k E N°), and let x = (xk) E CB be a given sequence which satisfies at least one of the following conditions: (i) There exists an index sequence

such that

ll

F, bkxk

k=0

E c and

111

that lim E bkxk 0 limB x. k=0

(ii) sup E bkxk I

V

00.

k=0

Then there exists an index sequence (k=) with k° = 0 such that for every step 1-block sequence with respect to (k;) there exists a subsequence (y(J))

such that for each subsequence (yti°>) of (yit>) the statement yx f cB h,,y(?-) (coordinatewise sum) and holds where y is any sequence in S. Before we prove this theorem, we use it to deduce Theorem 9.1.4.

Proof of 9.1.4. Let Y be a sequence space containing V and having the SIGNED P_oSCP, and let B be a matrix with Y C CB. To prove Y C Ag we assume that there exists an x E Y C cB with x I AB A. Then we have x E IB \ AB or x f IB. In the first case condition (i) of 9.1.11 occurs, whereas in the second case (1) and (ii) of 9.1.11 take place, if the sequence

of the partial sums of Ek bkxk is bounded or unbounded, respectively. Therefore, Theorem 9.1.11 is applicable in any case. Thus we may choose an index sequence (k;) with k° = 0 in accordance with 9.1.11 and, since Y has the SIGNED P_osCP, a step 1-block sequence in accordance with the definition of the SIGNED P-osCP. For the latter we choose a subsequence in accordance with 9.1.11 and finally-according to the definition of the in S with SIGNED P_oSCP-a subsequence (y(3 )) and a sequence

466

Consistency of matrix methods

yx E Y where y :_ E

Now, by 9.1.11, we get yx 0 cB which contradicts the assumption Y C cB. Proof of 9.1.11. We make some preliminary remarks. In both cases (i) and (ii), we will choose an index sequence (ki) with ko = 0 depending on x E cB and on statements deduced from V C CB. For all index sequences (ki) with ko = 0 and (ni) and every sequence z = (zk) E wB we will use the notation

57,bnikZk = Ai + A, + A + Ci

(i E N`°),

k

where k;-1

Ai =

k;-1 (bn:k - bk)xk

,

A:

k=0

k=0

k;+1 _1

Bi

57, bk zk 00

bn,kzk

and

Ci :_ E bn,kxk.

k=k;

k=k;+1

In both cases (i) and (ii) we will construct index sequences (ki) with ki = 0

and (ni) such that (Ai)Ec0 and (C,)Eco with z := yx where y will be chosen as described in the theorem. From this we get that each of the following statements implies z CB (a) (Az) E c and (Bi) c. (b) (A,*,.) V c and (Ba,) E co, where (p?) is suitably chosen.

Now, let x = (xk) E cB, and let

with i o = 0 be any index

sequence. (Later we will fix on the basis of the conditions (i) or (ii).) On account of V C cB and x = (xk) E cB we may choose index sequences

(ni), (ki) and (vi) as follows. For vo := 0 and ko := 0 we may find an no E No such that ko

L Ibnk - bk I Ixk I < 2-0 I

(n > no).

k=0

Then we choose a v1 > vo such that for k1 := q, the inequality E bnkxk

< 2-1

(n < no, k1 < 1 < L ; 1, L E N°)

k=1

holds. Having chosen ni_1 and vi we fix for ki := 17,,; an ni > ni_1 with k;

L Ibnk - bkI Ixkl < k=0

2_i

(n > ni) ;

(9.1:3)

Consistency and theorems of Mazur-Orlicz type

467

further we choose a vi+1 > vi such that for ki+1 := ijv,+, the inequality L

(n < ni, k=+1 -< 1 < L; 1,L E N°)

E bnkxk < 2-(%+i)

(9.1:4)

k=1

is satisfied. Now, let (y(n)) be a step 1-block sequence with respect to (k1). Then we consider in both cases (i) and (ii) the subsequence (y{20). In particular, if (ij) is chosen in accordance with the definition of a step 1-block sequence, we have (see Figure 9.1.1) y(2v)

_

if k < kis or k > ki6i+3 if k<

0, 1,

{v, k E 1`N°),

which implies for y = (yk) :_ Ev y(2v) (coordinatewise sum) the identity

yk-

1 0, 1

if

k<

(v,kEN°).

ifk.1 6'+1 < . k
Noting in addition that IIyII, < 1 and applying these observations to z = (zk) := E,, hvxy(2& ), where by E S is arbitrarily given, we obtain in both cases that ki

IAiJ <_ L Ibnik - bkI IxkI

t-+ 0

k=0

on account of (9.1:3) and (cf. (9.1:4)) 00

Icil =

k,.+i -1

00

E bnikzk k=ki+t

:5 E

co

bn,kxk

r=i+1

k=k,.

< E 2-r L.:t 0. r=i+1

Obviously, the corresponding statements are also true if we consider z :_

Ej h, xy("J) where (h;) is any sequence in S and (µj) is a subsequence of (2v). Now, we fix (77,,) appropriate to the cases (i) and (ii), respectively. If (i) holds, then we may choose a sequence (riv) with 710 = 0 and '?.-I

a := lim v

bkxk # limB x =: d. k=0

Moreover, we may assume bkxk <

2-v

(v, 1c E N°).

(9.1:5)

k=nom

For this (q,) let (ki), (ni) and (vi) be chosen as described above. Let (y(n)) be any step 1-block sequence with respect to (ki), let (p.) be any

468

Consistency of matrix methods

subsequence of (2v) and let z = (zk) := Ej hjxy(JAIwhere (hj) is any sequence in S. We now prove z 0 cB. For this we may assume z E WB, since otherwise there would be nothing to prove because cB C CB.

Let p, A E W with 0 < A < p be given. This, together with (9.1:5), gives us kP-1

kA-1

bkzk -

bkzk k=0

k=0

ko-1

vo-1

v0-1 '1v+1-1

bkzk < k=ka

2-A 'Z0,

bkxk µ=va

k=n,.

R=va

which implies the existence of limi Ek;01 bkzk and therefore (A;) E C. We now verify (Bi) 0 c. By construction we have

hjxk, if kj3MJ+, < k < ki3,j+2

zk = { 0,

E NO).

if kis(n;+i) < k < kis(v;+i)+i

Thus we obtain Bi3(µi+l) = 0, that is (Bis(n;+,)) E co. Hence (Bi) 0 c is proved, if (Bi3,,.+,) 0 co is satisfied. However, for i := i31t;+1 this is an immediate consequence of ki+1-1

k;+t -1

E bnikxk = hi E bnikxk

Bi

k=ki

k=ki

:

ki-1

hj E bnikxk -

bkxk

k=0

k

ki-1

- hj

00

(bnik - bk) xk - hi

bnikxk k=ki+i

k=0

on account of (9.1:3), (9.1:4) and of ki-1

00

E bnikxk k=o

- F bkxk

Z-

d--a00.

k=o

Thus, we have proved (Al) E c and (Bi) 0 c, that is z V CB by (a). In the case of (ii) we may assume without loss of generality that

(ti)

sup R E kx= oo. k= 0

Consistency and theorems of Mazur-Orlicz type

469

(Otherwise we consider -x instead of x if K = IR, and -x, ix or -ix instead of x if K = C.) Therefore, we may choose an index sequence such that '1o = 0 and

(Ti-)

0v -1

1P+

E bkxk

"-+1 _1

(v E N)

IbkxkI k=0

k=0v

is satisfied. This implies

bkxk J

>0

(v, r E N°)

(9.1:6)

(v E N°, r E N).

(9.1:7)

and therefore

>v+

bkxk

k=q

IbkxkI k=0

As in the first case, we choose for this sequence the sequences (k1), (ni) and (vi). Moreover, let (y(n)) be an arbitrary step 1-block sequence

with respect to (k;), let (;c,) be any subsequence of (2v) and let z = (zk) := FJ. h1xy(µj) with arbitrary h,, E S. We now prove*z 0 cB. We may again assume z E WB. For i := pi := i3(,1+1) we obtain, by the definition of z, the statement Bpi = 0, that is (Bpi) E c° i and also (AP,) 0 c because koj -1 IA=1

=

bkxk

k=0

k'3(Yj+1) -1

2:

bkxkyj"

jh11

-

k'3Kj+2-1

bkxk k=k;3µ j +1

bkxk

k=0

k=k;3,.j }3

>R(

k,s, j+1 -1

- 1:

Ibkzk) ? ?

1

oo,

k=0

where we note in the second inequality (9.1:6), (9.1:7) and (5) in Definition 0 9.1.2. From (b) we now obtain z cB, which completes the proof.

Next we give the outstanding proof of 9.1.7. For this we show that WE has a `strong' version of the SIGNED P_oscP if E is an FK-space containing W. We first give two lemmas for which we need the notion of the convex hull of a subset of a linear space.

470

Consistency of matrix methods

Definition and Remark 9.1.12 (convex hull). Let A be a subset of a linear space X over K. Then n

convA:_

n

a=at

l

InEN,aiEA,A1>0,.1Z=1}

z=i

==i

JJJ

is called the convex hull of A. It is an easy exercise to verify that cony A is a convex subset of X and that

conv A = n m with JC := { M C X I ACM and M is convex} . MEK

Thus, conv A is the smallest convex subset of X which contains A. A Lemma 9.1.13. Let (E, r) be an FK-space containing gyp, and let (77v) be an index sequence. For each x E WE we have x E conv {x(17-1 j v E No}, and there exists a sequence (x(r)) of `convex combinations' t,.

(rtr E No with 8r < tT < Sr+t,

X(r) 8r

tr

0 < Pr., < 1, t2rt,. j4 0 and

E prv = 1

v=s,.

of sections of x such that x(' -> x in (E, r). Proof. Let x = (xk) E WE be given, that is x(n1 --> x in (E,o(E,E')). Now, if (ri,,) is any index sequence, then

x('7,1 --* x (v > p, v -+ oo) in (E, o(E, E')), because (x1'7-J),,>,, is a subsequence of (x(n)) for every p E No. Thus x is, for each p E No, in the o(E, E')-closure of conv {x(n-1 j v E No , v > p} . Because closures of convex sets are independent of compatible topologies (cf. 6.6.13), we get x E conv {x(n-1 j v E NO , v > p} for each p E No. In particular, with the case p = 0 the first statement in the lemma is proved.

By induction we now determine a sequence (x(')) of the desired kind. We assume that r is generated by a (countable) family (pn j n E N°) of semi-norms. Further, we use the notation My := conv {x1' "1 I v E No , v > p}

(p E N°).

If r = 0, then we choose x(0) E Mo with po (x - x(°)) < 1. Since x(°) is a convex combination we have

x(0) =

to

Evx[n..] V=80

for suitable so, to E NO with so < to and for suitably chosen Pop with 0 <_ Pov <- 1, pot. 0 0 and Ev=so uo,, = 1. If x(o), ... , x(r-i) with Sk, tk E No,

Consistency and theorems of Mazur-Orlicz type 471

pk, (k E N°_1) according to the second statement are determined in the case of an r E N, then we put sr := tr_1 + 1 and choose an x(r) E M8, with p, (X - x(r)) < r1 (0 < l < r). Since x(r) is a convex combination, we have

X(r) _

pr"x(17-1

v=s,

for suitably chosen tr E NO with tr_1 < Sr < tr and prv with 0 < prv < 1, Art, # 0 and Ev=8r pr,, = 1. The chosen sequence (x(r)) obviously has the desired representation anti satisfies x(r) ---+ x in (E, r) because 1

pt (x - x(r)) < r I

1

for all l E N° and r>1

0

in accordance with the choice of (X(r))

Lemma 9.1.14. Let (E, r) be an FK-space containing gyp, y = (yk) E w, and let (y(j)) be a block sequence E with y = Ej y(j) (coordinatewise sum) and supj Ily(j) IIbV < oo. Then for each x E WE the statement

yx E E and yx = E y(?)x in (E,a(E,E'))

(9.1:8)

j=o

implies yx E WE.

Proof. Let x = (xk) E WE, and let y = (Yk) E w and y(j) be chosen in accordance with the formulation of the lemma such that (9.1:8) is satisfied. To prove yx E WE we consider an arbitrarily given f E E' and verify

f

((yx)("))

---* f (yx)

(v -> 00).

(9.1:9)

By definition, x E WE implies (xk f (ek)) E cs, in particular

sup 37, xu f (e"`) -3 0 (v -+ co). k>v

1

(9.1:10)

µ=k "0

Now let (yj) be chosen in accordance with the definition of a block sequence. If v E NO is given, then we choose a j E NO with ryj < v < ryj+l. This gives

.

7,+t-1

E ykxk f (ek) = f k=O

A=O

and, applying 2.1.14, ykxkf(ek)

k=v

(j-

y(u)x) j-*

f (yx)

(on account of (9.1:8))

472

Consistency of matrix methods

7j+1-1

<

sup xj, k>v {c=k

f( el)

{ E I yk - yk+1 1 + I yv-1 I + IY-ri+1-1 I)

v

0

k=v

on account of (9.1:10) and of supi Ily(i)< oo (which implies (yk) E m). Therefore, -fj+1-1

7j+1-1

V

ykxkf (ek) - E ykxkf (ek) ':±T- f (yx) - 0.

E ykxkf (ek) = k=0

k=v+i

k=0

Hence (9.1:9) holds.

17

Definition and Remark 9.1.15 (ABSOLUTE SP_OSCP). Let X be a se-

quence space containing W. Then, by definition, X has the absolute strong pointwise oscillation property (ABSOLUTE SP_OSCP), if in Definition 9.1.3 the sequence (y(i°)) can be chosen such that for each sub-

sequence (y(".)) of (y(i-)) and every sequence (hg) in S the statement yx E X holds, where y := Em h, y("M) (coordinatewise sum). Obviously, the ABSOLUTE SP_OSCP implies the SIGNED P_OSCP.

z

Theorem 9.1.16 (ABSOLUTE SP_OSCP). If (E, r) is an FK-space with E D gyp, then WE has the ABSOLUTE SP_OSCP.

Proof. Let x E WE and an index sequence (k=) with ko = 0 be given. Moreover, we assume that (pk j k E N°) is a family of semi-norms gener-

ating the FK-topology r of E. We put

j0 ifv=0 kv-1 ifv>0. Applying Lemma 9.1.13 we may choose an (x(r)) in cony {x(n-] I v E No } such that

x(r) -+ x in

(9.1:11)

(E, 7')

and t,.

Sr, tr E N° with Sr < tr < sr+1, V=Sr

(9.1:12)

t,.

/frt. 0 0 and E Arv = 1

0 _< Arv < 1,

v=8,.

are satisfied. By (9.1:11) there exists an index sequence (ri) with pk(x(r)

- x(r+µ)) < 2-j-1

(it E No

, r > ri and k < j).

(9.1:13)

We now define z(°) := x(r0) and z(i) := x(r2j)

- x(r2j_1) (j E N) and z

x(i) (coordinatewise sum).

i

Consistency and theorems of Mazur-Orlicz type 473

On account of-the representation of x(r) in (9.1:12) there are sequences y(i) = (yu)) (j E N°) defined by z(i) = y(i)x. In particular (note the definition of 77v ) 0,

yk?)

if k < k8-2j_1 or k > kt,2J

11, if kt,23_, < k < k8*2j . Obviously, (y(i)) is a step 1-block sequence with respect to (k=). We prove

that for each subsequence (y(iµ)) of (y(i)) and each sequence (h,) in S the sequence (E ° h,,y(iµ))N is a Cauchy sequence in (E, r). Then we get by the completeness of (E, r) and by coordinatewise convergence the identity

yx =

h,xy(iµ) E E with W:= E h,,y('µ) (coordinatewise sum).

Since y and (h,,y(iµ)) satisfy the assumptions in Lemma 9.1.14, yx E WE is verified, that is WE has the ABSOLUTE SP_OSCP.

So, let (y(iµ)) be a subsequence of (y(i)) and (h,,) be a sequence in S. We have to prove that (FNo h y(?µ)x) is a Cauchy sequence in (E, r). N For that let k E No and e > 0 be given. We choose an no E N such that no+v

2_0

57,

< e (v E N°) and k < 2jno - 1.

f=no

For v E No we get no++v

A µ=no E

hrly(iM)xl

J

=A

no+v

(nO+V

h,,z(iµ}l

J

l+=no

<E

Pk(z{iµ})

v=no

no+v

E ryfk(x(r21M 0 -x(r21p)) 1+=no

no+v

<

57, 2-2iµ st=no no+v

S

[(9.1:13) applied to 2j,. - 1 and r2;µ_1 instead of j and r, respectively]

1: 2'1` <e. µ=no

That completes the proof.

Proof of 9.1.7. By the following remark, the statement in 9.1.7 is an immediate consequence of 9.1.16.

474

Consistency of matrix methods

Remark 9.1.17. Let X and Z be sequence spaces with w c Y := X n Z. If X has the SIGNED P_GHP and Z the ABSOLUTE SP-OSCP, then Y has the SIGNED P_OSCP. A

Proof. Let x = (xk) E Y and an index sequence (k1) with k° = 0 be given. Since Z has the ABSOLUTE SP_OSCP, we may choose a step 1-block

sequence (y(i)) with respect to (k1) in accordance with the definition in 9.1.15 and 9.1.3, respectively. Now, let a subsequence of (y(i)) be given. Since Z has the ABSOLUTE sP_oscP there exists a subsequence (y(J )) of it with the properties fixed in 9.1.15. Because the step 1-block sequence (y(i)) is also a block sequence satisfying IIy(j) IIb < 2 (j E N°) and since X has the SIGNED P_GHP, by Definition 9.1.5 there exist for the subsequence

Ws-)) of (y(3)) a subsequence (y("µ)) and a sequence (ha) in S with yx E X where y := Et, hµy("µ) (coordinatewise sum). Since Z has the ABSOLUTE SP_osCP we have yx E Z, that is yx E X n Z. Thus, Y = X n z

0

has the SIGNED P_OSCP.

Finally, let us remark that we proved 9.1.4 exclusively using classical methods (in particular, with gliding hump arguments) and 9.1.13-9.1.16 by applying functional analytic methods. Thus we can say that in the proof of 9.1.7 we have combined classical with functional analytic methods which is a first justification of the topic of Part III. Furthermore, we note that this section is largely contained in the joint paper [40] of Fleming, Leiger and Boos and partially in previous joint papers (for example, in [41] and [42]) of Leiger and Boos. The idea of considering in connection with consistency theorems sequence spaces with suitable oscillation properties is due to Snyder [221]. He dealt with the basic question of this section, but under very restrictive assumptions, with the consequence that his theorems of the type

XnWECWB

XnWECCB

hold only for a relatively small class of spaces X.

Exercise 9.1.18. Prove that bs has the SIGNED P_GHP. Hint: Define inductively for a given x = (xk) E bs and for any block sequence (y(i)) with yU) = (ykJ)} a sequence (h,) in s by h° := Sgn F,, xkyk°) and Sgn

Exkyki)

Sgn E by v=°

k

(? E N), k=,,,

xkyk

where (yy) is chosen as in Definition 9.1.2 and Sgn a :=

sgn a

fI

if a# 0

ifa=0

(aEK}.

p-bounded sequences and domains 475

Exercise 9.1.19. Show that fo has the SIGNED P_GHP by the following steps:

(i) If (yj) is any index sequence, then there exists a subsequence (-(j.) such that n+IsupvENo p

I p
I

for all v, n, p E NO and JMI denotes the number of elements, if M is a finite set. Hint: Choose (b,) such that yja := yo and y,,+, -

v + 1 (v E N°) is

satisfied.

(ii) For every block sequence (yWWW) with sup? II y(') IL < oo there exists

a subsequence (y(?o) of (y(2)) such that yx E fo for each x E bs, where y := E" yip-) (coordinatewise sum). (iii) In the case of the sequence y chosen in (ii) the statement yx E fo holds for every x E fo. Hint: bs is dense in (fo, 11 11.) (for a proof see [28]). (iv) fo has the SIGNED P-GHP.

Exercise 9.1.20. Let A be a matrix with cp C CA and let JAI

{(xk)Ew

I

sup 57 Iankxkl < n

c}

k

be the set of all absolutely A-bounded sequences. (a) Prove the following statements: (1) JAI has the SIGNED P_GHP. (2) JAI n CA C FA.

(b) Let B be another matrix with cB D IAJnCA. Determine a limit formula for limB on JAI fl CA. Hence, deduce a consistency theorem and compare it with that in 2.6.16. Bibliography: [40]; [41], [42], [26], [28], [221], [191]

9.2 it-bounded sequences and domains In the next two sections, as generally in summability theory, 'p-bounded' sequences and domains will play an essential role. As a preliminary, we introduce in this section some basic notation and prove some elementary results about `p-boundedness'.

476

Consistency of matrix methods

Notation 9.2.1. For any (arbitrarily given) real-valued sequence p = (pa) with 0 < uk < 1Ak+1 and supk ick = oo (briefly, 0 < µk f oo) we put

{x=(zk)Ew

71'bµ

11

111x1'-skplak<}

(set of 1A-bounded sequences), cop

_ J

()Eco} , 1Pkykl < oo j

y= (y k) E w 111 y11 p

P(FD) t

j

k

(set of absolutely p-summable sequences).

Theorem 9.2.2 (p-bounded sequences). For any sequence is with 0 < uk t oo the following statements hold: (a) m C coo < mp < w. (b) (mp,11 11p) is a solid BK-AB-space. (c) (co,, 11 11p) is a solid BK-AK-space. (d) cp C P(p) < P, and (P(µ),11 11p) is a solid BK-AK-space.

(e) For every S E {a,y} we have cot = P(p), and the mapping T : ((cop,1111p)', 1111) -4 (P(p),

11o), f -4 (f (ek))

is an isometric isomorphism. In particular, coµ = P(µ).

Proof. Since p is strictly increasing and diverging to oo, we obviously have m c cop. Further, considering the diagonal matrix D := diag (f) , we get

COD = Cog, mD = mp and 111100 o D = 11110. Therefore, cop and mp are sequence spaces and cop C mp because co C m. Moreover, (mn,11 11k) and (cop, 11 11k) are BK-spaces on account of the second part of 8.1.4(c) applied to the triangle D. The statement that (mn,11 11p) is an AB-space and that (cop, 11 11,) is an AK-space follows immediately from the definition of 1111.. Also the solidness of cop and mp is trivial. Thus we have proved (a)-(c). The proof of (d) is similar to those of (a)-(c) and is left to the reader. (e) To prove that T is an isomorphism, it is sufficient to verify

f(p) = cop' (= cof = coy)

[coo is solid, cf. 7.1.10(b)J

(cf. 7.5.4). However, this is a consequence of the following chain of equivalent statements for any y = (Yk) E w : y E co,

V (xk) E cop

:

(ykxk) E P

p-bounded sequences and domains 477

4- V (xk) E cot, 4=f V (zk) E co

(ILkYk-.) E l Ak

(pkykzk) E Q

:

[D : cow --> co is bijective]

=f (µkyk) E P [since c0 =1] Y E 2(p).

Thus (e) is proved if we verify IIfII = II(f(ek))IIµ

(_ j{(pkf(ek))jji)

for each f E co`n.

If f E c°u and x = (xk) E cog, then

If W1 =

E xkf (ek)

[coo is an SAK-space]

k

k

pkf(ek)

<_

IIxiL,.II(f(ek))II",

therefore If II < II(f (ek))Iiu. Moreover, we obtain for x(n)

n

:= j` p sgn f (ek) ek k=0

the statements IIx(n) II < 1 (n E N°) and n

suplf(x(n))I = SupEIELkf(ek)I = n

n

II(f(ek))II°_

k=0

0

Thus, we have proved IIf II = II(f (ek))II'`

Remark 9.2.3. Since solid sequence spaces X with V C X have, in particular, the SIGNED P_GHP, thus also the SIGNED P_OSCP (cf. 9.1.6), and since coo and mo are solid by 9.2.2(b) and (c), we may apply 9.1.4, 9.1.7,

9.1.8 and 9.1.9 to Y := X := coo and Y := X := mo, where p = (pk) with 0 < pk T oa is given. (We will explicitly formulate the case X := mo in 9.3.4.) tl The following lemma is a useful tool for the handling of p-bounded sequences.

Lemma 9.2.4. Let µU} _ (p )k, j = 0,1, ..., be finite or countably many sequences with 0 < µk?) Too (k

oo). Then there exists a sequence

A = (pk) with 0 < pk oa such that for each j there exists a k? E No with µk < Ak' (k - k, ). In particular, in such a case we have mo C mo(s) (j = 0,1,...). Moreover, in the case of finitely many sequences A(j) we may put pk infi=o,i,... Ak?} and k? = 0 (j = 0,1, ...).

478

Consistency of matrix methods

Proof. Without loss of generality we may assume that we are dealing with

countably many sequences p() (j E N°). (Otherwise we repeat the last one.) We inductively choose an index sequence (ki)?EN0 such that

bk>ki : u( ")>j+2 is satisfied. Now, if we put Ak

kp+l

j + 1 + ki+t kaki

if k < ko k1 < k < kj+1 J E 1J°

(k E N°),

then p := (Pk) has the desired properties. The remaining statements of

0

9.2.4 are obvious.

The preceding lemma tells us that we may proceed from finite and countably many m,(J) to a common mµ where p is smaller than p(j). In the next part of this section we deal with `p-bounded domains'.

Definition and Remark 9.2.5. If p = (Ak) with 0 < µk Too and a matrix A are given, then mµ n CA is called the p-bounded domain of A. Trivially we have m n CA c co, n cA C m n cA (cf. 9.2.2(a)).

A

Analogously to the case of bounded domains, in connection with consistency and the comparison of matrix methods, we make use of the following notation.

Notation 9.2.6 (p-comparison and p-consistency). Let p = (iak) with 0 < µk T oo be given, and let A and B be any matrices. Then B is called p-stronger than A if m. n CA C cB; in such a case A is defined to be p-weaker than B. Further, B is called p-equivalent to A, if mµ n CA = myy n CB- Moreover, A and B are p-consistent, if limA x = limB x

(x E m, n cA n CB).

In general we speak about p-stronger, p-weaker, p-equivalent and pconsistent, if the corresponding property is satisfied for at least one sequence A. In this sense we will use also the terminology p-comparison and p-consistency. The introduction of these notions gives rise to a series of questions, forexample:

Characterization of p-coercive matrices, that is of matrices A with mp C CA for at least one p, the connection between b- and p-comparison, sufficient conditions for p-consistency (analogous to the bounded consistency theorem), characterization of the p-comparison.

p-bounded sequences and domains 479

In the next section we will deal extensively with the last two questions. Now, we give an answer to the second one.

Remark 9.2.7 (b- and p-comparison). (a) If B is p-stronger than A, then B is also b-stronger than A (which is trivial by 9.2.5). (b) If 0 < a < /3, then m fl cc. = m fl ccs (cf. 4.1.16), but, as was shown by Meyer-Kong and Zeller (1958),

for each p = (pk) with 0 < µk too,

mu fl cc ¢ mu fl cc.

that is b-comparability does not necessarily imply p-comparability. The proof of this statement is very technical, and we refer the interested reader to [171, Satzl]. IL In the following theorem we answer the first question: the 'p-coercive' matrices are exactly the coercive matrices.

Theorem 9.2.8 (extended Schur theorem, see 2.4.1). If A is a matrix which is conservative for null sequences, then the following statements are equivalent: (a) A is coercive, that is m C CA.

(b) h(A) = 0.

3.u=

too

Co.

(d) 3 p = (Pk), 0 < Pk t oo : h(A diag (Pk)) = 0 and (ak) E t(p)-

(e) 3p=(Pk), 0
limA x =

Lr akxk

(x = (xk) E mp)

k

where ak denotes the limit of the kth column of A.

Proof. (a)

(b) : This implication is a part of Schur's theorem as it is formulated in 2.4.1. (b) . (c) : If (b) holds, then, because IIAII < oo,

j:iakl <-IIAII

and h(A)=O,

(9.2:1)

k

we may choose index sequences (np) and (kp) with no > 0 and k° > 0 such that for all p E No the inequalities lank - akl < and

2_p-'

(n E No with n > np)

(9.2:2)

480

Consistency of matrix methods 00

E (IankI + Iakl) <

4_P_1

(n E No with n < nP)

(9.2:3)

k=ky

are satisfied. If we now put n-1 := 0 =: k_1 and pk := 2P (4_1 < k < kp) for every p E NO, then, on account of (9.2:1)-(9.2:3), we obtain for all p E NO and n E No with nP_ 1 < n < nP the following: ky-1

ky-1

E pk lank - akI +

pk IankI

pk IakI + L: pk lankI

k=0

k

00

k=0

k=ky

ko-1

2PElank -akI+ EIakl

<

k=0

k

00

lakl + E 2v+1

2 2v+1

k=k

v

V=P

E Iankl k=k.

00

<

2-v-1 + E 2-v-1 <

1 + IIAII +

IIAll + 3.

V=P

Therefore, IIA diag (pk) I I < oo is proved.

(d) : Let p = (pk) with 0 < pk too and IIAdiag(pk)II < oo be (c) given and define p = (pk) by Pk := pk (k E No). Since for each k E NO the kth column of the matrix B := A diag (ilk) converges with limit akpk, the matrix B is conservative for null sequences and (akpk) E £, that is (ak) E P(p) C 2(p); moreover, II(akpk)II1 < IIA diag (pk)II. To prove h(A diag (pk)) = 0, let e > 0 be given. We choose a ko E N with 6

1

pko

< 4(1+2IIAdiag(pk)ll)

and then an no E N such that ko

E pklank - akI <

E for each n > no. '6

k=0

Now, for each n > no we get pkIank - akI k

1

ko

pkIank - akI + sup k=0

<

v>ko

2IIAdiag(pk)IIc c 2 + 4(1 + 2 IIA diag (pk)II)

00

pv k=ko+1

<s

Thus the desired identity h(A diag (pk)) = 0 holds.

pk(lankl + lakl)

p-bounded sequences and domains 481

(d) = (e) : If (d) is satisfied, then the matrix B := A diag (pk) is obviously coercive by Schur's theorem, that is m C cB, which is equivalent to mp C cA because the matrix map diag (pk) : m -3 7n p is an isomorphism. Applying the limit formula in the Schur theorem to the matrix B, we get, by the same reason, for each x E mp the identity limA x

1miB

I

\\

Pk) =

k

Pkak JTk =

k

akxk,

which is the desired limit formula. (e) (a) : This implication is trivial since m C m,,.

0

For the next two sections it is important to know

MA CWA or man CACIA where A = (Ak) with 0 < Ak 1' oo is suitably chosen. For certain proofs it is important to be able to pass from a given matrix A to a row-finite and p-equivalent matrix A. We now show how that can be done.

Lemma 9.2.9. Let N E N, and let Air} = (a;;k)) (j = 1,...,N) be matrices which are conservative for null sequences and with column limits

(k E No, j E NN ). Then there exist an index sequence (Ph) and an increasing sequence (ph) in N° with suph Ph = oo and a sequence ak?)

A = (Ak) with 0 < Ak t oo such that the following conditions are satisfied:

(hEN). (i) Ph-Ph-1 <1 (ii) Ph < Ph (h E N°). (iii) Ak Iahk I < oo (h E No and j E NN). k

(iv)

Ak

Iaka}l

< oo

(j E NN)_

k

(V)

Ph

limsup h

k=o

CO

Ak Ia(r) hk

hk - afirk) I +k=P,, E Ak Ia{r)

I

=0

(j E NN)

Proof. Because WAIT < oo and (a(k?))k E 2 (j E NN) we may inductively choose an index sequence (Ph) such that Ph > h + 1 and 00

(n
hold for every h E N°. Putting

ifk
>

Ak :=

t2"+l+p\ if Ph
we obtain for all h E N and j E NN the inequalities

(9.2:4)

482

Consistency of matrix methods

Ak lank I k=Ph

P +i-1

00

00

00

< E (2n + 2) E Iah c I < E 2.2n - 4-n = 2.21-h k=P

n=h

n=h

(9.2:5)

and 00

Ak

IakJ)

< 2 2'-h.

(9.2:6)

k=Ph

Now, we make use of the existence of the column limits. We choose an index sequence (hr) such that for all r E NO and j E NN the condition r

[: Ak la(j)

- ak1) I

<

kk=O

1

r+1

(h > hr)

(9.2:7)

is satisfied. If, in addition, we put Ph

J0 ifh
if hr < h < hr+1 (r E NO),

(9.2:8)

then (Ph), (ph) and A = (Ak) satisfy the conditions (i)-(v): (i) and (ii) follow from (9.2:8), (iii) and (iv) are contained in (9.2:5) and (9.2:6), whereas 0 (v) is an immediate consequence of (9.2:7) and (9.2:5).

Exercise 9.2.10. Let p = (pk) with 0 < pk T oo and S E {a,;8,-y} be given. Show mu =1(p) and £(p)S = m,. Exercise 9.2.11. Let A and B be given matrices, and let 0 0 M C w. Then A and B are called absolutely equivalent relative to M if M C WA fl wB and M C (co)A-B are satisfied. Prove in the case of matrices A and B which are conservative for null sequences and of a sequence µ = ({AA:) E w with 0 < pk T oo that the following statements are equivalent:

(a) A and B are absolutely equivalent relative to m,. (b) (i) V n E N° : (ank) k , (bnk) k E 1(p). (ii) A - B satisfies (Spo). (iii) h((A - B) diag (µk)) = 0. Exercise 9.2.12. Verify, in the case of matrices A and B being conservative for null sequences, the equivalence of the following statements:

(a) A and B are absolutely equivalent relative to m. (b) There exists a sequence p = (pk), 0 < pk T cc, such that A and B are absolutely equivalent relative to mU.

Bibliography: [34]; [196]

p-consistency and p-comparison

9.3

483

µ-consistency and p-comparison

The main subject of this section is the comparison of p-bounded domains of matrices A and B in terms of the matrices A and B and also in terms of continuity statements for the matrix map B : mu fl CA - c. We will make use of results on FK-spaces and their duals and of the Hahn-Banach theorem and the separation theorem. As in the proof of the main result of section 9.1 we will handle coarse steps of the proofs with functional analytic methods and then we have to perform the fine work by relatively extensive analytical methods. To get an idea of the characterization of p-comparison in terms of given matrices we recall the starting point in 8.7.1. Starting with the comparability of the domains, that is CA C cB, we deduced a quotient representation

B = CA + D with suitably chosen matrices C and D.

(9.3:1)

If in the case of regular matrices A and B the matrix C were regular and column- and row-finite, and if CA C COD held, then we could conclude from

(9.3:1) that CA C cB

and

limn IIA = limA .

In the case of the p-comparison we will first deduce from m, fl CA C CB a quotient representation (9.3:1) using functional analytic methods quite similar to those in 8.7.1. After that, analytical fine work will give us a quotient representation with a column- and row-finite regular matrix C and a `p-coercive' remainder D which is regular for null sequences. The characterization of p-comparison by continuity statements requires a certain semi-norm which we call an `i-semi-norm':

Definition and Remarks 9.3.1 (i-semi-norm). Let A be a matrix that is conservative for null sequences and which has column limits ak (k E NO). Then IIxIIA := limsup Dank - ak)xk n

(x = (xk) E m)

k

defines a semi-norm II IIA on m. In the case A := I we write II 11= instead of II II' and call it an i-semi-norm. Moreover, in the case of regular matrices A we have IIAxII= = IIxIIA for every x E m. Obviously, the following statements hold: (a) Kern 11 IIA = m f AA . (b) II IIA = 0 if and only if A is coercive. (c) IIxIIA < 2IIAIl IIxIIA for each x E m.

In the next theorem we characterize p-comparison of regular matrices.

Before we start on the very technical and extensive proof of that theorem, we deduce from it a consistency theorem of Mazur-Orlicz type in the

484

Consistency of matrix methods

case of p-comparable matrices, which we will contrast with the consistency theorem 9.1.9 in the particular case X := mA.

Theorem 9.3.2 (p-comparison). For regular matrices A and B the following statements are equivalent: (a) 3 p = (pt), 0 < pk f oo mP n CA C CB including consistency.

(b) 3p=(pk), 00 VxE If Bx'Do < MIIAxII, + Lf IxElx.

(d) There exist matrices C and b with

B=CA+D, IICII
G oo,

where A = (Ak) with 0 < Ak fi oo is suitably chosen.

(e) For B there exists a `quotient representation' B = CA + D, where C is a column- and row-finite regular and D is a coercive matrix (which is necessarily conservative for null sequences). (f) 3 K > 0 V x E m : IIBzII < KlIAxII,.

The equivalence `(e) q (f)' is due to H. Baumann [13], whereas the remaining equivalences go back to Boos. Theorem 9.3.2 contains the following consistency theorem of MazurOrlicz type which was proved independently (in a slightly modified version) by G. M. Petersen (1963) and J. Copping (1966) (cf. also [196, Theorem 4.1.8]).

Corollary 9.3.3 (p-consistency). If A and B are regular matrices and MA n CA C cB holds for a sequence p = (pk) with 0 < pk f oo, then there

exists a sequence p = (pk) with 0 < pk T oo and pt < pk (k E N), such that A and B are consistent on m p n CA. (In particular, mp n CA C CB. ) Proof. The statement is an immediate consequence of `(b) =:> (a)' in 9.3.2 0 taking 9.2.4 into consideration. As far as the applicability of 9.3.3 concerned, the transition from m,ncA

to mQ n CA is necessary in general since-as J. Copping [71] provedTheorem 9.3.3 fails in general for p := A. However, we can save that change from p to p, if we consider m, n FA instead of m,, n CA and recall 9.1.8 and 9.1.9 (cf. 9.2.3 too):

Theorem 9.3.4 (limit formula and consistency on m,, n FA). Let p = (pk) with 0 < pk t oo be given and let A and B be matrices with cpCmAnFACcB. Then mµnFA =(m,nwA)ED (u) and, as far as u E IB is satisfied,

with uE {0}U(mµnFA \WA)

u-consistency and Et-comparison

limB x = a

485

(limAx_Eakxk) + 1: bkxk (x = (xk) E m , f1 FA) k

k

with

a=0, if u=0, and a=AA(u), if u#0. In particular, A and B are consistent on mµ fl FA if they are consistent on cp ® (u). (For example, if A and B are regular, then we may choose

u.= e.) Proof. Apply 9.1.8 and 9.1.9 to the special case `X := mi,' and note that MA is solid and has consequently the SIGNED P_GHP (cf. also 9.2.3).

There is a further application of 9.3.2 in connection with the characterization of `m n CA = c' (cf. 8.5.1 and 8.5.4) in 9.3.7. We will spend the remaining part of this section on the very long and

technically hard proof of 9.3.2. First of all we give the simpler steps of proof, namely `(a) = (b) (c) = (d)', `(e) (a)' and `(e) = (f)'. After that, (e)' which requires very technical and analytical methods. Finally we will prove the implication `(f) (d)' using both functional analytic and analytical methods. we will give a proof of `(d)

Proof of 9.3.2. (a) = (b) is trivial since (b) is included in (a). (b) = (c) : Let m fl cA C cB for a fixed sequence p = (p,) with 0 < Ilk t oo. Then m,, fl CA is an FK-space as an intersection of the FK-spaces m,, and CA, and its FK-topology is generated by the semi-norms 11

1 1 0 00A ,1 11 1p and pn In E N° )

where pn is defined as in 8.1.3 (cf. 7.3.9, 8.1.5(a) and 9.2.2(b)). Then

B:mµflCA -rc,x---+ Bx, as a matrix map between FK-spaces, is continuous; that is, there exist M > 0 and N E No such that N

IIBxII.< M (IIAxIloo+ EPn(x) + IIxIIt.

(9.3:2)

n=0

for each x E MP fl CA. Applying Lemma 9.2.9 to A we obtain a sequence

a=(ak) with 0
Using the notation

(n E N-).

486

Consistency of matrix methods N

L:=M [ i+(N+1)cN+EEQklankl n=0 k

and Ak := min{ok,µk}, we have 0 < Ak t oo and we get by (9.3:2) for every x = (xk) E mA n cA C mµ fl CA, in particular for each x E gyp, the inequalities N

{IIxII..

IIBxIJoo

< MII Axll oo + M

E ankxk Ixnl + sup Ik="O

+ n=0 N

(IIXII,\

< MIJAxJJ. + M

l

+ E (O'nlXnl + E cklankI IT n=0

k

< MI JAxil. + Lllxllx. Hence (c) holds. (d) : Let (c) be satisfied for a sequence A = (Ak) with 0 < Ak t 00 (c)

and for positive constants M and L, that is IlBxiloo < M IIAxJJoo + L Ilxll,\

(x E gyp).

Therefore, if we define

fn:tp--*IK, x -i

bnkxk

(n EN ),

k

then (fn) is by 6.5.16 an equicontinuous sequence of linear functionals on the locally convex space (gyp, (11Iloo o A, JJ Jl, )) . By the Hahn-Banach theorem (cf. 6.5.4) there exists for each fn an Fn E (co,\ n cA, (11Iloo o A, pn (n E N°), IJ Ila))` such that

Fn(x) = fn(x)

(x E ')

and

I F0(x)I < M JJAxlloo + L JIxIIA

(x E cfla fl CA).

Analogously to the proof of 8.7.1 we conclude the existence of hn E c' and 9n E coa with

Fn(x) = hn(Ax) + g. (x) Ihn(y)I S M Ilylloo

(x E cm n CA), (y E c)

and

J9n(x)I < L IlxtJA With this and 9.2.2(e) we obtain

(x E cox).

(9.3:3) (9.3:4)

487

p-consistency and p-comparison

(gn(ek))k E P(A)

and

EAkl9n(ek)I = II9nII S L k

for each n E N°. Therefore, the matrix D = (dnk) condition

(gn(e')) satisfies the

IlDdiag(Ak)II = supj:Akldnkl < L < oo. k

Further, on account of hn E c' and (9.3:4), we may choose for every n E NO a t(n) = (n) )k E E and a An E K such that

hn(y) = An lira yk +

tkn}yk

(y = (yk) E c)

k

and

Ipni + IIt(n}II1 = IIhnII :5 M.

(9.3:5)

Now, if we define C = (cnk) by cnk := tkn} (n, k E N°), then we obtain IICII < co by (9.3:5) and for all n, k E N°, by (9.3:3), the identity bnk = Me

k)

= Fn(ek)

= hn(Aek) +gn(ek) t

)a,,k + link

[since Aek E co ]

I,

[CA]nk +dnk.

That is, B = CA + D. Hence, the matrices C and b have the desired properties. (e) (a) : If (e) holds, then we may choose a column- and row-finite regular matrix C and a coercive matrix D such that B = CA+D; we note that D is necessarily regular for null sequences, which implies particularly dk = 0 (k E N°) for the column limits of D. Now, we apply the `extended Schur theorem' `(a) * (e)' in 9.2.8 together with the limit formula to the

matrix D and get a sequence p = (pk) with 0 < pk T oo such that mp C COD. Thus, we obtain mp fl CA C CB with consistency since

Bx = (CA + D) = (CA)x + Dx = C(Ax) + Dx

(x E mp fl CA)

is satisfied because C is row-finite (cf. 2.2.5) and regular. (e) (f) : Let B = CA + D where D is coercive and C is a columnand row-finite regular matrix. Because D is coercive and

dk = bk - limo Aek = 0 we have

(k E N°)

488

Consistency of matrix methods

lim sup E I dnk l = h(D) = 0. n

k

by Schur's theorem 2.4.1. Furthermore, since C is column-finite, we may choose an increasing sequence (vn) in NO such that (9.3:6)

SUP Mn = 00 n

and

cnv = 0

(n, v E N° with v < vn).

(9.3:7)

We now obtain IIBxII= < KII AxIIi

(x E m and K := IICII + 1)

by (9.3:6) and by the following estimation which holds for all x = (xk) E m

andnENO:

E xk

E bnkxk k

cnvavk + v

k

avkxk + IIxUIoo E IdnkI

Cnv k

:5

dnkxk k

[cf. 2.2.5]

k

+ llxlloo E IdnkI v>v Ek avkxk k

11C11 sup

[cf. (9.3:7)].

Thus we have proved that (e) implies (f). (d) = (e) : First we show that for each matrix C = (cn,,) with IICE) < 00 there exists an index sequence (nq)gENo such that for every q E N° the submatrix (cnv)n>nq, 'ENO of the matrix C may be separated into finitely many submatrices C(q, t) = (c(q, t, r, v))r,VENO

(t E iVtq , tq E No suitably chosen)

which satisfy the following properties:

(a) The set of the elements of the with column of C(q, t) has for each v E N°q a diameter' smaller than or equal to q+i For all fixed q E NO and n E No, n > nq, there exist a t E Ntq and an r E Nl° such that Gnv = c(q, t, r, v) for every v E No . For this we cover for every q E No the complex plane with pairwise disjoint semi-closed squares Sp (p E N°) with edge length 2(a+r} . Because 1101 < 0o we may choose the-squares such that' at most finitely many of them contain coefficients of C. ' If 0 0 A C K, then supo

Ja - bj is called the diameter of A.

n-consistency and u-comparison

489

If n_1 := 0 and n0,... , nq_1 are already determined for a q E NO, then we choose an nq E NO with n9 > nq_1 such that for every p E NO and v E NO, the set S, contains none or infinitely many coefficients Cnv of the `subcolumns' That choice is possible since otherwise-in contradiction to the above discussion-the coefficients of at least one of the subcolumns (cnv)n>n (v E NQ) would be distributed on infinitely many (different) sets Sp. Without loss of generality we may assume that exactly the squares (jq E NO suitably chosen)

So, ... , S?Q

contain for every v E Nq (infinitely many) coefficients cn with n > n4. Now, for every (q+1)-tuple (po, ... , pq) E (N°q )9+1 we consider N(*Po,...,P,) :_ In E No

I

n > nq and

env E Sp. (v E Nq0)}

and then choose an nq E NO with nq > nq such that for each (q+l)-tuple (PO, ... , pq) E (N° ) q+l and each n E N(P0 Pq) the relation n < nq holds, if N(PO .Pq) is a finite set. In particular, either N(P0....,Pq) :- {n E N° I n> nq and n E N{P......Pq}} is empty or N{Po

Pq)

is an infinite set. Since for every n E No with n > nq

there obviously exists(PO,a-

. , pq) E (N° ) a+1 with n E N(p( ..... Pq) I the

set

H :_ {N(0 ,...,pq)

I

(p0, ... , p4) E (Njq )

q+1

and N(Pa,...,Pq) # @}

is a partition of {n ENO I n > nq } into, say, tq pairwise disjoint infinite sets. Thus, if we map each N(Po..... Pq) E H into the row submatrix ENO of C, then these tq row submatrices of C satisfy the conditions (a) and ($). After this preliminary discussion we give the_proof _of `(d)

(e)'. For this

let A and B be regular matrices, and let B = CA+D with IICII < oo and IID diag(Ak)II =: K < oo be satisfied for a suitable sequence A = (Ak) with 0 < ak T oo. According to the preliminary discussion we choose an index sequence (nq) and for each q E NO a partition of the matrix (i nv)n>nq,vENO

into row submatrices C(q, t), t E Ntq , such that (a) and (0) hold. Because

IIC(q,t)II
490

Consistency of matrix methods

easily verify. Because B = CA + D and since A and B are regular, the corresponding row submatrices of b satisfy the condition A,lrD(q, t)I < K := IlD diag(Ak) ll < 00,

(9.3:8)

where r5 (q, t) denotes, for v E No, the limit of the with column of the corresponding row submatrix of D_ .

Now, we define the matrices C = (cnv) and D = (dnk) by

_

if0
cnv c(9>

cn` '

t

r,

v) -

r.,v(q, 4 t)

if qn < n <

n,+,,

q E NO

(n,v E N`0}

(9.3:9)

(where the connection between n and (q, t, r) is given by (/3)) and by

b := (C - C)A +A

(9.3:10)

In accordance with the definition of C and b we have

B=CA+D and

IICII<211C11
Since rv(q,t) is the limit of the with column of a (suitable) row submatrix of C(q, t), we obtain by (a) and (/3) the relation 1cn v

I_1c(gtrv)-rv (gt)I< >

>

>

1

q+1

for every n E No with nq < n < nq+i and v E IN. In particular, the columns of C converge to zero, that is C enjoys (Spo) and is consequently regular for null sequences. Therefore, b is also regular for null sequences because B = CA + D and since _A and B are regular. In the next step we verify 1)Ddiag(Ak)Jf < 2K, which implies that D is coercive by `(a) C* (c)' in 9.2.8. This follows from the inequalities

E Ak

Idnk l

: E Ak J(cnv - Cnv)avk

k

v

k

+ E Ak I (tnk I


<

[cf. (9.3:10)]

k

f

[cf. (9.3:9)]

k

and

1: Ak I k

dnk I

< E Ak k

rv (q, t)avk I + E Ak I ink l k

[cf. (9.3:9)]

p-consistency and p-comparison

491

< 2K, ifnq
trices C of C and D of D with column limits rv (C) and rv (D) (v E NO) the identity

(kEN°) V

is true because B = CA + D and since A_ and B are regular. Now, we verify (Zsj) for the matrix C, which proves the regularity of C. We obtain (Zsi) from the equalities

cnv-1 =

Cnv 11- E avk + E E Cnvavk - 1

l

v

=

k

(1 -

Cnv

k

avk) +

v

(bnk - 1) - F_ 11nk

since the last term converges to zero on account of (Sp°) for b and since h(D) = 0 (cf. 9.2.8), and because the middle term converges also to zero on account of (Zsl) for B, and, finally, since the first term converges to zero on account of (Zs1) for A and because C is regular for null sequences. In the next step we pass from C and D to matrices C and D which satisfy the conditions in (e). For this we `truncate' the matrix C by applying 9.2.9 on both sides and replace the coefficients of the truncated parts by zeros. Namely, for C there exist, by 9.2.9, an index sequence (Pn) and an increasing sequence (pn) of non-negative integers with

pn < Pn (n

N°)

and

sup pn = oo n

and (note cv = 0 (v E N°) since C is regular) Dn

Icnvl + OOE

limsup v-0

n

0.

(9.3:11)

v=P +1

If we now define C = (cn,) and D by

_ cnv

Cnv

if pn < v < Pn

0

otherwise

(n, v E NO )

and

D:= ((5 -C)A+D, respectively, then by (9.3:11) the regularity of C implies that of C (cf. 2.3.711). Moreover, D is coercive because b is and since h((C - C)A) = 0

492

Consistency of matrix methods

holds for the matrix (C - C)A (which is obviously regular for null sequences) because

E

P.

E(Cnv -

cnv)avk

<-

llAil

v

(Rh+ v-0

(n -* oo)

0

00

i;Fnvi)

[cf. (9.3:11)].

Further, B = CA + D, and C is row- and column-finite. Thus, we have proved that C and D satisfy the conditions in (e). (d) : Since in this part of the proof we will make use of the sep(f) aration theorem, we assume K := R for convenience. In particular, all sequences and matrices have real coefficients. (The case K := C can be reduced to the case K := R quite similarly to the proof of the Hahn-Banach theorem and requires some technical tricks.)

First we introduce some notation. If A = (ank) is any matrix, then a(n) :_ (ank)k denotes the n" row of A, and if IIAII < oo and K > 0, then let

HK(A) :_

{h:=Etna (n) t = (tn) E e and IItIII <- K I , I

n

where we understand the convergence of the series coordinatewise. The set HK(A) has the following properties as we will verify: (i) HK (A) is well-defined, and HK (A) C e. (ii) Ka(") E HK(A) for every v E N°. (iii) HK (A) = KHI (A). (iv) HK (A) is absolutely convex. (v) HK (A) is closed in (e, 1111 1) if A is regular.

(i): Let t = (tn) E e, IItIII < K, and let h = (hk) be defined by hk E. tnank. (The series converge since t E e and (ank)n E m.) Then for each N E N° we have N

N

< E ItnI

Ihkl = k=0

k=0

n

kI s 11Alllltill; k=0

thus h E e, and therefore HK(A) C P.

(ii): For v E N° and t = (tn) := Ke" we have IItIII < K and also (iii): Consider the map defined by t - K t (t E e) which maps the closed ball in (e, 11 111) with radius K and centre 0 into the closed unit ball of V'11 Ill). (iv): Note, the closed unit ball of (e, 11 111) is absolutely convex.

he-consistency and p-comparison

493

(v): The proof of this statement is a nice application of Theorem 2.3.6 and is left to the reader (cf. Exercise 9.3.8). As usual, we use for x E 2 and 0 54 S C t the notation

d(x,S):=inf{IIx-yII1 I yES} for the distance of x and S. Now, we assume that (f) holds, that is IIBxIIi < KIIAxIIi

(x E m)

(9.3:12)

for a certain K > 0 which we assume to be fixed. Then by `(a) q (c)' in 9.2.8 the statement in (d) is proved if we can show the existence of a coercive matrix D which is regular for null sequences and the existence of a matrix C with IICII < K and B = CA + D. For. this, by Exercise 9.3.9 it is sufficient to prove lim sup d(b(n), HK (A)) = 0.

(9.3:13)

n

We further assume that there exists an e > 0 with limsup d(b(n),HK(A)) > e.

(9.3:14)

n

Now, we construct suitable index sequences (nj) and (kj). We put ko :_

0 =: n_1 and choose (inductively) for each j E No an nj E N with nj_1 < nj, ki

L(Iankl + Ibnkl) < 2-j

(n > nj)

[A, B satisfy (Spo)]

(9.3:15)

k=0

and

d(b(ni),HK(A)) > e

[cf. (9.3:14)]

(9.3:16)

[note IIAII, UBII < oo].

(9.3:17)

and for nj a kj+1 E N with kj+1 > kj and 00

E (Iankl + Ibnkl) < 2-j (n < nj) k=kj+i

Now, if

U'

It E e I

IIb(ni)_tII <E}

(jEN°)

is the open e-ball in (t, II 111) with centre b(ni), then J.

u,

HK(A) = 0

(j E N°)

'[cf. (9.3:16)]

is obviously true. Since the non-empty set HK(A) is absolutely convex and U- is open and convex, we may apply the separation theorem 6.5.14(a) to

494

Consistency of matrix methods

HK (A) and Uf. For every j E No we obtain a (real-valued) functional

gj E eand an aj E R with gj(y) < aj < gj(y)

(y E HK(A) and j E U').

(9.3:18)

Now,

0 < aj

(j E N°)

[since 0 E HK(A)],

(9.3:19)

and we may assume without loss of generality (j E N°).

IIgjII = 1

(9.3:20)

(Otherwise we may consider ;8jgj and 8jaj with 3j := IIgjU-; instead of

gj and aj.) Since and b(n') - z E Uj

b(n9) + z E UE

hold for every z E t with IIzIII < e, we obtain by (9.3:18) and (9.3:19) (b(n'))

Igj(z)I < gj

(z E e with Ilzlli < e)

for every j E N° and consequently gj(b(nj)).

IIgj1I

(9.3:21)

Because Ka(n) E HK(A) (cf. (ii)) and S

b(n1)

1 -

E UE

2IIBII

(jEN°)

we get by (9.3:18) (and (9.3:19)) that (i E N°)

191(a(n))1 < K ( 1 - 2IIBII) g,(b(nj))

(9.3:22)

which implies in particular, since 0 < e < JIBIJ (cf. (9.3:16)),

0<1-2IIBII <1.

E w for each j E NO by x(j) := gj(ek)

Now, we define x(j) (/r E N°) and obtain x{?} E m,

(9.3:23)

i!x(j) = IIgjII

xk'')tk

and gj(t)

(t = (tk) E e)

k

by 7.5.4. Hence, for every j E No we can summarize

lix(?)IL = 1

[cf. (9.3:20)],

(9.3:24)

p-consistency and A-comparison

gj (b(ni))

_

bn1

kxkj) > E

[cf. (9.3:20) and (9.3:21)],

495

(9.3:25)

k

IIAx(') 11. <-

K' (1-

2IIBII 16

)

[: bnjkxkj)

[cf. (9.3:22)]. (9.3:26)

k

In the next step we aim to fit together `parts of suitable x(j) 'to obtain a sequence x E in with IIBzIIi > KIIAxIIi (which contradicts the assumption in (9.3:12)). For this we recall the chosen index sequences (nj) and (kj) and define y = (Yk) according to 2.5.6 by

(kj < k < kj+l and j E N°).

Yk := fj

(9.3:27)

Motivated by the properties of y and f = (f j) (cf. 2.5.6 and Figure 9.3.1) we introduce further index sequences (rq) and (s(q)). For each q E NO let rq := k(q+1)(q+2)

and s(q) :_ (q + 2)2 - 1.

We now define x = (xk) E m with the desired properties by

(rq < k < rq+l and q E NO).

xk := ykxk8(q))

(9.3:28)

Obviously we have

x E m and IIxfl. < 1 [since IIyII. < 1 (2.5.6(i)) and (9.3:24)]. (9.3:29) To verify IIBxIIi > KIIAx!Ii, we fix an arbitrary n E No with n > n2 and

determine the unique j E N with nj < n < nj+l. For j there exists a unique q E N° such that rq < kj+1 < kj+2 < rq+1

We have to distinguish the cases `rq < kj' and 'rq > kj'. First, let us consider the case

rq < kj < kj+1 < kj+2 < rq+l . We estimate essentially IEk ankxk) by IEk bna(a)kxkl and deduce from it that IIxIIB > KIIxIIA. For the chosen n and x we have

1: ankxk k

ki-1 < k=0

<

00

ki+2 -1

IankxkI + 57 )ankxk) + E ankxk k=k, k= kJ+2

2-j + 2-j-1 ki+2 -1

ki+2-1 ank x(8(q))

+fj

k

k=ki

+Ifj+l -fjl

(s(q))

ankxk

k=ki+'

496

Consistency of matrix methods

C

N

Fig. 9.3.1: Gliding hump

0

µ-consistency and µ-comparison

497

(cf. (9.3:15), (9.3:17), (9.3:27)-(9.3:29) ]

ki+z-i

<

2-j+1 + E ankxka(q)) + Ifj+1 - fjI IIAII k=kj

[because 0:5 fj < 1 (2.5.6(1)) and (9.3:24)]

<

2-j+1 + IE ankxk8(q)) I + 2-j+1 + I fj+i - f i I IJAII k

(cf. (9.3:15), (9.3:17) and (9.3:24)]

<

2-j+2 +

Ifj+i - fjl IIAII +

K C1-

2IIBII)

k fE

(cf. (9.3:26)].

Because

E

bn.(q)kx(ks(q))

k

ka(q)+, -1

<

1:

2-s(q) +

bn,(q)kxka(q))

+2-8(q)

k=k,(q)

[cf. (9.3:15), (9.3:17) and (9.3:24)]

2-8(q)+i

E bn,(q)kxk !ks(g-i I

I

[on account of (9.3:27), (9.3:28), s(q) =(q+1)2_1 and 2.5.6(iv)]

<

2-e(q)+1 + it bn,(q)kxk I + 2-8(q) + 2-8(q) 1k [cf. (9.3:15), (9.3:17), (9.3:29)]

=

2-8(q)+2

+

E k

we obtain the inequality E ankxk

<

2-j+2 + Ifj+i - fj I IIAII

(9.3:30)

k

+K

(2_s(+2 +

l1

21IBIl)

The estimation in the second case, that is the case

kj < kj+i = rq < kj+2 < rq+1 ,

k

498

Consistency of matrix methods

is more trivial than in the first case:

IEankxkl k

ki -]

E k=0

ki+2 -1

CO

Iankxkl +

Iankxkl +

E ankxk k=ki

k=ki+2 ki+2 -1

<

<

2-i+l + fi+1

akx(8(q)) L k=ki}

[(9.3:15), (9.3:17), ff = 0 because j = (q + 1)(q + 2) - 1, 2.5.6(iii)] 2-?+1 [since j + 1 = (q + 1)(q + 2)] + f(q+1)(q+2) JJAII (i -+ oo) [because f(q+1)(q+2) = 9+2 ]

-0

As n tends to oo, that is j -* oo and s(q) -+ oo, we obtain from the above and from (9.3:30) the inequalities flxIlf

= <

limsup n

k

limsup K (1 s(q)->o

l1

I

- 21IBJI/

ll:bn.(v)kxk k

JIBxlli < 21JBII

JIBxii2

[cf. (9.3:25), (9.3:23)],

K

and thus the required inequality IIBxi!i > KIIAxIli. Thus we have proved Theorem 9.3.2.

Considering in 9.3.2(d) the case b = 0 (thus K = 0) and checking the proof of `(d) = (e)' in this case, we get b = 0. This implies the following theorem due to J. Copping [701 (1958).

Theorem 9.3.5. If _B = CA holds for regular matrices A and B and for a matrix C with JICIJ < oo, then there exists a regular matrix e with

B=CA. Proof. Put K := 0 in the proof of `(d)

(e)' in 9.3.2.

By refining the proof techniques one can show that Theorem 9.3.2 remains true in the case of coregular matrices A and conservative matrices B. We state the corresponding theorem which emphasizes the meaning of multiplicative matrices.

Theorem 9.3.6 (Ii-comparison). Let A be a coregular and B be a conservative matrix. Then the following statements are equivalent:

p-consistency and p-comparison

499

(a*) 3 P = (Pk) , 0 < At t000 : mp n CA C CB and hmB x = x(Aj hmA x + L,k {bk - x4 ak) Xk (x = (xk) E mp n CA).

(b) 3p=(pk), 0
(c) 3a=(Ak), 00 `dxE{p JJBxIj. < MIJAxII. + Lllxlla(d) There exist matrices e and b with B = CA + D, IICJJ < oo and K := JJDdiag(Ak)II < oo, where A = (.\k) with 0 < Ak 1 00 is suitably chosen.

(e*) For B there exists a `gaotient representation' B = CA+D, where D is a coercive and C is a row finite and column-finite, z(A)-multiplicative matrix.

(f*) 3 K> 0 b x E m :

IIxHIB < KJJxjl;'.

Proof. We refer the interested reader to [36, Satz 1].

0

We remark that Theorem 9.3.2 can be very usefully applied to questions concerning singularities in Section 9.4.

Exercise 9.3.7. Let A be a coregular matrix. Verify the equivalence of the following statements:

(a) mnCA=C. (b) m n CA is closed in the FK-space CA. (c) c is closed in the FK-space CA. (d) There exist a column-finite and row-finite,

x A -multiplicative matrix C and a conservative matrix D with I = CA + D. (e) 3 K > 0 V x E m IjxhI= < KJJxfh;'

(f) 3p=(JUk), 0
Exercise 9.3.8. Let A be regular. Prove that HK(A) is closed in (t, II II,) (which is statement (v) in the proof of `(f) (d)') in 9.3.2. (h(*)) Hint: Choose a sequence in HH(A) with h(*) -+ h in (i', I V') with defining sequences t{'') and t, assume (without loss of generality) t{''i -+ t in (w, r.) and apply T = (t(')) to the columns of A (cf. 2.3.6).

Exercise 9.3.9. Let A and B. be regular matrices such that IIBxjjj < K JJAxII2 (x E m) for a certain K > 0. Show the equivalence of the following statements:

(a) There exist a coercive matrix D which is regular for null sequences and a matrix C with IICII < K and B = CA + D. (b) lim sup,, d (b(v), HK (A)) = 0. Bibliography: [36], [13]; [196], [71], [70]

500

Consistency of matrix methods

9.4

Singularities of matrices

Domains of regular matrices have to satisfy certain conditions. For example,

a regular matrix cannot sum all bounded sequences by Schur's theorem. However, by a result due to G. M. Petersen there are regular matrices A and B such that m C CA + CB. Therefore, in an attempt to capture as many sequences as possible, it is an obvious idea to consider simultaneously

several regular matrices, say A('),..., A(N). Proceeding in this way, the following questions, among others, arise. (1) Are A(1}, , A(N) pairwise consistent? (2) If x E EN 1 CA(i}, that is x = x(j) with x(?) E CAC,>, is `limit' of x defined by limA(,> x(?) unique? In such a case, we speak about `simultaneous consistency'. EN1

EN1

(3) For given regular matrix methods AM,-, A(N) does there exist a `roof method', that is a regular matrix B, which is stronger than and consistent with each A(?) (j E NN )? (In such a case we may work

with B instead of AM,-, A(N).) In the case of a negative answer to (1)-(3) we encounter so-called 'singularities' of (regular) matrices. Naturally, the above questions also make sense (in a modified version) if one considers only parts of domains, for example bounded or /A-bounded domains. Since the questions are closely related to the comparison and consistency of matrix methods we restrict our interest in these questions to the cases mentioned above as has also been done in the literature. First we introduce the notion of simultaneous consistency which plays a central role in this section.

Definition 9.4.1 (simultaneous consistency). Let AM,-, A(N) be matrices, and let 0 qE L C w. Then AM,-, AM are called simultaneously consistent relative to L, if `d x(i) E L n CA(J) (j E NN) :

(X=0 E N

N

j=1

j-1

limAw x(i) = 0

holds. In the cases L := w, L := m and L := mu (0 < pk t oa) we use the notation simultaneously consistent, simultaneously b-consistent and simultaneously A-consistent, respectively. Remarks 9.4.2. Let AM,-, A(N) be matrices, and let L < w. (a) In the case N = 2 the notions of consistency relative to L and simultaneous consistency relative to L are equivalent. (b) If the matrices AM,-, A(N) are simultaneously consistent relative to L, then they are also (pairwise) consistent relative to L. However, the converse implication is not true in general, as the (obviously) regular matrices

A, B and C, defined by

Singularities of matrices

a2n,3n = 1 =: a2n+2,3n+1 b2n,3n+1 C2n,3n+2

1 1

b2n+1,3n+2 C2n+1,3n+3

501

and ank:= 0 otherwise, and bnk := 0 otherwise, and cnk:= 0 otherwise,

prove. Namely, cA n CB = cB n cc = CA n cc = c, which implies the pairwise b-consistency on account of the regularity. Moreover, the simultaneous inconsistency (relative to m) follows easily from consideration of the

sequences x :_ (1,1, -2,1,1, -2,1,1, ...), y :_ (-2,1,1, -2,1,1, -2, ...) and z := (1,-2,1,1,-2,1,1 ....), which satisfy x E CA, Y E cB, z E cc,

x+y+z=0 and limAx+limBy+limcz=3. (c) The matrices

are simultaneously consistent relative to

L if and only if for each x E with x(j) E L n CA(,), the value representation of x, that is

EN1(L n CAM), that is x = ? 1 x(j) IimAU) x(j) is independent of the FN1

N

N

N

F : I(L n CAu)) - > III, x = 1: x(j) -.y E hmA(,) x(j) j=1

j=1

j=1

a

is well-defined.

The proofs of the remarks are straightforward and are left to the reader. The characterization of simultaneous b-consistency of regular matrices is very simple as we now show.

Theorem 9.4.3 (simultaneous b-consistency). In the case of regular A(N) the following statements are equivalent: (a) A('),..., A(N) are simultaneously b-consistent.

matrices A(1),

(b) e O

,

EN1(m n CoA(j) ).

Proof. (a)

. (b) : Assume that (b) fails. Then there exist N

Y(j) E m n cOA(,) (j E NN) with e = E y(j). j=1

Putting x(1) := e - y(1) and x(j) := -y(j) (j = 2, ... , N) we obtain N

N

E x(j) = 0 j=1

and

E "MAW x(j) = limAO) e = 1, j=1

thus A('),..., A(N) are not simultaneously b-consistent. (b) . (a) : If A('),..., A(N) are not simultaneously b-consistent, then there exist x(j) E m n cA(j) (j E NN) with N

N

E x(j) = 0 and a

E "MAW XU) 36 0.

j=1

j=1

502

Consistency of matrix methods

Because Air) is regular, there exist aj E K and Y(i) E m fl C°A(;) with x(j) = y{?) + a?e (j E NN), which implies

EyU) _- (ai) e and 0¢a=Eat. N

N

N

j=1

j=1

j=1

Since e = - 1: 1 yt?) and y(i) E mflC°A(;) (j E NN) hold, we obtain

0

1(m fl c°A(;) ), that is (b) fails.

eE

We now formulate question (3) in the case of bounded and 2-bounded domains, respectively. The remainder of this section will essentially be devoted to it.

Question 9.4.4. Let A('),..., A{N) be given regular matrices. Does there exist a regular matrix B which enjoys N

CB D E(mflCA(;))

(9.4:1)

?=1

or even N

(m, fl CA(;)) for a certain

cB D

(µk) , 0 < {Ak T oo ?

(9.4:2)

i=1

First, we work on the necessary conditions in terms of ABM) and their domains for the validity of (9.4:1).

Remarks 9.4.5. Let B, A('),..., A(N) be regular matrices such that condition (9.4:1) is satisfied. (a) Then A(1), ... , AU") are pairwise b-consistent since, by the bounded consistency theorem, B is b-consistent with each AU) (j E Na). (b) For the same reason, A('),..., A(N) are also simultaneously b-consisA tent which is equivalent to e 0 EN1(m fl C°AM) by 9.4.3. By Example 9.4.9 we will prove that simultaneous b-consistency of reg-

ular matrices AM, ... , A{N) is not sufficient in general for the existence of a regular matrix B with CB D EN1(m fl CA(;) ). Prior to that we deduce a further necessary condition for it. For this, the following remark will be useful.

Remark 9.4.6. If A is a matrix with FIAII < oo, then (m fl CA, 1111.) This is is a BK-space and, in particular, m n CA is closed in (m, 11 so because m fl CA is an FK-space as the intersection of the FK-spaces (m, 11 11.) and (CA, {pj I j E N° } U 11111. o A}) and its FK-topology is

Singularities of matrices

generated by the semi-norms II 8.1.5 and 7.3.9). However, IIAxII0 <- IIAII 11x1100

II

,

503

and pj (j E Nl°) (cf. 7.3.2(b),

II

and pj(x) < (1 + IIAII) IIx1I00

(x E m)

(9.4:3)

obviously holds and therefore the FK-topology of m n CA is generated by II III alone (cf. 6.4.13). We now reflect on the (non-)singular behaviour of finitely many regular

matrices A('),..., A(N). For this, provided it exists, let B be a regular matrix with cB DEN 1(m n cAU) ). From 9.4.6 we may conclude N

II

III

CB D E(m n C°AU))

(9.4:4)

j=1

where the closure is taken in (m, 11 11.). On account of the continuity of the matrix map

B : (mncB,11 II00) -+ (c, II II00), x --; Bx and of lim E c' we obtain limB Imf)ca E (m n CB, II II00)`. Moreover, N

II

II00

e 0 1:(mnC°Aw)) j=1

n CA(j) ). Otherwise we would have

follows from CB D

N

V e> 0 3 x(j) E m n c°A(J) (j E NN) : lie

- E x(j) 11 j=1

< C.

(9.4:5)

00

Therefore for every e > 0 and such x(j) we would have on the one hand (note, B and A(j) are b-consistent, see 2.6.12) N

N

lime e - E lima x(?) = lima e - E limA(WW x(j) = limB e j=1

j=1

and on the other hand

lime e -

N N lima x(j)

j_1 N

N

lima

e - E x(j) j=1

<-

IIBII

e - ExU) j_1

-<

IIBII F

00

(cf. (9.4:5) for the last inequality). That would be a contradiction in the case of e :=

1 21W .

504

Consistency of matrix methods

Now, we can interpret the statement in (9.4:5) which is equivalent to II IE

N

e EE(mflcOAw) j=t

as follows. The sequence e, which has a `big' transform under a regular matrix (namely, a sequence with limit 1), can be approximated (relative to 11 III ) by bounded sequences with a `small' transform (namely, by jN -1 x(j) with x(j) E m fl coac>> ) This behaviour becomes even clearer if we consider the statement V e > 0 3x(j) EmflcAw (jENN) N

N

<e and EIlimA(j)x(j)I <E,

e - E XU) j=1

j=1

00

which is obviously weaker than (9.4:5) and which is equivalent to (9.4:5) as we will verify in the next theorem. In such a situation we call the behaviour , A(N) `singular': of regular matrices A(1),

Definition and Theorem 9.4.7 (singularity S1). Let A('),..., A(N) be regular matrices. By definition, they have a singularity S1 (briefly, (A(2)) E S1), if V e> 0 3

E m fl CA{i)(j(j E NN) N

N

e - Ex(j) j=1

< e and E IlimAM X(j) I < E (9.4:6) j=1

00

is satisfied. Then the following statements hold:

(a) (AU)) E S1 4=f e E

EN1(m fl coAw )

plf

(b) If (AU)) 0 Si, then A('),..., A(N) are simultaneously b-consistent. (c) If there exists a regular matrix B with CB D EN1(m fl CAM), then (A(j)) 0 S1. Proof. (a) The implication '4--' is true since (9.4:6) follows obviously from (9.4:5). Conversely, if (AU)) E Si, that is (9.4:6) holds, then for a given e > 0 we may choose sequences x(j) E m fl CAM;) (j E NN) with N

e-E x(j) j=1

< 00

E

N and

E IlimAC,) X(j) I < j=1

2

(9.4:7)

Putting aj := limAU) x(j) (j E NN) we have for every j E NN the identity

505

Singularities of matrices X(j)

= y(j) + aje for a suitably chosen y(j) E m fl COA{j}

(9.4:8)

because A(j) is regular. Hence, we get the estimation N

N

N

e - Ey(j) j=1

<

e - EX(j)

+ E (x(j) - y(j)) j=1

j=1

Ca

N

<

2 + E aj < e

[by (9.4:7) and (9.4:8)];

j=1

that is, y(j) (j E NN) satisfy the condition (9.4:5) and ` .' is proved. (b) is an immediate consequence of (a) and 9.4.5(b). (c) Using (a), (c) follows from the discussion prior to this theorem.

13

The following characterization of S1 in the special case of two regular matrices will be useful in connection with the promised example.

Theorem 9.4.8 (S1 for two matrices). If A and B are regular matrices, then A and B have a singularity S1 if and only if

be>0 3xEmflcA 2yEmflcB : lix-yjj,,.<e and ilimAx-limByI>1.

(9.4:9)

Proof. Since it is a good exercise, the proof is left to the reader (cf. 9.4.15).

0 By the following example we show that the converse implication in 9.4.7(b) is not true in general. Then, by Theorem 9.4.7(c), it is also proved that the simultaneous b-consistency of matrices A(1), , A(N) is not sufficient for the existence of a regular matrix B with cB D EN j=1 m fl cA{j>. The example is due to G. G. Lorentz and K. Zeller (1958). It is highly nontrivial and its construction requires careful working since the properties which are equivalent to N

N

e f 57 (m fl coA{j}) and e E 1: (m fl cOAtj}) j=1

(9.4:10)

j=1

lie very closely together. Lorentz and Zeller described that situation as follows. For the construction of (b-consistent regular) matrices A and B (with (A, B) E S1) we make use of an idea of Petersen (cf. [195]) to find pairs of consistent matrices which are `embryonicly almost b-inconsistent',

and then we `condense' this property to `almost b-inconsistency' in the sense of (9.4:10) (cf. [153, p. 429]).

Consistency of matrix methods

506

Example 9.4.9 ((A, B) E Sl , A and B b-consistent).

We start

our construction with the following (2 x 3)-matrices where 0 < p < 1 : A(p) :=

(1+

1+p

0

-P)

B(p) := (0

,

01

1 0-P)

The essential properties of A(p) and B(p) for our use are 11 A(P)

A(p)

0

1

1

1

= Ci) = B (p)

(9.4:11)

1

= ( 1 + P) and B(p)

p

= (P) .

(9.4:12)

0

Therefore, the (2 x 3)-matrices A(p) and B(p) have (in the sense of (9.4:9)) a'singularity Sl on a small scale' since in the case of a small p the 3-tuples (1, 0, 0) and (1, p, 0) have the distance p (they are close together) whereas the transform according to (9.4:12) gives a distance equal to one. To get an idea of the necessity of the following as well as the extensive and difficult construction we advise the reader to deal first with Exercise 9.4.16. Then it should become clear that a simple `condensation process' fails.

For a `refined condensation process' we modify A(p) and B(p) :

For all tt E N and p E N. let A'A be a (2p x 3)-matrix which contains each row of 2-PA(4-p) exactly p times. (We do not yet fix the arrangement of the rows.) Analogously, we define BPS` with 2-PB(4-F). Now, we define the matrix A = and analogously the matrix B = by the `block matrix' A2 0

A :_

0

0

0

0

0

0

0

0

0

Al

A2

0

0

0

0

0 0

0

All

0 AZ

A3

0

0

0

0

0 A16

A16

0

0 0 0 A16

0

...

0 0

...

A16

where the arrangement of the rows of Ay1, is not yet determined. (In de-

tail, in the ntn `row block' of A we have zero matrices and the blocks 2'.)

l..,An A2".

Since every Art' contains exactly two different rows, we may arrange the 2n .. , Asuch that each combination of the rows of rows of the blocks I A2"'.

2-' A(4-1),..., 2-nA(4-n) appears exactly once. (Note, there exist exactly 2n combinations and A is well-defined if we fix the order of the rows of the blocks in that way.)

Singularities of matrices

507

Now, we show that the matrices A and B have the desired properties. Applying 2.3.711 we verify the regularity of A; similarly we may prove that of B. Obviously A satisfies the column condition (Spo). Since each of the

matrices Apu has for each row the row sum 2-p and the row norm less than 2_p+1, A has in the pth row of the `block matrix' A (in every row) the row sum F,'_1 2_p and the row norm less than Ep_1 2-D+1, which implies that A satisfies (Zsl) and (Zn). Thus, A is regular. In the next step we prove (A, B) E Si by verifying (9.4:9). For this we arbitrarily fix an r E N and put a := 2-'+1 Then we define a bounded sequence x which consists oiltriples (0, 0, 0) and 2'(1, 0, 0), corresponding to the columns of the `block matrix' A, as follows (ef. Figure 9.4.1): If a `column block' of A contains a matrix A, , where p is suitably

chosen, then we put in x at the corresponding position the triple 2'(1, 0, 0) and otherwise (0, 0, 0).

On account of (9.4:5) and the definition of A and that of x for n > r the nth row block of A transforms the sequence x into a 2'2-tuple with the

constant value 1 + 4-', that is limA x = 1 + 4-'. Further, we define a sequence y E m by replacing in x the triple 2'(1, 0, 0) with 2'(1, 4-r, 0). Then, by (9.4:12), the matrix B sums y to 4-'. Hence, we have

IIx - yJI,, = 2-' < e and I limA x - limB yI = 1; that is, by 9.4.8, the matrices A and B have a singularity Si. It remains to verify that A and B are b-consistent. For this we partition into pairwise disjoint infinite subsets

Np :_ {k E No I

contains a column of Ay0 for a suitable p E N }

(p E N), and show (x3k+1 - x3k+2)3kEN,, E co

for each (xk) E CA and P E N.

(9.4:13)

We assume the existence of x = (xk) E CA and p E N with lim sup k-+co, 3kEN9

Ix3k+1 - x3k+2I

3>0

(9.4:14)

and choose an r E NO with 3r E Np

and

Ix3r+1 - x3r+2I >

2

(9.4:15)

Moreover, we determine n E N such that A2" hits the 3rt' (thus the (3r + i)th and the (3r + 2)th) column of A. (Note, Ay" sits in the nth column of the `block matrix' A.) By the order of the columns in the blocks Ai" , ... , An" there exist within then th row block of A exactly two columns of A which have different entries for k = 3r+1 and k = 3r+2. If they have

oa.od.oa.oa. I

I

I

I

n

s,

n

.a.oa.oaoao

Cl

I

f

i i i i iii i t t t t t t t t

ti

O V O y. O d. O e. I

I

I

O yi O

O y. O

.

O

I

I

7

avv. .

7 7 7 7 7 t t t t t t t t a7

O

N

O a. O d. O@ O y I

I

I

.LOQOTO

O

O

I

I

I

O

I

I

I

1

1

1

1

1

tttttttt

ea .e

t t t +

O

O

Fig. 8.4.1

Singularities of matrices 509

the row index yr and ur, respectively, then we obtain by the definition of Ap' the equalities

E avrkxk - E aµ,.kxk k

k

Dav,.k - ap.k)xk _ 8-P IX3r+1 - x3r+2! =

e

k

8-PS 2

'

Since on account of (9.4:14) there are infinitely many r E NO satisfying condition (9.4:15) we get 8-Pa lim sup r

E avkxk - E ap.kxk k

2

k

>0

contradicting x E cA. Quite similarly one may prove that (x3k - x3k+2)3kEN, E co for each (xk) E cB and p E N.

(9.4:16)

Now, let x = (xk) E m fl cA fl cB be given. By the statements (9.4:13) and (9.4:16) we get

E (avk - bk)xk -> 0 (v -* oo),

(9.4:17)

kEN,

since, if the with row of A sits inside the nth row of the `block matrix' A and A2' hits the 3kth row of A, the identity E (avk - bvk)xk

kEN,

2-P ((1 + 4-P - 4-P)x3k - 4-Px3k+1 - (1 - 4-p)x3k+2) or 2-P ((1 + 4-P)x3k - x3k+1 - 4-Px3k+2)

is satisfied. Further, for every v E N° we have 00

E

kfN1U...UN.

2-P = 2-r+i

l avk - bvkI < 2 p=r+1

Therefore, we obtain for each r > 1 and v E No the inequalities E(avk

- bvk)xk

k

<

(avk - bvk)xkI +

I

kEN,u...UN.

E

kN,u...UN.

(avk - bvk)xk

510

Consistency of matrix methods

E

(a.,k - bvk)xk

+ 2-r+lllxII,,.

kENiu...uN,.

That is, lim sup V

57(avk - b.,k)xk

= 0,

and thus limA X = limB x

k

by (9.4:17). Thus, the constructed matrices A and B are b-consistent and have a singularity S1. LL Knowing from 9.4.7(c) that (in the case of regular matrices) the condition (A(2)) 0 Si is necessary for the existence of a regular matrix B with (m n CAW ), the problem obviously arises whether the condition is also sufficient. This problem is still open. In the following we deal with the second question in 9.4.4: If A('),..., A(N) are given regular matrices, then we ask for conditions in terms of A(j) (j E NNV) which are necessary and sufficient for the existence of a regular matrix B with cB D

N

CB D

(p = (Ilk), 0 < µk Too , suitably chosen).

E (m, n CAM) j=1

As in the case of bounded sequences, we work first on the necessary conditions where we deduce one of them from a consistency theorem (cf.

9.3.3) and then a further one from a theorem on the p-comparison (cf. 9.3.2).

Theorem 9.4.10. Let B and A('),..., A(N) be regular matrices with N

cB 3 E(mu n CAW)

(p = (Ilk) , 0 < Ilk T oo , suitably chosen);

j=1

then there exists a A = (Ak), 0 < Ak T oo, such that AO),. .. , AM are simultaneously A-consistent.

Proof. Using 9.3.3 we may choose, by the assumptions, for every j E NN a sequence A = (Ak) with 0 < Ak t oo such that

mx C mu and

lima x = limA(j} x

(x E m,\ n cA(j) ),

where A can be determined independently of j E NN (cf. 9.2.4). Now, if N

0 = T'x(j) with j=1

x (1)E m,\n cA(j)

Singularities of matrices

511

is satisfied, then

N N N 0 = limB > x(j) = >2 limB x(?) = >2 limAC,) x(j) j=1

j=1

j_1

which proves the simultaneous A-consistency of A(1),

,

0

A(N)

In the case of regular matrices A(1), ... , A(N) the characterization of pcomparison given in 9.3.2 shows us a further necessary condition (in terms of A(1), , A(N)) for the existence of a regular matrix B with N

(p = (pk) , 0 < µk T oo , suitably chosen).

eB D E(m,, fl CA(,)) j_1

Namely, in such a case there exists a K > 0 with

(x E m and j E NN). (9.4:18) Now, motivated by the definition of the singularity S1, on account of IlBeII= = lima e = 1 we may guess that (under the assumptions) the seJJBxJJi < KII A(s)xII,

quence e cannot be `approximated' by a finite sum of bounded sequences x(j) with a `small' transform under A(j). We proceed with the definition of the (non-)singular behaviour that we sketched and the proof of the conjecture concerning it.

Definition and Theorem 9.4.11 (singularity S2). Let A('),..., A(N) be regular matrices. Then, by definition, they have the singularity S2 (in short, (A(i)) E S2), if

Ve>0 3x(),...,x(N)Em N

N

e - > x(j) j=1

<e

and 1: IIA(')x(i) II, <,E. j=1

00

The following statement holds:

If there exists a regular matrix B and a p = (µk) with 0 < pk T oo and cB D

E=1 (mu fl cA(,) ), then (A(?)) 0 S.

Proof. Under the assumption of the theorem we may choose a K > 0 such that (9.4:18) is satisfied. Assuming (A(3)) E S2, for any given s > 0 we may determine x(1), , x(N) E m such that N

N

e - E x(j) j=1

11

<e

and

(9.4:19)

>2 II A(j)x(j) II, < C. j=1

00

Then using (9.4:18) we get IjBetli

:

N

B

(e - E N x(i) j=1

+B

H ,

N

E x(j) j=1

,

512

Consistency of matrix methods

N <-

N + E 11 Bx(1) i

e - E XU)

IIBII

j=1

1100

;=1

N

<

IIBII e + K E IIA(J>x('i {J .

<

(IIBII + K) e.

In particular, if a :=

2

[cf. (9.4:18) and (9.4:19)1

1+x , we obtain IIBeII1 < 1 which contradicts

1 = lima e = IIBeIIt.

0

In the following theorem we show that the necessary conditions which we obtained in 9.4.10 and 9.4.11 are also sufficient for the existence of a regular matrix B with the desired properties.

Theorem 9.4.12 (singularity S2). If At'),... , A(N) are regular matrices, then the following statements are equivalent:

(a) There exists a regular matrix B and alt = (µk), 0 < Ilk t oo, with CB D E31, (ma fl CAM)-

(b) There exists a A = (Ak), 0 < Ak Too, such that A(),..., AW are simultaneously A-consistent.

(c) (A('}) 0 S2.

Proof . The implications `(a) * (b)' and `(a) = (c)' are just the statements in 9.4.10 and 9.4.11, respectively. To prove `-,(c) => -,(b)' and `(c) (a)' we refer to the book by G. M. Petersen (cf. [196, Theorems 5.1.3 and 5.2.3, 0 respectively]); these proofs are technically rather involved.

We now deduce a corollary from Theorem 9.4.12 and make some remarks on the relations between S1 and S2. Namely, on the basis of the result in 9.4.12 we may answer question 9.4.4. However, the answer is not very satisfying since we have to pass from the regular matrices A('),..., A(N) to suitable b-equivalent regular matrices B(1>, , B(N). Hence it is still

an open problem whether the existence of a regular matrix B with cB D

1(m n CAM) implies (A('}) f S2.

Corollary 9.4.13. If A('),..., A(N) are regular matrices, then the following statements are equivalent: (a) There exists a regular matrix B such that cB J E 1(m fl CA(S) ). (b) For each j E NN there exists a regular matrix B which is b-equivalent

to A('i such that (B(pi) 0 S2.

Proof. Noting the b-equivalence of A(l) and B('}, `(b) = (a)' is an immediate application of `(c) = (a)' in 9.4.12 to (B(J)).

Singularities of matrices

513

If B is chosen in accordance with (a), then for every j E NN we define B{'} _ (bnk) by means of B = and Air} = (an k) by (i}

bnk

1 ask

tbk .

for each k E No, if n = 2v (v N°) for each k E No, if n = 2v + 1 (v E 1J°).

One may easily check that for every j E NN the matrices AU and BU) are b-equivalent and that jjBxjj, < IIBi?}xII= (x E m) is satisfied. On account of `(f) (b)' in 9.3.2 and `(a) (c)' in 9.4.12 the last inequality implies

the statement (BU)) 0 S2; thus (b) holds.

0

By means of 9.4.12 and 9.4.13 and of a result of Copping, which we do not prove here, we are able to differentiate the singularities SI and S2.

Remarks 9.4.14 (S1 and S2 ). By the definitions we immediately get that regular matrices A('),..., A(N) which have a singularity Sl also have a singularity S2. The converse implication fails in general. Namely, by a result due to J. Copping (cf. [71, Theorem 2(ii)]) there exist b-equivalent regular matrices A and B which are not A-consistent (thus not simultaneously A-consistent) for each sequence A = (Ak), 0 < Ak ? 00; consequently, (A, B) E S2 by `(a) q (c)' in 9.4.12, whereas (A, B) Sl follows from 9.4.7(a) since e 0 m n COA = m n COATI N°° (by an argument as in 9.4.6) and since W FM coAll 11- = (m n COA) + (m n coB)II II°°

b-equivalent).

(note, A and B are A

We close this section with some remarks on the history and development of singularities of regular matrices. The idea of singularities and the questions concerning them are already contained in the early papers of G. M. Petersen (1956, 1959). The essential development of singularities of matrices started in 1958 with a decisive paper by G. G. Lorentz and K. Zeller [153], in which (in the case of two

matrices) the authors dealt with the singularities Sl and S2, gave the important example 9.4.9 and proved (essentially) Theorem 9.4.12. The further developments, which are connected with the names J. W. Baker and G. M. Petersen (1965-1967), had their starting point in the following two possibilities of extending the questions in 9.4.4: The question in 9.4.4 also makes sense in the (more general) case of a sequence of regular matrices Ate} (j E N). This idea resulted from the connection with a singularity S3 in the papers of J. W. Baker and G. M. Petersen (cf. [10]) to a statement corresponding to that in 9.4.13 and in connection with a singularity S4 in the paper [37] of Boos to a statement analogous to that in 9.4.12. A further interesting extension of the question in 9.4.4 arises if one

requires in addition, respectively, s V CB and s E cB for a fixed s E m\F?_1(mncA(,) ). This is connected with singularities S=(s), i E {1, 2,3, 4}, where s takes the role of e in the corresponding definitions.

514

Consistency of matrix methods

The essential development of `singularities of regular matrices' is sketched in [37] (cf. also [33]) from a consistent point of view with reference to the literature.

Exercise 9.4.15. Verify the statement in 9.4.8.

Exercise 9.4.16. For every 0 < p < 1 let the (2 x 3)-matrices A(p) and B(p) be defined as in 9.4.9. Show that the (diagonal block) matrices

/A(4-1)

0

A(4_2) C :_

and

A(4-3)

B(4-1)

0

B(4-2)

D :_

B(4-3)

0 are regular and not b-consistent. In particular, (A, B) E S1.

Exercise 9.4.17. Let AM,-, A(N) be given regular matrices such that (AU)) V S2, and let x E m with V e> 0

3X(1),.-.,x(v)Em N

N

x - E x(j) j=1

< s and E II A(j)x(j) II z <e j=1

00

be given. Prove the existence of a regular matrix B which satisfies N

x E cOB

and CB D E(mu fl CAM) j=1

for a suitably chosen p = (µk), 0 < Pk too. Bibliography: [196], [153], [267]; [37] and [193], [194], [10]

10

Saks spaces

and bounded domains In Section 2.6 we proved the bounded consistency theorem by applying the gliding hump method. In this chapter we will develop functional analytic tools which allow us to deduce the bounded consistency theorem among other results. However, we should note that we still cannot do this without applying gliding hump methods. We are in the situation where, for instance, we cannot apply the (general version of the) Banach-Steinhaus theorem. The bounded domain m n CA of a conservative matrix endowed with the supremum norm 11 11',, is a BKspace and the topology TAImncA on m fl cA induced by the FK-topology of CA is coarser than its BK-topology. Therefore the topological structure of CA disappears in the BK-topology T11 II on m fl CA. As a consequence

we have that, for example, in the case of a regular matrix A satisfying m fl cA $ c, the space cp ®e is not dense in (m n CA, 111 loo) since c is a closed subset of it. Moreover, in the case of a conservative matrix A with m fl cA # c the BK-space m fl CA is not separable whereas the FK-space CA is separable. The way out of this situation is to consider the subspace mflWA (note that mflcA = (mflWA)(D (u) with a certain u E mflcA) and with the finest locally convex topology y on m fl WA which coincides with we call this a mixed topology and TA on the unit ball of (m fl WA, 11 we will show that it is finer than TAI mfWA and coarser than T11 III As we will also show, a further important property of ry is that the v-convergence of a sequence W")) in mflWA is equivalent to the (( II0-boundedness and TA-convergence of (x(1)). In this connection we speak about two-norm

convergence of (x(")). The idea of mixed topology goes back to A. Persson [192] and A. Wiweger [258] and is based on the consideration of two-norm convergence and

two-norm spaces, due to A. Alexiewicz [3] and to A. Alexiewicz and Z. Semadeni [4], and of Saks spaces, due to W. Orlicz [191]. The principal aim in these papers is to make available some kind of Banach-Steinhaus-type theorems.

In Section 10.1 we give a short introduction to the Saks space theory based on the consideration of mixed topologies on Saks spaces. We aim to

516

Saks spaces and bounded domains

establish a theorem of Banach-Steinhaus type in the case of Saks spaces and obtain some corollaries of it which are fundamental for the subsequent sections of this chapter. We apply, in Section 10.2, the results in Section 10.1. For any FK-space (E,TE) containing co we essentially prove that (m f1 WE, rE{mnwE, 1111".)

is a Saks space and that (m f1 WE, 7) is a separable locally convex space with sequentially weakly complete dual. Moreover, as an application of a result in this section we give a further proof of Theorem 8.5.3 which is not based on the gliding hump method. The principal aim of Section 10.3 is to show-applying the results in

Section 10.2-that in the case of a given FK-space E containing co the implication

mf1WECF=*m0WECWF holds for each separable FK-space F. With this we get a further proof of the bounded consistency theorem which is relatively `soft' in comparison to that in 9.1. The main subject of Section 9.3 is the comparison of la-bounded domains

of regular matrices A and B in terms of the matrices A and B and also by continuity statements for the matrix map B : mµ nCA -+ c. Moreover, we characterized m , n CA C cB by quotient representations of the type B = CA + D where C is regular and D is `small' in a certain sense. Our considerations in Sections 10.2 and 10.3 enable us to compare in Section 10.4 the bounded domains of regular matrices in a similar way.

10.1

Saks spaces and mixed topologies

In this section we introduce the notion of Saks spaces endowed with the mixed topology and the notion of two-norm convergence. Moreover, we will show that-under certain hypotheses-a theorem of Banach-Steinhaus type holds. As corollaries we than deduce for Saks spaces a closed graph theorem, the weak sequential completeness of the dual space and, if the space is separable, the statement that the mixed topology is just the Mackey topology. We follow the presentation of these ideas by J. B. Cooper in the first chapter of his book [69] which is motivated by the papers of A. Alexiewicz on two-norm spaces, of W. Orlicz [191] on Saks spaces and of A. Persson [192] and A. Wiweger [258] on mixed topologies.

Definition and Remark 10.1.1 (Saks space). Let G be a linear space and let

and r be a norm and a locally convex topology on G, respec-

tively.

(a) Then (G, 11 11, r) is called a Saks space if the unit ball B {x E G I f lxII < 1} is r-bounded and r-closed. If r is generated by a certain family P of semi-norms, then we also write (G, 11 11, P) instead of (G, 1111, r).

If, in addition, r is metrizable, that is r is generated by a

para-norm ! !, then (G, 1111,! !) is called a two-norm space.

Saks spaces and mixed topologies

517

(b) If (G, 111 I,r) is a Saks space, then the strongest locally convex topology

T on G such that FIB = TI B is called the mixed topology (on G by virtue of II II and r); we denote it by ¶ or, more precisely, by ¶G. (c) Let Bi := 2i-1B (i E N) and let U, be a neighbourhood basis of zero in (G, r) consisting of absolutely convex sets. Then (cf. [69])

U := UE(UiflBs) I UiEU,(iEIN)} n=1 i=1

JJJ

and

U7:= j Ua fl n (Un + Bn) I Un E U, (n E NO) } n=1

are neighbourhood bases of zero in (G, y).

11

We do not prove the statements in (c), but we give an example which illustrates the nature of mixed topologies.

Example 10.1.2. Let (m, 11 I j E 1N°)) with ii(x) := Ek=1lxkI (x = (xk) E m) be given. Then (qi I j E 1N°) generates r4, on m and

U:_{Ui1n:_{xEmI gj(x)
(m, 11 11., (4i I j E N°)) is a Saks space (and a two-norm space as well).

Following the notations in 10.1.1 we have

Bi = 2`-1B = {(xk) E m I Ixkl < 2`-1 (k E N°)}

(i E N).

The members U7 of U have, as one can show, the following form: U7 is an infinite-dimensional cuboid with finite edge lengths (some are small and the supremum taken over all is oo), whereas Ulan is an infinite-dimensional cuboid with at most finitely many edges with finite lengths. 0 We give here some characteristic properties of spaces with mixed topologies.

Properties and Definition 10.1.3. Let (G, II II, r) be a Saks space and ry the appropriate mixed topology. (a) r c y C i where rII It denotes the topology generated by II 11

II

(b) V(xn) in G bxEG : x,, -+ x (y)

xn -* x (r) and sup IIxnII < oo n

.

:

xn -+x('y).

If r is a metrizable locally convex topology, then the last convergence is called two-norm convergence, and we denote it as y-convergence.

518

Saks spaces and bounded domains

(c) A subset M of G is y-bounded if and only if it is II II-bounded. (d) A subset M of G is 7-compact if and only if it is II II-bounded and r-compact. (e) (G, 7) is sequentially complete if and only if (B,TIB) is sequentially complete.

(f) If (G, y) is barrelled, then r = y = T1l 11.

Proof. (a) By the definition of y we obviously have T C y. Further, we immediately get y C T11 from the r-boundedness of the unit ball B, if we write the r-boundedness of B in terms of r-continuous semi-norms. If xn -+ x (T) and sup,, IIxnII < oo, then x,, -r x (y) since (b) 11

obviously TI rB = 7IrB where r := 1 + sup,, IIxnII. ' ': Let xn -+ x ('y). Then x,, --3 x (r) because r C y. We assume that

supra IIxnII = oo. Without loss of generality we may assume that x = 0. Then we may choose for each i E N an ni E N such that Ilxn; II > 2i-1,

that is xn, V Bi = 2i-1B. Because Bi is -r-bounded, we may further choose for each i E N a Ui E UT with xni V Bi + 2Ui and Ui + Ui C Ui_1. Hence, for every k E N we obtain (cf. 10.1.1(c)) W

U((Ul nBl)+...+(UnnB,,))

U

n=1 00

U((Ul nBl)+...+(UnnBn)) n=k CO

C U(A+...+Bk-1)+(Uk+...+Un)) C Bk+2Uk Axnk. n=k

Thus xnk V U (k E N) which contradicts x,, -+ x (y) since U E ;F(c) If M is a II II-bounded subset of G, then it is also 77-bounded since 11. If M is not II II-bounded, then there exists a sequence (xn) in M such that IIxnII ? n2. Consequently, the sequence (n-1xn) is also not 1111bounded; thus it is not y-convergent by (b). Hence, M is not 77-bounded by 6.4.19. (d) This is an obvious consequence of TI B = 71B and part (c). (e) This immediately follows from (b) when we note that a sequence in G is a 7-Cauchy sequence if and only if it is II II-bounded and a T-Cauchy sequence (cf. Exercise 10.1.12). 0 (f) We refer the reader to [69, Proposition 1.14]. y C Tj1

The fact that (G, y) is not barrelled if r # y (cf. 10.1.3 (f)) motivates the examination of theorems of Banach-Steinhaus type for (G, y). For that, we characterize the equicontinuity of families of linear maps on (G, y). We start with a lemma due to A. Grothendieck.

Saks spaces and mixed topologies

519

Lemma 10.1.4. Let (E, r) and F be locally convex spaces, let X be an absolutely convex subset of E and T : E -+ F be a linear map. Then T I x is Tlx-continuous if and only if it is rix-continuous at zero.

Proof. Obviously, rIx-continuity at zero is necessary. Conversely, let Tix

be rIx-continuous at zero, x E X, and let V be an arbitrarily given absolutely convex neighbourhood of zero in F. We choose an absolutely

convex neighbourhood U in E such that T (X n a U) C i V. Then, if y E ((x + U) n X), we have x - y E X - X = 2X and T(2X n U) C V. 0 Thus T ((x + U) n X) C T (x) + V, that is T i x is continuous at x. Next, by preparing a B'anach-Steinhaus-type theorem in the case of Saks spaces, we characterize the equicontinuity of a family of linear maps on a Saks space.

Definition and Theorem 10.1.5. Let (G, 1111, r) be a Saks space and ry the appropriate mixed topology. By definition, a Saks space (G, (j

11,,r)

satisfies the condition (El) if

Vx0EB VUEU, 3VEU, : vnBC((xo+U)nB) - ((xo+U)nB) where U,. denotes a neighbourhood basis of zero in (G, r). Let (F,TF) be a locally convex space with neighbourhood basis UF of zero (consisting of absolutely convex sets), H be a set of linear maps from

G to F, and let HIB :_ {TSB I T E HI. Then the following statements are equivalent: (a) H is ry-equicontinuous. (b) HIB is TIB-equicontinuous. (c) HIB is rIB-equicontinuous at zero. If (G, 1111, r) satisfies the condition (E1), then we have the following additional equivalent statement:

(d)VWEUF3xoEB3UEU,VTEH: T((xo+U) nB) CT(xo) + W. Proof. Since FIB = 71B we obviously have `(a) a (b)

(d)'. Fur(c) ther, `(c) =:> (b)' follows from Grothendieck's lemma (cf. 10.1.4). To prove `(d) . (c)' we assume that (G,1111, r) satisfies (E1).

Let W E UF be given. Then, by (d), there exists an xo E B and a U E U, such that T((xo + U) n B) c T(xo) + 2W for each T E H. Because (G, 1111, r) satisfies (E1), we may choose a V E U, such that

V n B c ((xo + U) n B) - ((xo + U) n B). Hence we obtain

T(V n B) c T(((xo + U) n B) - ((xo + U) n B))

520

Saks spaces and bounded domains

c (T(xo) + ZW) - (T(xo) + ZW) = W. Therefore, HIB is rlB-equicontinuous at zero.

0

We are now ready to prove a theorem of Banach-Steinhaus type in the case of Saks spaces.

Theorem 10.1.6 (of Banach-Steinhaus type). Let (G, fl I(, r) be a Saks space satisfying (E1) such that (B,rlB) is a Baire space, let F be a Locally convex space and let (Tn) be a pointwise convergent sequence of ;7-continuous linear maps Tn : G -t F. Then (T.) is ;y-equicontinuous and

T: G -4 F, x -4 T(x) := limT,,(x) n is (linear and) 7-continuous.

Proof. The proof that the y-continuity of T follows from the yequicontinuity of (Tn) is straightforward and is left to the reader. To prove that (Tn) is y-equicontinuous, it is sufficient to show that 10.1.5(d) holds in the situation at hand. Now, let q be a continuous semi-norm on F, c > 0

be arbitrarily given, and let V4 := {y E F I q(y) < e). We have to find xo E B and U E Ll, such that T((xo + U) fl B) C T(xo) + VQ. For that we put

A,.:={xEB I Yn,s>r : q(T,(x)-Tn(x)) <4} (rEN). Since T8 - Tn is y-continuous and because B is r-closed, A, is r-closed for each r E N. Moreover, because (Tn) is pointwise convergent, it is a (pointwise) Cauchy sequence with respect to q; thus B = U,, A,.. Further, (B, rIB) is a Baire space; thus we may choose an no E N such that Ano is somewhere dense. Therefore, there exist an xo E Ano and a U E U, with (xo + U) fl B C Ano r = Ano and (since Tn is continuous)

q(TT(x) - Tn(xo)) < 4

(n < no and x E (xo + U) fl B ).

Moreover, for any x E (xo + U) n B and n > no we obtain q(Tn(x) - Tn(xo))

< q(T,(x) - Tno(x)) + q(Tno(x) - Tnu(xo)) + q(Tno(xo) - Tn(x0))

<

< s.

4

Hence Tn(x) E Tn(xo) +V9 for all x E (xo + U) fl B and n E N.

0

We draw three corollaries from this theorem of Banach-Steinhaus type.

Saks spaces and mixed topologies 521

Corollary 10.1.7. Let (G, 1111, r) be a Saks space as in 10.1.6. Then the space (G', v(G', G)) is sequentially complete where G' : = (G, y)'.

Proof. The statement is an immediate consequence of 10.1.6 since a sequence (f.) in G' is a a(G', G)-Cauchy sequence if and only if it is pointwise convergent. By 10.1.6 the pointwise limit f of (f,) is ;y-continuous,

that is f E G'. Obviously, fn -* f in (G', a (G', G)). Now, we apply Kalton's closed graph theorem (cf. 6.7.19) to the situation of Saks spaces. We only sketch the proofs of the following closed graph theorem and its corollary since we need tools which are not available in the context of this book.

Theorem 10.1.8 (closed graph).

If (G, 11 11, r) is a Saks space as in 10.1.6, then a linear map T from G into a separable Frechet space (F, TF) is -y-continuous whenever its graph is closed.

Proof. Let T : (G,7) --+ (F, TF) have a closed graph. Then T (G, r(G, G')) -+ (F,TF) has a closed graph too and we may apply Kalton's closed graph theorem (cf. 6.7.19) since (G', a(G', G)) is sequentially complete by 10.1.7. Thus T : (G, T(G, G')) -* (F, TF) is continuous, therefore weakly continuous (cf. 6.6.20(a)). Without loss of generality (cf. (12, p. 185 and p. 112]) we may assume that F is a (separable) Banach space with a Schauder basis: that is, there exists a sequence (xk) in F such that each x E F has the unique representation

x=I:akxk with akEK (kEN) k

(with respect to the norm of F) and such that for each k E N the projection on F, defined by x -> ak, is continuous. (Note, for instance, (co, II II.) is a Banach space with Schauder basis {ek I k E No } .) We now consider the sequence (Tn) where n

Tn:=PnoT with Pn:F-+F : x=Eakxk -*Eakxk (nEN). k

k=1

Obviously, (Tn) is pointwise convergent. Thus, by 10.1.6, the continuity of

T : (G,;7) -> (F, TF) is proved if Tn : (G, y) -4 (F, TF) is shown to be ;7-continuous (n E N). However, Pn is continuous and T is weakly continuous. Thus, Tn is weakly continuous. Hence, Tn : (G, y) --+ (F, TF) is continuous since dim Tn (G) < oo and all locally convex Hausdorff topologies on finite-dimensional spaces coincide (cf. 6.5.11).

Corollary 10.1.9. Let (G, II II, T) be a Saks space as in I0.1.6 and suppose that (G, y) is separable. Then (G, y) is a Mackey space, that is r(G, G') where G' = (G, y)'.

522

Saks spaces and bounded domains

Proof. We obviously have o(G,G') C C r(G,G'). Therefore, ry = r(G, G') if the identity map i : (G,77) -+ (G, r(G, G')) is continuous. (Note that i is obviously weakly continuous.) However (cf. [12, p. 185 and p.

112]), this is true if the inclusion map i : (G, y) -+ F is continuous where F denotes any Banach space containing G and inducing on G a topology weaker than r(G, G'). Using further known facts in functional analysis, we may conclude that i(G) (as the range of a separable locally convex space under a weakly continuous map) is a weakly separable, therefore a separable, subspace of F. Now, the continuity of i : (G,;7) -* F follows 13 from 10.1.8 since i has a closed graph as a weakly continuous map.

Exercise 10.1.10. Let (G, 11 jJ, ri) and (G, 11 II, r2) be Saks spaces with the same norm 11 11 and unit ball B, and let ; and ;2 be the corresponding mixed topologies. Verify that yi = y2 if and only if riIB = r2IB. Exercise 10.1.11. Prove that the mixed topology ry of the Saks space (m, 1{ 11., (q`j I j E N`°)) in Example 10.1.2 satisfies r,,, C -y C rr(II_.

Exercise 10.1.12. Let (G, !I 11, r) be a Saks space with mixed topology y, and let (x,,) be a sequence in G. Show that (xn) is a ;7-Cauchy sequence if and only if it is 11 11-bounded and a r-Cauchy sequence. Bibliography: [69]; [3], [4], [191], [192], [258]

10.2

The Saks space m fl WE

Motivated by and based on the papers of A. Alexiewicz and Z. Semadeni dealing with two-norm convergence, W. Orlicz [191] considered Saks spaces.

In particular, he proved by Saks space methods the well-known bounded consistency theorem due to S. Mazur, W. Orlicz and A. L. Brudno. Using the notion of mixed topologies and with the aim of giving the results of W. Orlicz a general functional analytic scope, G. Bennett and N. J. Kalton [26] investigated the space m fl WE for FK-spaces E containing co. They basically proved that the elements of m fl WE may be characterized by two-norm convergence of modified sections (cf. Theorem 10.2.3). As a consequence of their results, which we will present in Theorem 10.2.6, they obtained a theorem of Mazur-Orlicz type and the bounded consistency theorem. In this section we apply the results on- Saks spaces, presented in the

foregoing section, to the space m fl WE with the aim of obtaining the results of G. Bennett and N. J. Kalton that we mentioned above. We start with a characterization of those FK-spaces which contain co.

Theorem 10.2.1. Let E be an FK-space containing W. Then co C E if and only if Ef C P.

Proof. Let co C E and f E E. Then, since FK-topologies are monotone (cf. 7.3.4), f (co E c°'; thus (f (ek)) E Co' = 2. Hence, Ef C .

The Saks space m n WE 523

Conversely, let Ef C e. Then the inclusion map i

: W -f E, x -4 x

is a(W, E')-continuous, therefore also r(W, e)-T(cp, E')-continuous by 6.6.20(b). However, r11 II j,, = r(tp, t) since metrizable locally convex spaces are Mackey spaces (cf. 6.6.21) and since (gyp, jj = { f j,o 1 f E co } because {p is dense in the FK-space co. Thus, since E is complete, we may

continuously extend the inclusion map i : p -- E to a continuous linear map i : (co,jj jj.) -} (E,TE) where rE is the FK-topology of E. Now, i(x) = x for each x E co since convergence implies coordinatewise convergence. Hence, co C E and the theorem is proved. Using 'two-norm convergence' we obtain a characterization of those bounded sequences which are weakly sectionally convergent in an FKspace E. As an immediate corollary we obtain a result which says that cp is `dense by virtue of y-convergence' in the bounded null domain of any matrix A that is regular for null sequences since in this situation we have m fl COA = m fl WA. We prepare this result by the following lemma, which we prove by applying the Silverman-Toeplitz-type theorem 7.4.6.

Lemma 10.2.2. Let E be an FK-space containing co, let x = (xk) E m If x(n) --- x (y), then x(n) and X(n) = (xkn})k E m fl E (n E x (a(m, t)), that is xkyk = lim1: xkniyk for each (yk) E e. n

k

(10.2:1)

k

Proof. Since in FK-spaces convergence implies coordinatewise convergence, it is sufficient to prove the lemma in the case E = w. Let (Yk) E e be given. Because x(n) -+ x (y) we have (cf. 10.1.3(b)) M:= supllx(n)lloo < oo

and

n

xknl n-4

xk (k E N°).

To prove (10.2:1) we consider the matrix A = (ank) with ank := xkn (n, k E N1°). Then jlA11°O = supn,k jan,kj = M < 00 and A satisfies (Sp) where xk is the limit of the kth column. Then, by 7.4.6, e C CA and xkyk = limA y = lim E xkn}yk n

k

for each y = (Yk) E e,

k

which is (10.2:1).

Next, we characterize the members of an FK-space which are bounded and weakly sectionally convergent.

Theorem 10.2.3. If (E, rE) is an FK-space containing co, then for every x E m fl E the following statements are equivalent: (a) x E WE.

(b) 3 (x(')) in tp

:

x(n) -+ x (TE) and

(c) 3 (x(n)) in co

:

x(n)

x (y).

llao :5 jjxjjoo.

524 Saks spaces and bounded domains

Proof . Let x E m n WE. Since (cf. 6.6.13 z('1 - x (a(E, E')) implies x E cony {xln] k n E NO} re we may obviously choose a sequence (x(') ) in V satisfying the conditions in (b). The implication `(b) = (c)' is trivially satisfied. To prove `(c) = (a)', let x E m n E be arbitrarily given and let (x(' )) with x(n) = (xk"l )k be a sequence in co such that x(n) -+ x (y)

Further, let any f E E' be given. Then (f (ek)) E 2 since E D co (cf. 10.2.1). Thus

E f(ek)xk

f(ek)x(n)

= firm

k

k

by 10.2.2. However, because X(n) E co C WE and since x(n) -+ x (rE), we obtain

f

AX) = lim f (x(n)) = lim n

(ek)xkn)

k

= E f(ek)xk k

0

and therefore x E WE. In the remaining section we consider m n WE as a Saks space.

Theorem 10.2.4. Let (E, rE) be an FK-space containing co, and let r rE I mnWE Then

(a) (m n WE, II Iloo, r) is a Saks space satisfying (E1). (b) (m n WE, II II,o) is a BK-space.

Proof. (a) The unit ball B = {x E m n WE I

Ilxlloo 5 11 is -r-closed

since r-convergence implies coordinatewise convergence. To prove the r-

boundedness of B, let p be a rE-continuous semi-norm on E. Because FK-topologies are monotone and since co C E, we get p(x) < MIlxll00

for every x E Co.

In particular, p is also a continuous semi-norm on (co, Il Iloo). Thus

p(x) < Mllxll0

for every x E m n WE

because (cf. 10.2.3) for each x E mnWE we may choose a sequence (x(n) in W such that

x(n) _,y x (rE)

and

supllx(n)j!c" < IlxlIoo n

This proves (m n WE, Il II00, r) to be a Saks space. We now show that (m n WE, 11

110,r) satisfies (E1). For that let P = {p? I i E N} be a

The Saks space m n WE 525

countable set of semi-norms generating r such that pn < pn+l (n E N) and n

V n E N 3j n E N V x= (xk) E m n WE : E Ixk l< pjn (x). (10.2:2) k=1

(For (10.2:2) note that the coordinate functionals are continuous.) From the definition of a Saks space, we may choose for each j E N an

Mj > 0 such that pj (x) < Mj IIxIIoo (x E m n WE).

(10.2:3)

As usual we consider the basis of neighbourhoods of zero

U:={Uj 1 e>0 and jEN} with U£ :={xEmnWE I pj(x)<e}. Now, let x E B, j E N and e E ]R with 0 < e < 1 be given. Using 10.2.3 we may choose a v = (Vk) E V such that

pj(v - x) <

and

(10.2:4)

IIvD oo <- Ilxllao

We put n := min { k E No I vi = 0 (i > k) } and choose jn according to (10.2:2). Then, for jn and j we may choose an r E N such that pjn(y) < pr(y)

and pj(y) < pr(y)

(y E m n WE)

(10.2:5)

and an Mj > 1 according to (10.2:3). Let

V := U6 with 5 :=

6

4M,(1+v II III )

and w :_ (1- S)v.

We now prove

v n B C ((x+Ue)nB) - ((x+Ug)nB). For this let y E V n B be given and z := -y + to. Since vk = wk=0 (k > n) and

Iwkl=(1-5)Ivkl<(1-S)IIxII, <1-S

(k
(cf. (10.2:4)) and lykl <- pjn(y) < Pr (Y) < S

(k < n)

(cf. (10.2:2) and (10.2:5)) we obtain

IIzIIm = II-y+wll < 1, and thus z E B. Further we get (cf. (10.2:4) and (10.2:3)) pj(x - w)

<

pj(x - v) + pj(v - w)

526

Saks spaces and bounded domains

+ pi((1 - (1- 8))v) 4 + 8p,, (v) <

+ 8M? IIvII

2

and therefore (cf. (10.2:5))

pj(x - z) = pi(x-w+y) < pj(x - w) + pr(y) < 2 +5 < s,

that is x-w E UE and x-z E Uf. Since y=x-(x-w)-(x-(x-z)) the desired inclusion is proved. (b) We now show that m n WE is closed in (m, II which requires that (mnWE, II III) is a BK-space. For this, let (x(r)) be a sequence in mnWE

converging to an x in (m, II Iloo). Because r is weaker on m n WE than (x(r)) is a Cauchy sequence in the FK-space E, that is x E m n E. T1I III + Arguing as in the proof of '(c) #, (a)' in 10.2.3 we get x E WE.

Theorem 10.2.5. If (E,TE) is an FK-space containing co, then the following statements hold:

(a) (mnWE, 7) is separable. (b) (m n WE)' := (m n WE,7)' - P by virtue of the isomorphism

T:t -+ (mnWE)', (yk)--4g with g(x) := 57YkXk for each x E m n WE. k

Proof. (a) In Theorem 10.2.3, '(a)=:,,, (b)' implies that the sequences in 10.2.3(b) are obviously 7-convergent. Now, to prove that (m n WE, 7) is separable, we can proceed quite similarly as in the proof of 6.7.18. We leave the proof to the reader.

(b) We prove that T is an isomorphism. For this, let (Yk) E P be given. Because m n WE C m, the functional g is well-defined. Further 91B is sequentially TEIB-continuous because each TE-convergent sequence in B is also a(m, t)-convergent (cf. 10.2.2) and since g is obviously 0'(M, t)continuous. Thus gIB is TEIB-continuous since TEIB is metrizable. Hence g is 7-continuous (cf. 10.1.3(f) and 6.8.3) and T is well-defined. The map T is linear and injective since cp C mnWE and surjective because 7FI, C TU III

and co' - P by virtue off -* (f (ek)).

Now, we are able to apply the results in 10.1.7-10.1.9 to the Saks space

mnWE where E is an FK-space containing co. Theorem 10.2.6. Let E be an FK-space containing co. Then the following statements hold: (a) (t, a (t, m n WE)) is sequentially complete. (b) (m n WE, 7) is complete. (c) 7 = 7(m n WE, 2).

A theorem of Mazur-Orlicz type 527

Proof. (a) This statement follows immediately from 10.2.5(b) and 10.1.7. (b) On account of 10.1.3(b) the space (mnWE, y) is complete if and only if (B, TI B) is complete. However, the last statement is true since B is closed in the FK-space E, see the proof of 10.2.4(b). (c) This is an obvious corollary of 10.2.5(a), 10.2.5(b) and 10.1.9. r(mnWE, f), cf. Theorem 10.2.6(c), Using essentially the fact that we obtain a result closely related to Corollary 8.5.4.

Theorem 10.2.7. If (E, rE) is any FK-space containing co, then the following statements are equivalent: (a) co is closed in (E,TE).

(b) co = WE.

(c) co =mnWE. (d) (m n WE, r(m n WE, P)) is barrelled. Proof. `(a) (b)' holds since co C WE C iT C co follows from (a) and (7.2:9). Obviously, (b) implies (c). If (c) is satisfied, then we have r(m n WE, e) = r(co,1) = T11 I_. Hence, (m n WE, r(m n WE, B)) is a BK-space, and thus barrelled (cf. 6.8.3). Thus, (d) holds. Now, let (d) be true. Then, since 7 = r(m n WE, 1), we get TEI mnwE = 'Y = 711 11= by 10.1.3(f). Thus, since co C WE and (co, II 1I<) is a BK-space, co is closed in (E,TE). As a corollary we get Theorem 8.5.3 which we proved in Section 8.5 by means of the gliding hump method.

Corollary 10.2.8 (cf. Theorem 8.5.3). If E is an FK-space and if con E is not closed in E, then E contains bounded divergent sequences, that

is (mnE)\c

0.

Proof. Since co n E is not closed in E, it follows from 7.3.23 that co is not closed in the FK-space co + E. Thus, by 10.2.7, co C m n Wco+E from which it follows that E contains a bounded divergent sequence.

Exercise 10.2.9. Let (E, TE) be an FK-space containing cp and let K C m n WE. Prove that K is relatively T(m n WE, 2)-compact, if and only if it is jj III-bounded and relatively rE-compact.

Exercise 10.2.10. Let A be a matrix which is conservative for null sequences and let B be a matrix such that m n WA C cB. Show that limB is a(m n WA, t)-continuous on m n WA. Bibliography: [69], [191], [26], [120]

10.3

A theorem of Mazur-Orlicz type

Applying Theorem 9.1.7, which is a theorem of Mazur-Orlicz type, to X

m we get the particular result that the implication

528

Saks spaces and bounded domains

mnWEccB

mnWECAB

(10.3:1)

holds for each matrix B and any FK-space E containing V. Now, using the following simple proposition, which is a connecting link between summability and functional analysis, we get an idea of how to prove that, in the case of FK-spaces E containing cc, the implication in (10.3:1) remains true if

we replace AB by the smaller space WB. This is of interest since this method of proof is relatively `soft' in comparison to that of 9.1.11. (Note, we made use of 9.1.11 in the proof of 9.1.4 and 9.1.7.)

Proposition 10.3.1. For any sequence space Y containing W the following statements are equivalent: (a) (Ya, a(Y', Y)) is sequentially complete. (b) Y C CB implies Y C AB for each matrix B.

Proof. If (a) holds and B = (bnk) is a matrix with Y C CB, then b(") (bnk)k E Y'3 since Y C wB and (b(1)) is a o(Y'3,Y)-Cauchy sequence because Y C CB. Thus, by (a), the sequence (b(")) is o(Y3, Y)-convergent to a certain b = (bk) E Y' . Since cp C Y, the topology o(Y'3, Y) is a K-topology which implies that bk = lim bnk (k E N°) n

and

liras x =

bkxk (x _ (xk) E Y). k

Conversely, let (b) be satisfied and let (b(n)) with b(') := (bnk)k be a o(Ya, Y)-Cauchy sequence in Y'3. In particular, (b(n)) is coordinatewise convergent, say to b = (bk) E w, since o(Y'3,Y) is a K-topology. Then the matrix B = (bnk) fulfils Y C cB and bk is the (existent) limit of the kth column of B. Hence, by (b), we obtain Y C AB , that is b E YO and bnkxk

n

k

bkxk

for every x = (2k) E Y

k

which is the o(Y'3, Y)-convergence of (b(n)) in YO.

O

Applying Proposition 10.3.1 to Y := m n WE, where E is an FK-space containing co, we get independently of the discussion in Section 10.3 the second statement (that is, (10.3:1)) in Theorem 9.1.7 as a corollary:

Corollary 10.3.2. Let E be an FK-space containing CO. Then mnWE C CB implies m n WE C AB .

Proof. (mnWE)'3 = P, and

is sequentially complete (cf. 0 10.2.6(a)). Hence, the statement follows from `(a) (b)' in 10.3.1.

Remark 10.3.3. Proceeding as in 2.6.10 (see also 9.1.8) we again obtain the consistency theorem 2.6.11 (see also 9.1.9) and as a corollary the bounded consistency theorem 2.6.12. Hence, we gave a further proof of the bounded consistency theorem. A

b-comparison through quotient representations

We now use further information on the Saks space (m n WE, II

529

r)

where r := TEI mnwE -namely, the fact that the mixed topology is the Mackey topology of the dual pair (m n WE, f) -to prove, as promised, that the implication in (10.3:1) remains true if we replace AB with the smaller space WB. More generally, we also replace CB with arbitrary separable FK-spaces.

Theorem 10.3.4. Let E be an FK-space containing co. Then m n WE C F

m n WE c WF

(10.3:2)

whenever (F,TF) is a separable FK-space.

Proof. Under the assumptions of the theorem we may apply Kalton's closed graph theorem (cf. 6.7.19) to the inclusion map

i : (m n WE, r(m n WE, 2)) --3 (F, rF), x - x.

Obviously, i has a closed graph since r(m n WE, t) and rF are Ktopologies, and it is consequently continuous by Kalton's closed graph theorem because (t, v(e, m n WE)) is sequentially complete (cf. 10.2.6(a)). Hence, i is weakly continuous (cf. 6.6.20(a)). Since (mnWE, o(mnWE, 2))

is obviously an SAK-space, we obtain by the weak continuity of i that X("] -) x (o (F, F')), that is x E WF for every x E in n WE. Exercise 10.3.5. Carry out the details for Remark 10.3.3. Bibliography: [26]

10.4 b-comparison through quotient representations As we mentioned in the introduction to this chapter, the main subject of this section is the comparison of p-bounded domains of regular matrices A and B in terms of the matrices A and B, by continuity statements for the matrix map B : m n cA -a c and by quotient representations of the type B = CA + D where C is regular and D is `small' in a certain sense. We start with a more general situation and characterize, in the case of an FK-space E containing co and a separable FK-space F, the inclusion m n WE C F by the comparison of two-norm convergence of sequences in W. This result is due to K. Zeller (cf. [259, Satz 10.6]) in the case of bounded domains of regular matrices and to Boos (cf. [36]) in the general case.

Theorem 10.4.1. Let (E, TE) and (F, TF) be FK-spaces containing co,

and let yE and yF be the mixed topologies of the Saks space (m n 11., rE) and (m n WF,11 11., rF), respectively. If F is separable, then the following statements are equivalent: WE, 1I

(a) mnWECF.

530

Saks spaces and bounded domains

(b) m n WE c WF. (c) V (x(")) in cp : x(") --3 0 (7E) In each case, 7E is finer than

x(") -+ 0 (ryF).

Proof. (a) (b)

(b) : This is an immediate consequence of Theorem 10.3.4. (c) : If m n WE C WF, then the inclusion map

is (mnWE,o(mnWE,P)) --4 (mnWF,a(mnWF,8)), x -4 x is obviously continuous. Because, cf. 10.2.6(c), yE = r(m n WE,1) and yF = r(m n WF, t), the inclusion map i is also 'yEJyF-continuous (cf. 6.6.20(b)). This proves the additional statement, and the statement (c) now follows from the fact that y- and 7-convergence of sequences are equivalent (cf. 10.1.3(b)). (c) (a) : Let rE and rF be generated, by the sequences (p,,) and (q,,) of semi-norms, respectively. We assume that there exists an x E m n WE

such that x F. By 10.2.3 we may choose a sequence (x(")) in v with x(") -* x {yE). Because (x(")) is coordinatewise convergent and since F is complete, (x(")) is not a Cauchy sequence in (F, rF). In particular, there exists an no E N such that (q, (x("))) is not a Cauchy sequence. Therefore there exist an e > 0 and an index sequence (vk) such that qn,0 (x("k) - x("k+1)) > e

(k E N)

and

pi (x ("k) - x("}) < k

(j:5 k and v > vk, k E N).

Putting y(k) := x("k) - x("k+') (k E N) we obtain y(k) --Y 0 (yE) and y(k) __74 0 (-F); that is, statement (c) is not satisfied.

0

We remark that we make use of the hypothesis `F is separable' in part `(a) = (b)' only and that (b) and (c) are equivalent without this hypothesis.

Since conservative matrices A satisfy m n CA= (m n WA) ® u for a certain u E m n CA and because domains are separable FK-spaces we get the following characterization of the b-comparison of conservative matrices as an immediate corollary of 10.4.1.

Corollary 10.4.2. Let A and B be conservative matrices and let u E m n CA be chosen such that m n CA = (m n WA) ® u holds. Then the following statements are equivalent: (a) m n CA C CB.

(b) mnWACWB and uEcB. x(V) -* 0 (in). : x(") --* 0 (yA) Here yA and yB denote the two-norm convergence on mnWA and mnWE, respectively. (c) u E CB and b (x(")) in cp

b-comparison through quotient representations 531

Next, we give-the easy proof is left to the reader (cf. Exercise 10.4.6)for each regular matrix A a row- and column-finite regular matrix which is b-equivalent and b-consistent with A. This will be an essential tool in the later comparison theorem. Remark 10.4.3. Let A = (ank) be a regular matrix. Then we choose an index sequence such that 00

lank I <

V+1

(n < v),

and we define matrices T = (tin) and G = (gnk) by

tin and

gnk

1

ifkn
0

otherwise

(n E N)

Jank ifk>knandnEN°

l0

otherwise,

respectively. Then

H := T(A - G) + F with F := ding (2-n) is a regular triangle and satisfies

mncH=mncA and limAx=limHx (xEmncA), that is H and A are b-equivalent and b-consistent. IL We now formulate the theorem which gives us different possibilities for the b-comparison of regular matrices. The proof is non-trivial and is based on tools which we developed in the foregoing sections. Again we combine `soft' (functional analytic) with `hard' (analytic) methods.

Theorem 10.4.4. Let A and B be regular matrices, and, for A, let H be chosen according to the previous remark. Then the following statements are equivalent: (a) m n CA C CB-

(b) Ve > 0 35>0 bxEW :IlBxllo < e(bIlHxlloo+llxlloo). (c) For each e > 0 there exist matrices C and b such that B = CH + D, IIG11 < oo and 11DI1 < e.

(d) For each 5 > 0 there exist a_ row- and column-finite regular matrix C

and a matrix b such that B = CH + D and IIDII < 5. (e) For each e > 0 there exist a row- and column-finite regular matrix C and a matrix D satisfying B = CA + D and h(D) < E. (f) V s> 0 3 5> 0 Y x E m : IIBxllz <_ e (I IIAxII= + IIx11,,.)

Additional statement. In the case that A is a triangle, the theorem remains true if we replace the matrix H with A in (b), (c) and (d).

532

Saks spaces and bounded domains

First we give a. technical lemma and note that the proof of it is based on an idea which we used earlier in the proof of Theorem 9.3.2.

Lemma 10.4.5. Let A and B be regular matrices such that B = CA+D, IICII < oo and IIDII < $. Then there exist a regular matrix e and a matrix D with IIDII < (4+6IIBIJ) IIDII and B = CA+ Proof. First we construct a matrix C which is regular for null sequences and a matrix D with IIDII < 211D II and B = CA + D. Note, if D = 0, then-as we will prove-C is necessarily regular. As we have shown in the proof of 9.3.2 there exists for C = (with IICJJ < oo) an index sequence (nq)gENo such that for every q E No the submatrix (c,lµ)n>n.,µENo of C may be divided into finitely many submatrices C(q, t) = (c(q, t, r, p))r,14ENo

(t E Ntq , tq E No suitably chosen)

which satisfy the following properties:

(a) The set of the elements of the ut' column of C(q, t) has for each ,u E Nq a diameter less than or equal to q+i .

(;0) For all fixed q E N° and n E No, n > nq, there exist a t E Ntq and an r E No such that cnµ = c(q, t, r, p) for every p E NO. Since IIC(q,t)II < IICII < oo is satisfied for all t E Ntq and q > Q°, for every pair (q, t) with q > qo and t E Ntq there exists a conservative row submatrix of C(q, t) with (existing) column limits ru (q, t) (p E N°). The corresponding submatrices of D are also conservative because B = CA+D and since A and B are regular. Now, we define the matrices C = (cnu)

and D by

cn

cnu

c(q, t, n, it) - rµ (q, t)

if n < nqo if nq < n < nq+1, q >- qo

n, p E N° )

and b := (C - C)A + D. On account of ((3) the matrix C is regular for null sequences. Furthermore, for corresponding row submatrices C' of C and D' of D we haves

rk(B) _ (limo. e -

limo, eu) rk(A) + Erp(C')auk +rk(D') fff

K

for every k E No. By this and by rk(A) = rk(B) = 0 (k E N°) and Ek Irk(D')I < IIDII we get IIDII < II (C - C)AII + IIDII <- 21IDII 1 If H is a matrix with V C cH, then rk(H) denotes the (existing) limit of the Oh column.

b-comparison through quotient representations

533

Moreover, if IIDII > 0, we find that there is an no E No such that for every n > no, the inequalities

e

F, cn, 1-Eaµk) +EE2Fnayk-1

E (i_a) k

IA

IIDII

E bnk - 1

+ E Ijnk I

k

k

IIDII +211DII <- 3IIDII

(10.4:1)

hold. The last inequality holds since B is regular and because the matrix e, which is regular for null sequences, transforms the null sequence (1 - Ek auk),. into a null sequence. If D = 0, then we get from the first inequality that the row sums of e converge to one; hence e is regular. Therefore, for the remaining part of the proof we may assume IIDII > 0.

We define C and b by cnp cnK

cnv)-1

cnµ (Ep

if n < no if n > no

(n, ,u E N°)

C)A+ D. Because

and by

E cnv

> 1- 3IIDII > 2,

(10.4:2)

V

the matrix C is well-defined. Putting qn := F,, cn , for n > no we get, by (10.4:1), (10.4:2) and IIDII < 6, the inequalities EIdnkl

<-

1 1j IIaAII

+ IIDII

k

< 6 DII (IIDII + IIDII) + IIDII < (61IB11 + 4)IIDII.

Noting that the coefficients of b and D are equal for n < no and k E No we conclude IIDII < (4 + 6IIBII)IIDII Thus, C and b have the desired properties.

Proof of 10.4.4. (f) = (a) : This implication is trivial, because it suffices to conclude m fl cOA C COB from (f). (We should note that x E m fl COA if and only if x E m and IIAxIIZ = 0.) (a) = (b) : Let m fl CA C CB be satisfied. By 10.4.3 there exists a regular triangle H with m fl cH = m fl CA. Thus CB D m fl cH = (m fl WH) ®(e). Consequently we obtain from Corollary 10.4.2 and the additional statement

in Theorem 10.4.1 that xiv> -+ 0 (7H) implies x(") -+ 0 (ryB) for any

534 Saks spaces and bounded domains

sequence (x(")) in m n WH. Since H is a triangle, TA is generated by II

Iloo o H (cf. 8.1.4(c)). Therefore for each s > 0 there exists a b > 0 such

that for each x E m n WH the conditions IIHxttoo < S and IIxIIoo 5 1 imply IIBxIIoo < 2. This gives

IIHxII. <_ 1 and iixll, < 1

IIBxlloo < 2 from

(10.4:3)

E

for each x E mnWH. Now, if we consider for any fixed s > 0 (and suitably chosen b > 0) the normed space (m n WH, a (II Iloo o H) + II Iloo), then by (10.4:3) the map B : m n WH -4 c yields the inequality IIBxIIoo <_

2

(31

IIHxIi. + IIxIIo) for any x E m n WH D W. (10.4:4)

(b)

. (c) : Let (b) be satisfied, that is (10.4:4) holds for each x E W. In particular, this inequality holds for the row functionals gn of B defined by gn(x) := Ek bnkxk (x E gyp, n E N°). From 6.5.7 and the Hahn-Banach theorem for each n E NO we get fn E cH and hn E m' such that

gn(x) = f(x) + hn (x) for each x E {p, Ifn(x)I <_

2s

IIHxIIoo (x E cH) and Ihn(x)I <_ 2 IIxIIoo (x E m).

On the one hand we obviously have bnkxk =

gn(x) = k

hn(ek)xk

fn(ek)xk + k

(x E tP)

(10.4:5)

k

for each n E N°. On the other hand, since fn E cH and H is a triangle, we have (see 8.1.8(ii))

A(X) = an limes x + t(n)Hx (x E CH) for certain an E K and t(n) E e with II t(n) II < E. . From that, and because

H is regular, we obtain

fn(ek) _

(n, k E N°).

and b = (dnk) by

Defining e = c

tµn) hpk

tµnl

(n, f& E N°)

and

dnk := hn(ek)

(rt, k ENO),

respectively, we have B = CH + D, II CII <- s < oo and II D II < 2 < that is C and b satisfy the conditions in (c).

b-comparison through quotient representations

(c) (d) : Let 6 > 0 with 2i4+61 choose matrices C and D such that

B=CH+D,

i

535

< s be given. Then by (c) we may

IICII
11,511 <

5

2(4+611B11)'

Applying Lemma 10.4.5 to this situation we obtain a regular matrix C and

a matrix r such that B = CH + D and s

IIDII 5 (4+6I1BII)11 DII < (4+6IIBII) 2(4+6IIB1I)

(Cnk) and We now switch from desired properties. Starting with ko and determine index sequences

2

to matrices C and b with the 0 and n-1 0 we inductively such that no > 0,

8

L Icnkl < 3v(1 + IIDII) k=O 00

s

b ICnkl < 31,(1 + IlHII)

(n > n,,)

(n <

We define e = (Cnk) by

c'+k

Cnk

if k,

0

otherwise

and


(n, k, v E N°)

and put b :_ (C - C) H + D. The matrix e is-as one may easily checkregular and column- and row-finite. Further we get

B=CH+D and IIDII 0 be given. Then we may choose

acolumn- and row-finite regular matrix C and a matrix b such that IIDII <,F and B = CH +A-Since all matrices occurring here have finite norm, we obtain (cf. Remark 10.4.3)

B = CH+D = C(T(A-G)+F)+D = (CT)A- (CT)G+CF+D. Putting C := CT and D :_ -(CT)G + CF + D, we see that C is a column- and row-finite regular matrix and B = CA + D. It remains to prove IIDII= < s. We certainly have IICFIIi < IIFiI=IICII = 0. Because

536

Saks spaces and bounded domains

P := CT is column-finite, for sufficiently large no and all n > no the integer fmax{ia E N I pnk = 0 f o r

ifpnk=0forallkENo

n is well-defined and lln

00 as n tends to oo. This gives 00

E

00

: E E IPnit9µk I= E IPnp I E Iapk I is

k

k

E IpI,

J+=µn

k=kµ+1

Jt=Jun

f-n

+1

#An

+1IIPII

for all n > no. Hence II (CT)GIIi = 0. Thus we have proved IIDIIi < II

(CT)GIIi + IICFIIL + IIDII < e, that is (e) is satisfied.

(e) (f): Let e > 0 be fixed and let C and D be chosen in accordance with (e). Then, because C is column-finite, we have for each x E m the inequalities IIBxIIi <_ IIC(Ax)IIi + IIDxIIi <_ IICII

IIAxII1 + E IIxII00

Therefore b := 1T' fulfils the condition in (f).

0

In closing this section, we remark that Boos and R. Neuser proved in [46] the b-equivalence of the regular Cesaro matrices C. (a > 0) by giving

an explicit quotient representation of the type C6 = QC,,,, + R where 1 < 3 < a. Using this idea, A. Tali (cf. [237] and [236]) characterized classes of generalized regular Norlund matrices which are b-equivalent.

Exercise 10.4.6. Give a detailed proof of Remark 10.4.3. Exercise 10.4.7. Let A and B be regular matrices such that there exists a matrix C with B = CA and IICII < oo. Show that there exists a regular matrix C with B = CA. Moreover, if A is of type M, then C is also regular.

Exercise 10.4.8. Let A and B be regular matrices. Prove that A and B are b-equivalent if and only if for each c > 0 there exist column- and row-finite regular matrices A and b with m fl cA C cA, m fl cB C cB and IIA - BII < E.

Bibliography: [36], [38], [13], [259], [237], [236]

10.5 Notes on Chapter 10 In Chapter 10 we used Saks space theory mainly to characterize the comparison of the bounded domain of regular matrix methods. As a corollary of it we got the bounded consistency theorem.

Notes on Chapter 10

537

Another topic of research during recent decades, concerning bounded domains of regular matrices, was the determination and the characterization of the factor sequences of regular matrices A (respectively, of their

bounded domains). A sequence y = (yk) is called a factor sequence (or multiplier) of (the bounded domain of) A, if yx E COA for each x E m fl ccA, that is y E (m fl coA, cOA). The set of all factor sequences of

A, which is an algebra, is called the factor algebra of A and is denoted by FA. The investigation of FA and the determination of its members in the case of special methods are connected, for example, with the names K. Zeller (cf. [262]), G. Brauer (cf. [50]), M. Henriksen (cf. [113] and also [114]), R. E. Atalla (cf. [7];181) and G. M. Petersen (cf. [197], [198], [199], [200]). As an effective tool Atalla and Henriksen used the Stone-Cech compactification 8N of N. They identify the space m of all bounded sequences with the space C(;8N) of all continuous functions on 1N, and m \ co with

C(,8N \ M. Then any matrix operator A : m - in, which is defined by a regular matrix A, induces a continuous operator A* : C(,8N) -* C(/3N). Now, applying general theory to A* and evaluating pointwise they get results on factor sequences and the factor algebra of the regular method A. Ohlenroth (cf. [189] 2), a doctoral student of Beekmann's, refined this method in his thesis and got a series of interesting results concerning the factor algebra of regular matrices. In particular, he gave a negative answer to a problem proposed by Brudno (cf. [53] and also [196, Problem 6]). J. Connor and J. Kline (cf. [65]) analogously applied the concept of Atalla, Henriksen and Ohlenroth in connection with statistical convergence.

In particular, in the case of two regular methods R and T with nonnegative entries they characterized the consistency of R-statistical convergence and T-statistical convergence. This result is reminiscent of the bounded consistency theorem. Both the thesis of Ohlenroth and the paper by Connor and Kline indicate that there are still more applications in summability and topological sequence spaces of the Atalla-Henriksen method based on the Stone-Cech compactification ON of N.

2 A copy of Ohlenroth's thesis is available from the author.

11

Some aspects of topological sequence spaces In this chapter we consider several aspects of sequence spaces which have their origin in summability or which are links between topological sequence spaces and summability. Some (see Sections 11.1, 11.2 and 11.4) are based on `inclusion theorems' where we show the equivalence of certain functional analytic properties of a sequence space Y with a statement about summability. For example, the sequential completeness of (YB,a(YO,Y)) is equivalent to `Y C cB implies Y C AB for any matrix B' (cf. Proposition 10.3.1). Concerning the foregoing implication, we stated in the MazurOrlicz-type theorem 9.1.7 that it is satisfied whenever Y has the SIGNED P_osc '. So, by the inclusion theorem, the SIGNED P_OSCP gives us both information about summability, namely the Mazur-Orlicz-type theorem, and information about topological sequence spaces, namely the statement that YO is weakly sequentially complete. In each case of inclusion theorems we proceed as follows. We prove for a sequence space the equivalence of a

certain functional analytic property and of a summability property and give a sufficient condition for the validity of the properties under consideration. The proofs of the inclusion theorems are done by functional analytic (modem) methods, whereas the proof of the sufficiency of the respective condition requires classical methods. In Section 11.3 we apply certain results from Section 11.2 to prove some theorems of Toeplitz-Silverman type. The starting point in Section 11.5 is Hahn's theorem 2.4.5 and, more generally, the investigations into potent matrices in Section 2.9. We extend the notion of potent matrices to the notion of `sequence spaces having the matrix Hahn property' and also investigate the stronger notion of `sequence spaces having the Hahn property'.

11.1

An inclusion theorem

The main aim of this section is to extend the inclusion theorem 10.3.1 by giving further equivalent statements. We apply Kalton's closed graph theorem 6.7.19. By that extension we find that 9.1.4 and 9.1.7 remain true if we replace AB by the smaller set WB. For consistency theorems this

An inclusion theorem

539

idea is irrelevant, but we obtain very general Mazur-Orlicz-type theorems which are of independent interest.

Lemma 11.1.1. Let Y be a sequence space containing W, (F, 7-F) be a K-space, and A : Y -> F be a matrix map. Then A: (Y, or (Y, YO)) -+ (F, r,,) is continuous and, hence, A : (Y, r(Y, YI)) -4 (F, TF) has a closed graph.

Proof. Let A : Y -4 F be a matrix map with defining matrix A = (ank). In particular, Y C WA, so (ank)k E Y's for each n E NO. Because of the closed graph lemma 6.7.12, the map A : (Y, T(Y, Y'3)) -* (F, TF) has a closed graph if A : (Y,a(Y,Yv)) -3 (F, -r,,) is continuous (and, hence, it also has a closed graph). Obviously, since r,, is the product topology, this is the case, if for every n E No the row functional

An :(Y,O(Y,YR))-;(K,I I), (xk)-a> ankxk k

is continuous. However, this is trivially true, since (ank)k E Yo.

0

Next we give the promised inclusion theorem which is due to Bennett and Kalton (cf. [26]).

Theorem 11.1.2 (inclusion theorem). For any sequence space Y containing (p the following statements are equivalent: (a) (Ya, o(Y'3, Y)) is sequentially complete.

(b) Each matrix map A : (Y,r(Y,Ye)) --3 F is continuous whenever F is a separable FK-space.

(c) The inclusion map i : (Y, r(Y, Y'3)) -3 F, x -t x is continuous for every separable FK-space F containing Y. (d) If F is any separable FK-space with Y C F, then Y C WF. (e) If B is any matrix with Y C CB, then Y C WB.

(f) If B is any matrix with Y C cB, then Y C AB . Proof. The implication `(f) = (a)' holds by Proposition 10.3.1. Further `(b) - (c)', `(d) - (e)' and `(e) = (f)' are obviously satisfied since the inclusion map is a matrix map, cB is a separable FK-space and WB C AB L) respectively. It remains to prove '(a) (b)' and `(c) = (d)'. (a) (b) : Let (Y'3, o(YR, Y)) be sequentially complete, and let (F, rF)

be a separable FK-space and A : (Y, r(Y, Ya)) -+ F be a matrix map.

Then the matrix map A : (Y,r(Y,Y0)) -+ (F,TF) is continuous by Kalton's closed graph theorem 6.7.19 as we now verify. By Lemma 11.1.1 the matrix map A : (Y, r(Y,Ye)) -3 (F,TF) has a closed graph. The dual Y' = (Y, r(Y, Y1))' equals the (3-dual Ys of Y up to the isomorphism (cf. 6.6.22(b))

T : Y3 -+ Y, z = (zk)

f., where

.,(y) :_ > zkyk (y = (yk) E Y). k

540

Some aspects of topological sequence spaces

Moreover, the weak topologies a(Ya, Y) and a(Y', Y) on Y,6 and Y', respectively, are generated by the semi-norms qy (y = (yk) E Y) defined by

qy(z) :_

=ifz(y)i (z=(zk)EYO );

therefore, (Y', a(Y', Y)) is sequentially complete since (Y,6, 0'(Y,6, Y)) is sequentially complete (cf. 11.1.5). Moreover, the image space is a separable FK-space. Thus, Kalton's closed graph theorem is applicable.

(c) = (d) : Let (c) be satisfied, and let F be a separable FK-space with Y C F. Now, if i : (Y,r(Y,Ye)) --) F, x -4 x is continuous, then it is weakly continuous by 6.6.20(a), that is i : (Y,a(Y,Ye)) -; (F,a(F,F')) is continuous. However, (Y,a(Y,Ye)) is an AK-space (cf. 7.2.16(a)); thus

by the (sequential) continuity we get for each x E Y from xl'1 _ x in (Y, a(Y, YO)) the statement xl") -+ x in (F, a(F, F')). That is, x E WF. This proves Y C WF. Theorem 11.1.2 enables us to extend the statement in Theorem 9.1.4.

Theorem 11.1.3 (of Mazur-Orlicz type). If Y is a sequence space containing cp and having the SIGNED P_OSCP, then Y C F implies Y C WF

for each separable FK-space F.

Proof. This is an immediate consequence of 11.1.2 and 9.1.4.

In this connection we also find a large class of sequence spaces with weakly sequentially complete fl-dual.

Theorem 11.1.4. If Y is a sequence space containing W and having the SIGNED P_OSCP, then (YS, a(Y'3, Y)) is sequentially complete.

Proof. Apply 10.3.1 and 9.1.4. For examples of sequence spaces with the SIGNED P_OSCP we refer to Section 9.1 (cf. 9.1.5, 9.1.6 and 9.1.16).

Exercise 11.1.5. Let (X, rX) and (Y, ry) be locally convex Hausdorff spaces which are topologically isomorphic. Prove that (X, rX) is sequentially complete if and only if (Y, Ty) is. Exercise 11.1.6. Show that the inclusion theorem 11.1.2 falls if we replace WB by LB in 11.1.2(e).

Bibliography: [27], [40]

11.2

Gliding hump and oscillating properties

In this section we deal mainly with the question whether Theorem 11.1.3 remains true if we replace Wp by the smaller set SF. In general-as Example 11.2.1 shows-the answer is negative. So the following problems arise:

Gliding hump and oscillating properties 541

Find a class-as large as possible-of sequence spaces Y such that Y C F implies Y C SF whenever F is a separable FK-space. (11.2:1)

Give a characterization-similar to Theorem 11.1.2-of those sequence spaces Y which satisfy (11.2:1).

Example 11.2.1. Let A be a matrix with W C SAC WA C CA. For example, let A be the Zweier matrix (cf. 8.4.3). Then WA has the SIGNED P_oscP and CA is a separable FK-space. However, (11.2:1) does not hold since Y := WA ¢ SA and F:= CA is separable. A

Note, the foregoing example shows on the one hand that the SIGNED_OSCP is not sufficient for a sequence space Y to satisfy (11.2:1) and on the other hand that the inclusion theorem 11.1.2 fails in general if we try to extend it by adding (11.2:1) as a further equivalent statement. First we give a solution of the second problem.

Theorem 11.2.2. Let Y be a sequence space containing W. Then the following statements are equivalent: (a) (Y, r(Y, Y'3)) has AK and (Y'3, o (Y'3, Y)) is sequentially complete.

(b) If F is any separable FK-space with Y C F, then Y C SF. (c) If B is any matrix with Y C cB, then Y C SB. Proof . The implication `(b)

(c)' is obviously valid since domains cB are separable FK-spaces. (a) . (b) : Let YO be a(YR, Y)-sequentially complete and (Y, r(Y, Yj6))

be an AK-space. Then the map i : (Y, r(Y, YO)) --> (F, rF) , x -* x is continuous by Theorem 11.1.2 and hence i(xN"i) -* z (rF) for each x E Y. Thus Y C SF. (c) =;). (a) : Let (c) be valid. Then Y0 is o(Y16, Y)-sequentially complete by Theorem 11.1.2. We assume that (Y,r(Y,Y')) is not an AK-space. Thus, by the definition of the Mackey topology, we may choose an x E Y and an absolutely convex a(YR, Y)-compact subset K of YR such that pK (x1n] - x) -74 0 (n --> oo)

where the semi-norm PK is defined by pK(Z) Sup(nk)EK IEk akzkI (z = (zk) E Y). Consequently, there exist an index sequence (ni) and a sequence

(b(i)) in K such that bk=lxk I > 77 > 0 I

(i E N°).

(11.2:2)

k=n;+1

Since K is u(YR, Y)-compact, the weak topologies o(Y0, Y) and o(YR, gyp) coincide on K and a(YF, gyp) = r,, is metrizable. Hence we may assume that (bil)) is a(YO, Y)-convergent to a certain b E K. (Otherwise we switch

542

Some aspects of topological sequence spaces

to a subsequence of (b{')).) If B = (bik) denotes the matrix defined by bik := bkE> (i, k E N°), then the last assumption tells us Y C CB. From (11.2:2) we get x 0 SB which contradicts (c). In the next part of this section we aim to find a large class of sequence spaces Y such that the implication (11.2:1) is true by strengthening the SIGNED P_osCP. Since this implication is closely related to sectional convergence, because the nth section x1' J can be written as the coordinate-

wise product e1ri>x and-as in the case of weak sectional convergencedifferences of sections play an important role, it is natural to replace, in the definition of the SIGNED_OSCP, the step 1-block sequences by block se-

quences which take only the values 0 and 1. Such block sequences and the corresponding oscillation property are given in the following definitions.

Definition 11.2.3 (1-block sequence). A block sequence

(y(A))

is

called a 1-block sequence if yi?i E X for each j E N°.

Definition 11.2.4 (SIGNED P_01-OSCP). Let Y be a sequence space

containing V. Then, by definition, Y has the signed pointwise 01oscillating property (SIGNED P_01-oscP), if the definition of the SIGNED P_oscp is satisfied for 1-block sequences (instead of step 1-block sequences). That is, for each x E Y and every index sequence (k=) with k° = 0 there

exists a 1-block sequence (y{2>) with respect to (k1) such that for each subsequence of (y{2>) there exist a subsequence (y(i°>) and a sequence in S with yx E Y, where y := E,, h.,,yijY> (coordinatewise sum).

Definition and Remarks 11.2.5 (ABSOLUTE SP-GHP). Let Y be a sequence space containing {p. Then, by definition, Y has the absolute strong pointwise gliding hump property (ABSOLUTE SP_GHP), if for each x E Y and any block sequence (y(3)) in Y with sups IIyWfI ,, < oo and for each subsequence of (y(3)) there exists a subsequence (y(2 )) such in S we that for each subsequence (y(.)) of it and each sequence have yx E Y, where y := Ev h,,y{iv> (coordinatewise sum). If we understand, for example, by ABSOLUTE SP_GHP the set of all sequence spaces having the ABSOLUTE SP_GHP, then we have the following inclusions: (a) ABSOLUTE SP_GHP C ABSOLUTE SP_OSCP C SIGNED P_OSCP.

(b) ABSOLUTE SP_GHP C SIGNED P_GHP C SIGNED P_01-OSCP C SIGNED P_OSCP.

The proof is similar to that of 9.1.6 (cf. 11.2.14). Next we give simple classes-of sequence spaces having the ABSOLUTE SP_GHP and the SIGNED SP_OSCP respectively.

Definition and Remarks 11.2.6 (monotone spaces).

A sequence space Y is called monotone if ux E Y for all u E X and x E Y. Obviously

Gliding hump and oscillating properties

543

each solid space is monotone, and m° is monotone but not solid. Further, as one may easily verify, each solid space containing cp has the ABSOLUTE SP_GHP, and each monotone space containing tp has both the ABSOLUTE SP_OSCP and the SIGNED P_01-OSCP.

The next theorem gives us a large class of sequence spaces having the ABSOLUTE SP_GHP.

Theorem 11.2.7. Let E be an FK-space containing cp. Then SE has the ABSOLUTE SP_GHP. In particular, if E is an FK-AK-space, then E has the ABSOLUTE SP_GHP.

Proof. From Theorem 7.6.3 and Exercise 7.6.10 we know that SE is an FK-AK-space. Further we may assume that the FK-topology of SE is generated by semi-norms (r E N and x E SE).

pr (r E N) such that p, (x) < pr+i(x)

(11.2:3)

Now, let x = (xk) E SE be given. Then

Sup pr ( xkek J -* 0 (n -3 co and r E N). v>n

(11.2:4)

k-n

Further let (y(J)) be a subsequence of a block sequence satisfying M sup? IIy(?}Ilav < oo. There exist index sequences (vj) and (µj) such that vj < 1Lj < vj+1 (j E N°) and JAj

yk }eki

y(2}

thus yk?) = 0 fork ¢ [vj,,uj].

k=vj

First we prove xy(j) -4 0 in E. For that end let r E N be given. Then $45

Pr(xy(3))

=

Pr

1: xkykJ}ek k=vj K

<

sUP Pr E xkek

K>v; (k=v;

yk?)

K

< M sup Pr E xkek K>v;

- yk+1

k=v;

?. 0

k=v;

by (11.2:4) which proves xy(j) -+ 0 in E. Now, since xy(j) -+ 0 in E we may choose a subsequence (y(M) of (y(J)) such that

2-r for all r E N.

544

Some

aspects of topological sequence spaces

Together with (11.2:3) this gives

pr (xy(j°)) < 2-" for all v > r and r E N. Hence the series E" h"xy(j°) converges in the FK-space SE . Thus yx E SE , where y := >" h"y(j-) and (h,,) E S. This obviously remains true for any subsequence of (xy(l°)). So SE has the ABSOLUTE SP_GHP.

0

For convenience we use in the following the notion of a strong subsequence (y(j,0) of a sequence (y(j)) in the sense that N \ { jk I k E N} is an infinite set.

Now, similar to our procedure in Section 9.1, we prove a 'nonsummability theorem'.

Theorem 11.2.8 (non-summability). Let B be a matrix with cp C CB and let x E cB \ SB be given. Then there exists an index sequence (1") such that for each strong subsequence (y(4')) of any 1-block sequence (yiµl) with respect to (lv) we have z := yx 0 cB where y := Ej hjy(.`j) (pointwise sum) and (hj) is an arbitrary sequence in S. We precede the proof with a useful characterization of those elements in a matrix domain which are sectionally convergent. The easy proof is left to the reader in Exercise 11.2.15.

Proposition 11.2.9. Let A = (ank) be a matrix with lp C cA and let x E cA. Then x E SA if and only if Ek ankxk converges uniformly for

nEN°. Proof of 11.2.8. We assume x E AB \ SB (otherwise we may apply Theorem 9.1.11). First we remark that, by Proposition 11.2.9, x 0 SB if and only if there exists an E > 0 such that for each v E No there exist index v; -1 bn, ; kxk > e. sequences (n"j) j and (J3"j) j with /3 o > v such that Ek=" Let e > 0 and the index sequences be chosen in that way. We put k1 := 0 and choose n1, k2 E N° with k2 > k1 such that k2 -1

0

Ibnk - bkIIxkI < 2-1 (n > nl)

and

E bnikxk

> C.

k=k1

k=0

Then we choose n2, k3 E No with k3 > k2 and n2 > n1 such that k3-1

k2-1

Ibnk - bk4 IxkI < 2-2 (n > n2) k=0

and

E bn2kxk >E k=k2

and

Ebnkxkl < 2-2 and k=I

<2-2 (n
Gliding hump and oscillating properties

545

Having chosen ni, .. , ni and k2i ... , ki+1i we choose ni+l, ki+2 E No with ki+2 > ki+1 and ni+1 > ni such that k;+1-1

ki+2-1

Ibnk - bkl Ixkl < 2-(i+1) (n > ni+1)

and

E bni+,kxk

> E

k=k;+i

k=0

(11.2:5)

and L

L

E bnkxk < 2-(+1) and k=I

bkxk <'2-('+') (n < ni, ki+2 < 1 < L). t4=1

Now, we put 1, := k2v-1 (v E N) and let (y(µ)) be a 1-block sequence with respect to and (y(µi)) be a strong subsequence of (y(µ)). Since it is a strong subsequence there exists an index sequence (jr) with uj, + 10 ,uj,+1 (r E N). For a representation of y(µ) (,u E N) we may choose an index sequence (vµ) with (µ) Ilk

= j1

if 1,,,, < k < 1,,,,+,

(p'k E M Thus, for y := Ej hjy(µj) (pointwise sum) where (hi) is any sequence in S we have {hj if1J (j EN) 0

otherwise

1

yk =

10

if k < 1,,,,1 or

-

< k < 1,,,,Jr}1 (r E N).

(11.2:6)

Using the same notation for Ai and AT as in the proof of 9.1.11 and k;+2-1

Bi := E bnikzk

00

and

Ci := E bn;kzk,

k=ki

k=ki+2

we get (Ai) E co, (A,) E c and (Ci) E co for z := yx similarly to the corresponding cases in the proof of 9.1.11. Thus, z cB is proved if (Bi) 0 c. But this follows from the following considerations: if r E N and i := 2vµ1r+1 - 2 then by (11.2:5) and (11.2:6) ki+2-1

IBi1=

k,+1-1

E bnikzk = 1hj,I 57 bnikxk

k=ki

k=ki

0 and if i := 1 then Bi = 0 for any r E N. On the basis of Theorems 11.2.2 and 11.2.8 we immediately get the following main result and some corollaries.

Theorem 11.2.10 (SIGNED P_01-oscP). Let E be a sequence space containing cp. If E has the SIGNED P_01-oscp, then E C F implies

E C SF for each separable FK-space F. In particular, this implication holds if F is the domain CB of any matrix B.

546

Some aspects of topological sequence spaces

Proof. In light of Theorem 11.2.2 it is sufficient to prove the theorem in the case of domains CB. Thus, let B be any matrix with E C cB. Then E C SB follows immediately from Theorem 11.2.8.

Corollary 11.2.11 (SIGNED P_01-oSCP). Let E be a sequence space containing V. If E has the SIGNED P_01-oSCP, then (En, v(E', E)) is sequentially complete and (E, r(E, EO)) has AK. Proof. Apply Theorems 11.2.10 and 11.2.2.

Corollary 11.2.12. Let Y be a sequence space and E be an FK-space with cp c Y fl E and F be a separable FK-space satisfying Y fl SE C F. Then Y fl SE C SF if Y has the SIGNED P_01-oSCP.

Proof. Use Corollary 11.2.10 and the fact that Y i SE has the SIGNED P_01-oscP (cf. 11.2.17).

Corollary 11.2.13. Let E be a separable FK-space containing W such that SEC WE. Then WE fails to have the SIGNED P_O1-oSCP.(whereas SE has the ABSOLUTE SP_GHP).

Proof. Apply Theorem 11.2.7 and Corollary 11.2.10.

Exercise 11.2.14. Prove the inclusions stated in the Remarks 11.2.5 and, moreover, that all of them are proper (cf. [39]).

Exercise 11.2.15. Prove.the statement in Proposition 11.2.9. Exercise 11.2.16. Show that bs has the SIGNED P_GHP but not the ABSOLUTE SP_GHP.

Exercise 11.2.17. Let Y, X be sequence spaces containing cp. Verify that Y fl X has the SIGNED P_01-oSCP if Y has the SIGNED P_01-oSCP and X has the ABSOLUTE SP_GHP.

Bibliography: [40], [27], [39], [221]

11.3

Theorems of Toeplitz-Silverman type via

sectional convergence and gliding hump properties In this section we use sectional convergence and the results of Section 11.2 to extend and prove some inclusion theorems of Toeplitz-Silverman type. This gives us a further method of proving such theorems and continues the investigations of Section 7.4 where we gave functional analytic proofs of theorems of this type. Here we make use of a combination of (classical) analytic and functional analytic methods.

The observation in Proposition 11.2.9 gives us a short proof of the following theorem which extends the Toeplitz-Silverman theorem characterizing the matrices which are conservative for null sequences (cf. 2.3.6 and 7.4.3).

Theorems of Toeplitz-Silverman type via sectional convergence and ...

547

Theorem 11.3.1 (CO C CA). For matrices' A = (ank) the following statements are equivalent: (a)

CO C CA.

(b) coCSA. (c)

W C CA and IIAII < oo.

Proof. The implication `(a)

(b)' comes from the AK-property of co

and the monotonicity of FK-topologies. (It follows also by Theorem 11.2.10 since co, as a solid space, obviously has the SIGNED P_01-osCP by 11.2.6.) Using standard estimates we may prove `(c) (a)' (cf. also the proof of 2.3.6).

(b) . (c) : Let co C SA. Thus O C CA. We assume IIAII = oo. Then we may choose a sequence (ni) in N° and index sequences (ai) and (0j)

with ai < /3j < ai+i (j E N°) such that' j2

(jEN°).

k=a;

Defining y = (yk) E co by i ?

Yk

sgn ant k

0

ifai
(k,jEN°)

otherwise

we get A;

Ian;kl ? j

an;kyk k=aj

(j E N).

k=as

Thus Ek ankyk does not converge uniformly for n E No which contradicts co C SA by Proposition 11.2.9.

Using the same method we also get a proof of a theorem containing a theorem of Hahn which we proved in 7.4.6. However, we should mention that the (purely functional analytic) proof of `(a) (c)', presented in Section 7.4, is more elegant.

Theorem 11.3.2 (1 C CA). For matrices A = (ank) the following statements are equivalent: (a)

t C CA.

(b) f C SA. (c)

548

Some aspects of topological sequence spaces

may choose sequences (ni) and (ks) in N° with k? < ki+i (j E N°) such that Ian,k,I ? j2

(j E N°).

Defining y = (Yk) E t by

yk :=

1 2sgnanikj if k = kg

to

otherwise

(k,j E N°)

we obviously get kj

L anikyk

= Ianjk,IT > 1 (j E N°).

k=ki

This gives us y 0 SA by 11.2.9 in contrast to 2 C SA.

0

Next we use this method to re-prove both the (extended version of the) well-known Schur theorem (cf. 2.4.1 and 9.2.8) and the Hahn theorem (cf. 2.4.5). (Recall that the Schur theorem characterizes the matrices summing all bounded sequences, whereas the Hahn theorem tells us that a conservative matrix, which sums all sequences of zeros and ones, also sums all bounded sequences.) In contrast to the proof of `(a) = (b)' in the case of the last theorems, we can no longer make use of the AK-property since the spaces under consideration do not have AK. We can, however, use Theorem 11.2.10 as an substitute.

Theorem 11.3.3 (extended Schur theorem, theorem of Hahn). Let A = (ank) be a matrix with m C WA. Then the following statements are equivalent: (a) m C CA.

(b) m c SA. (c) 3 IL = (µk) , 0 < ILk / oo : m}, C CA.

(d) 3 p = (µk) , 0 < Ilk / 00 : m1 C SA. (e) X C CA , that is m° C CA. (f) X C SA , that is m° C SA. (g) V C CA and Ek lank I converges uniformly for n E No .

(h) c° C CA and lim sup. Ek lank - akl = 0 where ak denotes the limit of the kth column of A.

0) VCCA and 3.u= (14),0<11k

/00

Ek Ilk I ank I converges uniformly for n. E No . W) CO C CA and 3 IL = (ilk), 0 < Ilk T oo llmsupn Ek ILklank - akI = 0.

We can choose a common sequence IL in (c), (d), (i) and (j).

Theorems of Toeplitz-Silverman type via sectional convergence and ...

549

Proof. We are going to check the following chain of implications: (c) iW (d) {i* (b) iI* (a)

((e)

(6)

ii* (f)

(10)

W7

(g)

(h) {1*(j)

(i)

(c).

The implications (2), (3) and (4) and-see the proofs of 2.4.1 and 9.2.8the equivalences (7) and (9) are obviously true. The implications (1) and (5) are immediate corollaries of Theorem 11.2.10 since m,, and mo have the SIGNED P_01-oscP as monotone spaces (cf. 11.2.6 and 11.2.5). For a proof of (8) and (10) we refer to 9.2.8.

Now, we prove (6). For this we assume that A is a matrix with real entries. (In the general case of complex entries we note that Ek lankI con-

verges uniformly for n E N° if and only if this is true for both the real and the imaginary parts of ank.) Let (f) be true, that is X C SA; thus V C mo C SA. If Ek lankl does not converge uniformly for n E NO, then we may choose an 77 > 0, a sequence (n?) in No and index sequences (a,)

and (j3) with a? < j3j < aj+l such that a; Tl

(jEN).

k=aj

We define y = (Yk) E mo by ik

yk :_ {sgnan 0

if aJ < k < I33 otherwise

(k, j E N°).

Since Q,

an;kyk k=aj

_ , lan,kI >_ rJ

(jEN°)

k=aj

the series Ek ankyk do not converge uniformly for n E No. Therefore

0 y ¢ SA by 11.2.9 contradicting X C SA. We remark that the proof is really short and simple, if we ignore the statements (i) and (j). (Then we do not need 9.2.8 in the proof.) Exercise 11.3.4. Let A = (ank) be an infinite matrix with (ank)k E tq, where 1 < p, q < oo and 1 + q = 1, and V C cA. Prove the equivalence of the following statements >;y applying the same method as in the proofs of 11.3.1 and 11.3.2 (cf. also Exercise 7.4.11): (a) 2P C CA.

(b) 2p C SA. (C) supn ll(ank)kllq < 00

Exercise 11.3.5. Let A = (ank) be a conservative matrix. Show that A is strongly conservative if and only if fo C SA (cf. also Theorem 2.4.9).

Bibliography: [39]

550

11.4

Some aspects of topological sequence spaces

Barrelled' K-spaces

The inclusion theorems 11.1.2 and 11.2.2 fail if we replace separable FKspaces by FK-spaces as we can easily show by considering Y := m and

F := m since Sm = Wm = co. So we may ask for a characterization of those sequence spaces Y (containing gyp) satisfying the implication

YCF

Y C SF whenever F is an FK-space.

(11.4:1)

Further we ask for a large class of sequence spaces Y satisfying this implication. The characterization, to which we are aiming, leads to sequence spaces Y such that (Y, z(Y, Y$)) is barrelled. To handle this situation we need a characterization of compact subsets in K-spaces. For this latter characterization we need the notion of (convergent) nets in topological spaces.

Definitions and Remarks 11.4.1 (nets). Let X be a non-empty set

and I be a directed set. Then a map f : I -* X is called a net (in X). If we take I := N (with its natural ordering), we see that sequences are special nets in X. Analogously to the case of sequences we also use for nets f notation like (f(a))11EI, (x«)QEI and (xa). A subset J of a

directed set I is called cofinal if for each a E I there exists a 6 E J with a < 8. Correspondingly, if x = (xa)aEI is a net and J is a cofinal subset of I, then (xp)PEJ is called a cofinal subnet of x. In particular, the subsequences of a sequence are just the cofinal subnets of it.

Definitions and Remarks 11.4.2 (convergence of nets). Let (X, T) be a topological space and f = (xa)aE1 be a net in X. Then f is said to be convergent in (X, T), or relative to T, if

3xEX VUEU(x) 3aoEI ba>ao : xaEU. In this case x is called a limit of f and f is said to converge, or be convergent, to x (relative to 7). Obviously, each limit of (xa) is a limit of every cofinal subnet of (xa). The convergence of sequences (cf. 6.1.13) is a particular case of the convergence of nets. Further, if (X, -r) is semi-metrizable or semi-normable, and if (X, T) is a locally convex space, then we can handle the convergence of nets analogously to that of sequences (cf. Sections 6.2-6.4). 1

Definitions and Remarks 11.4.3 (adherent point of a net).

Let (X, z) be a Hausdorffspace and f = (xa )aEI be a net in X. A point x E X is called an adherent point of (xa) if each neighbourhood of x contains a cofinal subnet of (x,,). Obviously, each cluster point of a sequence is an adherent point of it. 0 Compact subsets can now be characterized by the existence of adherent points. We refer the reader to [133, §3,1.(5)].

Barrelled K-spaces 551

Proposition 11.4.4 (compact sets and nets). Let (X, r) be a topological space and 0 0 K C X. Then K is compact if and only if each net in K has an adherent point in K. The foregoing proposition enables us to give a handy characterization of compact subsets in K-spaces.

Proposition 11.4.5. Let (E, r) be a K-space containing W and let K C E. Then the following statements are equivalent:

(a) K is r-compact. (b) K is sequentially r-compact.

(c) K is

r and r give rise to the same convergent

sequences in K.

(d) K is

and every sequence (x(')) E K which is r,,-convergent to an x E w is r-convergent to x and x E K.

Proof. (a)

. (b) : Let K be r-compact and (x(')) be a sequence in K. Then K is compact, and thus sequentially compact relative to the weaker and metrizable topology r (cf. 6.2.25). Therefore, there exist a subse-

quence (x(' i)) of (x(' )) and a unique x E K with x(ns) -* x By hypothesis (x(n>>) has an adherent point y in K relative to r, which is also an adherent point relative to r,,. So we have y = x. For the same reason, x is the only possible adherent point of any subsequence of (x('1>)).

Thus, (x('i)) converges to x relative to r. This gives us that K is sequentially r-compact.

(b) .. (c) : The implication '(c) . (b)' is obviously true (cf. 6.7.3). Now, let K be sequentially r-compact. Then K is r is weaker than r, and because r., is metrizable. Since (E, r) is a K-space, -r-convergence of a sequence implies

Conversely, let (x(n))

be a r,-convergent sequence in K with limit x E K. Hence, any (convergent) subsequence of (x(n)) can have only the limit X. So, if (x(' )) were not r-convergent to x, by the sequential compactness, a subsequence would be r-convergent to y $ x, and this is not possible. (c) . (a) : We assume that K is r and r, give rise

to the same convergent sequences in K. To prove that K is r-compact, let (xt°})aE1 be a net in K. Then, since K is r,,-compact, there exists an adherent point x = (xk) of (x(a)) relative to r,,. For each n E NO we now consider

Kn :=

{xin1

I x E K} and W,, :_ {[n] ( x E w} = 10+1

Then, for each n E No, (x(a)In1),E/ is a net in Kn and, since Kn is compact, there exists an adherent point u(n) = (ukn))k E Kn of it. We put ylni := (uon>>

Unn>,

xn+1, xn+2,

) and get yin> E E since x E E and

552

Some aspects of topological sequence spaces

cp c E., We can pick out a subsequence of (Y(')) which is coordinatewise convergent to an x E w; we again denote the subsequence by (y()) . Let a E I be fixed. Since U(n) is an adherent point of (x(°)ln,)aEI there exists for every n E NO an fn E I with /3n+1 > )3n* and x(0n)1")

- uk") I <

n

for each k E No

Applying the definition of y(1) we obtain that I x(0"1) -

y(i,") I

<

1

n

for each k E No

and therefore x(an) -+ x (r(j. Hence, by the hypothesis we can conclude

x E K and x0n) -+ x(r). (Note, x is independent of a E I.) We show that 2 is an adherent point of (x(°)). For that let U be a r-neighbourhood of i. Then for each a E I there exists an n° E No such that x(9L) E U. Obviously, (x(Rna))°EI is a cofinal subset of (x{°)) included in U. Thus

x is an adherent point of ((°)) which proves that K is r-compact (cf. 11.4.4).

(c) q (d) : The proof is left to the reader in Exercise 11.4.16. In the following theorem we characterize those sequence spaces Y which satisfy the implication (11.4:1).

Theorem 11.4.6 (inclusion theorem). For any sequence space Y containing V the following statements are equivalent:

(a) (Y,r(Y,YF)) is barrelled. (b) Each matrix map A : (Y, r(Y,YR)) -* F is continuous whenever F is an FK-space. (c)The inclusion map i : (Y,r(Y,Y' )) -+ F, x -+ x is continuous for every FK-space F containing Y.

(d) If F is any FK-space with Y C F, then Y C WF. (e) If F is any FK-space with Y C F, then Y C SF. (f) If B is any matrix with Y C mB, then Y C WB. (g) If B is any matrix with Y C mB, then Y C AB . (h) If B is any matrix with Y C mB, then Y C SB.

Proof. The implications `(b) = (c)', `(d) = (f) #- (g)', `(h) = (f)' and (h)' holds since S,,,H = `(e) = (d)' are trivially true. Moreover, `(e) ScB =: SB (because CB is a closed subspace of MB)(b) : Let (Y, r(Y, YO)) be barrelled, let F be an FK-space and let (a)

A : (Y, r(Y, YO)) -* F be a matrix map. Then A has a closed graph by Lemma 11.1.1 and is consequently continuous by the closed graph theorem for barrelled spaces (cf. 6.8.7).

553

Barrelled K-spaces

(c) = (d) : If F is a sequence space containing Y and the inclusion map i : (Y,T(Y,YO)) -> F is continuous, then i is weakly continuous by Theorem 6.6.20(a). Thus Y C WF since (Y,o(Y,Y'3)) is an AK-space. (g) (a) : Let (g) be satisfied, and let K be a bounded and closed subset of (Ye, a(YO, Y)). We first show that K is a(YO, Y)-sequentially compact. To see this, let (b(n)) with b(n) = (br) be a sequence in K. By selecting a subsequence we may assume without loss of generality that lim bkn) =: bk n

exists for each k E No.

(11.4:2)

So, if we put bnk := bkn> (n,k E N°), then the matrix B = (bnk) obviously satisfies V C cB by (11.4:2) and Y C mB since (b(n)) is o(Yp,Y)bounded. By (g) we consequently have Y C AB, that is

b := (bk) E YQ and binl -> b (o(Y'3,Y)).

Thus b E K, since K is closed in (Ye, a(YO, Y)), and K is proved to be o(Ya, Y)-sequentially compact. Then, applying 11.4.5 and noting that (Y0, o(Y, YO)) is a K-space, we get that K is a(YO, Y)-compact. Now, by 6.6.15(e), we have T(Y,YR) =;Q(Y,Y'), so that (Y,T(Y,Y'3)) is barrelled (cf. 6.8.2).

(a)

(e) : Let (Y,T(Y,YO)) be barrelled and let F be an FK-space

containing Y. Then (Y,T(Y,YR)) is an SAK-space (since (Y, a(Y, YO)) has AK). Thus it is both an AD- and an AB-space (cf. 7.2.12). For each n E NO

let in : Y - F, x -i x1n]. Then in (n E N°) is continuous since F is a K-space and because i : (Y, 7(Y, Y')) -4 F is continuous as we have already proved. Moreover, (in) is pointwise bounded since (Y, 7-(Y, Y-8)) is

an AB-space and it is obviously pointwise convergent to i on cp which is dense in (Y, T(Y, YO)) because (Y, r(Y, YO)) has AD. Now, applying the general version of the Banach-Steinhaus theorem (cf. 6.8.6) and noting the K-property of F, we get that (in) converges pointwise to i. This means 0 xIni -> x for each x E Y, and thus Y C SF. As a corollary we obtain a result connected with Theorem 11.2.2.

Corollary 11.4.7. Let Y be a sequence space containing

gyp.

If

(Y, 7(Y, Ye)) is barrelled, then it is an AK space and (YO, a(YO, Y)) is sequentially complete.

Proof. Let (Y,T(Y;Y' )) be barrelled. Hence, the condition 11.4.6(h), thus 0 11.2.2(c), holds. Therefore the assertions follow from 11.2.2.

Corollary 11.4.8. Every barrelled SAK-space is an AK-space with a weakly sequentially complete dual space.

Proof. Let (Y, T) be a barrelled SAK-space. By applying the closed graph theorem for barrelled spaces (cf. 6.8.7) we may prove, as in the case of FK-

spaces, that Y' = YO up to the isomorphism defined by f -* (f (ek)).

554

Some aspects of topological sequence spaces

Therefore, since barrelled Hausdorff spaces carry the Mackey topology (cf. 6.8.2), we have r = r(Y, Y,6). So we are in the situation of Corollary 11.4.7 0 and the assertions follow.

The next corollary gives an answer to the question whether barrelledness of (Y, r(Y, YR)) characterizes those FK-spaces Y which have the AKproperty.

Corollary 11.4.9. Let (Y, r) be an FK-space containing V. Then (Y, r(Y, YO)) is barrelled if and only if (Y, r) is an FK-AK-space.

Proof. An FK-AK-space carries the Mackey topology r(Y, Y0) and FKspaces are barrelled (cf. 6.8.3). Conversely, if (Y, r(Y, Y13)) is barrelled, then

the inclusion map i : (Y, r(Y, Y0)) --- (Y, r) is closed, hence continuous by the closed graph theorem 6.8.7, that is r C r(Y, Y0). Since the converse inclusion holds for each FK-space, we have r = r(Y, Y0). Consequently, 0 (Y, r) is an AK-space by 11.4.8. In Theorem 11.1.4 we found that for any sequence space Y the SIGNED P_OscP is sufficient for the sequential completeness of (YO, o(V, Y)). We analogously stated in Corollary 11.2.11 that the SIGNED P_01-OSCP is sufficient for both the sequential completeness of (Ye, o(YF, Y)) and the AKproperty of (Y,r(Y,YF)). Moreover, we gave in 11.2.13 (cf. also 11.2.1) examples of sequence spaces which have the SIGNED P_OSCP but not the SIGNED P_01-oscP. Now, we define a property of sequence spaces Y which is due to D. J. H. Garling (cf. [90], [91]) and which forces (Y, T(Y, Y13)) to be barrelled.

Definition 11.4.10 (B- and Bo-invariant set). Let M be a nonempty subset of w and B := {y E by I Mb, < 1} be the unit ball of the BK-space (bv, I (bv), cf. 6.3.18. Then M is called B-invariant if

M=BM:={yx=(ykxk) fy=(yk)EB,x=(xk)EM}. Further, if Bo denotes the unit ball of the BK-space (bvo, I lbv), then M is called Bo-invariant if M = BOM. For some easy observations concerning B-invariant sets we refer to Exercise 11.4.17.

Theorem 11.4.11. If Y is a Bo-invariant sequence space containing cp, then (Y, r(Y, Ye)) is barrelled and, consequently (cf. 11.4.7), (Y, r(Y, YF)) is an AK-space and (YI, o(YR, Y)) is sequentially complete.

Proof. By 11.4.6 the barrelledness of (Y,r(Y,Y16)) is proved if Y C SF holds for each FK-space F containing Y. Let F be an FK-space with Y C F, and let x = (xk) E Y be given. Then, since Y = BOY, there exist z = (zk) E Bo and y = (yk) E Y such that x = zy. The map

T:byO-*F, v -+vy

Barrelled K -spaces

555

is well-defined, linear and continuous (as matrix map between FK-spaces). Thus, since bvo is an AK-space, we obtain

x[n] = z[nl y = T (z(n]) -i T(z) = zy = x in F

0

that is x E SF, hence Y C SF.

In Corollary 11.4.9 we saw that the FK-AK-spaces Y are just the FKspaces Y such that (Y, r(Y, Y'3)) is barrelled. Moreover, Theorem 11.4.11 tells us that each Bo-invariant FK-space is an AK-space. We now aim to show that each FK-AK-space is Bo-invariant.

Lemma 11.4.12. If E is an FK-space containing gyp, then SE C B°E. Proof. Suppose that x E SE. Let (pi) be an increasing sequence of seminorms generating the FK-topology of E. There exists an index sequence (nj) such that

pj(x-x[n]) < 4-(j+1) for n > nj (j E N°). Let z = (zi) be defined by

for 0 < i < no xi zi '- t 2?+1 xi for nj < i < n,-.F1

(i, j E N°).

If 1 E N and ni 1, then pt (z131

- z[r)) k-1

< pt (z(ni+11 _ z[r]) + t

pt

(z(' '+i]

- z[n"1) +pi

- z[nk])

(z[$1

v=j+i

< 2j+lpi (x[ni+i) - x[r]) +

k-1

2v+1pv (x[n.+i] -

/

)

v=j+1

+ 2k+lpk (x(s) - x[nk]) 00

< E 2-(v+l) i 0. v=j

Hence (z[i1) is a Cauchy sequence in E, which converges to z, since E

is an FK-space. Thus z E E. By the definition of z we have x E Bo {z}, therefore x E BOE. 0

Theorem 11.4.13. If E is an FK-AB-space, then B°E = SE. Proof. By Lemma 11.4.12 it remains to prove B°E C SE. Suppose that x = (xk) E E and y = (Yk) E Bo. We put z := xy and prove that (z[n]) is a Cauchy sequence in E. (Then we may conclude that (z[n]) converges

556

Some aspects of topological sequence spaces

since E is an FK-space; in fact, it converges to z = xy, hence zy E SE.) For this, let p be a continuous semi-norm on E. Then for all n, v E N we get n+v

P(z[n+vJ

E - z[n-31 } = p (Ykxkek) k=n /n+v

k

n-1

n+v

= P E (yk - yk+1) 1: xeµ - yn 1: xµeµ + yn+v+l E xeµ k=n

supp r

(E µ=0

µ=0

xµeµ)

/

µ=0

IyµI n (IYk - yk+iI + 2 Sup µ>n

µ=0

0

(k'='On

since E is an AB-space and y E B0 C co. Thus (zlnl) is a Cauchy sequence

in E. We now integrate the foregoing results into the promised characterization of the AK-property in the case of FK-spaces.

Theorem 11.4.14. For any FK-space (Y, r) containing p the following statements are equivalent: (a) (Y, r) is an AK-space. (b) 7 = r(Y, YO). (c) (Y, r(Y, Y11)) is barrelled. (d) Y is Bo-invariant.

Proof. The implications `(d) = (c)' and `(c) = (a)' are contained in 11.4.11 and 11.4.9, respectively; `(a) = (b)' holds, since FK-AK-spaces carry the Mackey-topology and Y' = YO (up to isomorphism). Further `(b) = (c)' is true because FK-spaces are barrelled. Thus it remains to prove (d)'. For this let Y be an FK-AK-space. Then, applying Theorem 11.4.13 and the AK-property, we get BoE = SE = E.

`(a)

Remark 11.4.15. Obviously, the (arbitrary union) of Bo-invariant subsets of w is Bo-invariant. However, the union of a strictly increasing sequence of FK-spaces is not an FK-space (cf. 7.3.12). So there are a lot of Bo-invariant sequence spaces which are not FK-spaces, for instance Up>1 ep = OF l 2P

Exercise 11.4.16. Prove the equivalence of (c) and (d) in 11.4.5.

Exercise 11.4.17. Let 0 0 M C w. Prove the following statements:

(a) BM = n {N C w I M C N and N is B-invariant} =: B*(M). (b) If M < w, then BM and B0M are not necessarily linear spaces. (c) B` (M) is B-invariant. (d) A sequence space Y is B-invariant if and only if BoY is a linear space.

The sequences of zeros and ones in a sequence space

557

Exercise 11.4.18. Prove the following statements for any Y < w : (a) Y/3 and Y7 are B-invariant, but not necessarily Bo-invariant as one may show with Y:= 1, that is Ys = m. (b) BoY7 = BoY'3 C Y'. (c) If Y is Bo-invariant, then YR = Y. (d) In general, the SIGNED P_01-OSCP does not imply Bo-invariance. (Whether the Bo-invariance implies the SIGNED P_01-OSCP is an open question.)

Exercise 11.4.19. Let Y be an FK-space containing V. Show the equivalence of the following statements.

(a) Y is an AB-space.

(b) BOY=SE. (c) Y is B-invariant. Bibliography: [27], [90], [133]

11.5

The sequences of zeros and ones in a sequence space

In this section we continue the study, which we started in Section 2.9, of the role of all sequences of zeros and ones in a sequence space. For that we generalize the notion of potent matrices to sequence spaces. Recall that a matrix A is called potent if x fl CA C CB implies m fl CA C CB for each matrix B. This leads in the next definition to the notion of the `matrix Hahn property' of sequence spaces when we replace the bounded domain m fl CA by any sequence space E.

Definition and Remarks 11.5.1 (matrix Hahn property). By definition, a sequence space E has the matrix Hahn property if XE (x fl E) C CB implies E C CB for any matrix B. (a) Obviously, a matrix A is potent if and only if m fl CA has the matrix Hahn property. (b) If E is a sequence space with the matrix Hahn property, then E C m. [Assume E Q m. Choose x E E\ m and y E t\ {x}'. Then the matrix B with y in the first row and zero otherwise satisfies x 0 wB D ca and cB D m D XE Thus E does not have the matrix Hahn property.] IL

Examples 11.5.2. (a) Obviously, p has the matrix Hahn property. However, none of the spaces co, bvo, fp (1 < p < oo) and bs has the matrix Hahn property. [If E is any of those spaces, then XB = V. Thus, if we take any sequence y f E-6 and consider the matrix B that has y in the first row and zero otherwise, then XE = p C cB, but E sr wB D cB.]

(b) c and by do not have the matrix Hahn property. (The proof is left to the reader in Exercise 11.5.11.)

558

Some aspects of topological sequence spaces

(c) fo does not have the matrix Hahn property. [The proof is non-trivial, and we omit it. (For the construction of a matrix B with Xfo C CB and fo ¢ co one may use [266, Satz 1].)]

(d) If A is a strongly conservative matrix, then m n CA has the matrix Hahn property (cf. 2.9.5).

Ek`opk -+ 00, (e) If p = (p,,) is a real sequence with pn > 0 and P then, by 3.2.13, the bounded domain m fl cR,, of the Riesz matrix Rp has the matrix Hahn property if and only if (p) E co. (Note, there exist regular Riesz matrices which are not strongly regular, cf. 3.2.20.)

IL

As we remarked in 2.9.4 we proved:

Theorem 11.5.3. Let E be a subspace of m such that XE is dense in (E, 11 11,,). If B is a matrix with IIBII < oo and if XE C CB, then E C cB. (For instance, IIBII < co if T C CB by 2.3.8.)

The last theorem gives us a large class of sequence spaces having the matrix Hahn property.

Theorem 11.5.4. If f C E < m, then E has the matrix Hahn property. Proof. The statement is equivalent to that in 2.9.7(c). The definition of the matrix Hahn property and the corresponding result

together with our discussion in Section 11.4 give rise to the question of what happens to the matrix Hahn property if we replace in its definition the domain cB with any separable FK-space or with any FK-space. In the following we discuss very basically the case of FK-spaces.

Definition and Remark 11.5.5 (Hahn property). By definition, a sequence space E has the Hahn property if XE C F implies E C F for any FK-space F.

(a) Obviously, if E has the Hahn property, then it also has the matrix Hahn property and thus it is a subspace of m. For an example of a sequence space that has the matrix Hahn property, but does not have the Hahn property, we refer to [25].

(b) If E is an FK-space with the Hahn property, then XE is dense in E, and thus dense in (E, 11Iloo). [Assume XE contains XE, but not E. ]

E; then XE is an FK-space that

(c) If {EQ I a E I} is a family of sequence spaces all having the (matrix) Hahn property, then so does E:= EQEI EQ := (UQE1 EQ) . A The foregoing definition and remarks raise the question of characterizing

the (matrix) Hahn property. We now discuss the characterization of the Hahn property.

Proposition 11.5.6. If E is a Banach (respectively F-)space and Y is a dense subspace of E, then the following statements are equivalent:

The sequences of zeros and ones in a sequence space

559

(a) Y is barrelled (as a subspace of E). (b) If F is a Banach (respectively F-)space and T : F -+ E is a continuous linear map with T(F) D Y, then T(F) = E. Because we will apply only `(b) #, (a)', we prove only this part. For a proof of the other part we refer to [27, Proposition 1].

Proof of Theorem 11.5.6, (b)

(a) : We give the proof only in the

case when E is a Banach space. (The same method gives the corresponding result for F-spaces.) We use Mahowald's characterization of barrelledness (cf. 6.8.8). For that, let G beg Banach space and suppose that T : Y -> G is a closed linear map. Then Y is barrelled by Mahowald's theorem if T is continuous. To prove the latter, we consider on Y the norm

II IIY : Y -* R, x ---4 IIxIIY := IIxIIE + IIT(x)IIG Obviously, IIxIIE G IIxIIY (x E Y). Now, let F be any completion of (Y, II

IIY) and let J : F -+ E be the unique continuous extension of the

identity map (Y, II IIY) - (Y, II IIE). We show that J is bijective. Since

J(F)

Y and Y is dense in E we have J(F) = E. Further, J is injective, as we now prove. Let x E F with J(x) = 0 be given. Then, since F is a completion of (Y, II IIY), there exists a sequence (xn) in Y with x,,, -> x in (F, II IIF) In particular, (xn,) and (T(xn)) are Cauchy sequences in (E, II IIE) and (G, II IIG), respectively. Therefore, since J(x) = 0, we have

IIxnIIE -; 0, and, because T is closed, IIT(xn)IIG -+ 0, hence x = 0. Thus J is bijective. Since J is continuous, J and, because J is bijective, J-1 are closed. So, J-1 : E -+ F as a closed map between Banach spaces is continuous (cf. Theorem 6.7.13) and we have IIxIIY S IIJ_'II IIxIIE

(x E Y)

where IIJ-'11:5 1 since IIxIIE < IIxIIY (x E Y). Hence, IIT(x)IIG <-

(IIJ_1II

-1) IIxIIE

(x E Y)

by the definition of II IIY, so that T is continuous.

Applying the last theorem we get the desired characterization of barrelled dense subspaces of FK- and BK-spaces, respectively.

Theorem 11.5.7. Let E be a BK- (respectively FK-)space and let Y be a dense subspace of E. Then the following statements are equivalent: (a) Y is barrelled (as a subspace of E). (b) If F is any BK- (respectively FK-)space with Y C F, then E C F.

(c) If F is a BK-(respectively FK-)space with Y C F C E, then E = F.

560

Some aspects of topological sequence spaces

Proof. The implication `(b)

(c)' is immediate.

(a) = (b) : Let Y be barrelled and F be an FK-space containing Y. Then, by the closed graph theorem 6.8.7, the inclusion map i : Y -+ F is continuous. Now, if x E E, then there exists a sequence (x(")) in Y such

that xl"> -+ x. Thus (xi">) is a Cauchy sequence in the FK-space F. Then there exists a y E F with x(l) -* y in F. Since both E and F are K-spaces we have x = y. (c) = (a) : Here, we again restrict our interest to the Banach space case and apply `(b) = (a)' of Proposition 11.5.6. Suppose that G is a Banach

space and T : G --4 E is a continuous linear map with Y C T(G). The composition map II IIEoT is a continuous semi-norm on G, and the quotient space (Z := G/KernT, II IIz) (cf. 6.3.6) relative to (G, II IIEoT) is a Banach space. Now, we may identify T (G) and Z with the isomorphism T : Z -4

7'(G), y + KemT -> T(y). Then, as one may check, (T (G), II IIz o T-1) is a BK-space. Since the BK-space T (G) satisfies Y C T (G) C E, the 0 desired identity T(G) = E follows from (c). As an immediate consequence of the foregoing theorem we get a characterization of the Hahn property.

Theorem 11.5.8. If E is any FK-space, then E has the Hahn property if and only if XE is dense and barrelled in E. Now, we give a non-trivial example of a sequence space with the Hahn property.

Theorem 11.5.9. The sequence space m has the Hahn property.

Proof. In 2.9.6 we proved that X' = mo is dense in the BK-space (m,11 Iloo). So by Theorem 11.5.7 it is sufficient to prove the barrelledness of mo in the BK-space m. That proof is deep and we do not have the 0 tools available for it. We refer the reader to [252, 15-1.3]. Application of the last theorem gives us a large class of sequence spaces having the Hahn property.

Corollary 11.5.10. If 0 54 Q C X, then the solid hull E := SoIQ (mQ) of Q has the Hahn property.

Proof. On account of 11.5.5(c) it is sufficient to prove the statement in the case of a one-element set Q since E = EZEQ Sol {x}. Thus, let Q :_ {x} C X be given and let E = m{x}. Then XE = mo({x}) and, in an obvious way, one may identify (E, II IIc.) and (m, 11 11,,.). Therefore, we get

both the density and the barrelledness of XE in (E, II II,,) from that of mo in (m, 11 11.). Consequently, by 11.5.8 we obtain that E has the Hahn 0 property.

Exercise 11.5.11. Prove that c and by fail to have the matrix Hahn property.

Notes on Chapters 9 and 11

561

Exercise 11.5.12. Let E has the matrix Hahn property. Show (XE)a EO.

Exercise 11.5.13. Let E D W be a sequence space such that Xf1E is an at most countable set and such that XE C E. Verify the following statements: E does not have the matrix Hahn property. Moreover, if XE C E and E has the matrix Hahn property, then (E, 11 11.,,) is non-separable, and if E is an FK-space with the matrix Hahn property, then E is non-separable.

Exercise 11.5.14. Let E be a sequence space with tp C E C m. Prove the following statements:

(a) If E is solid, then XE is dense in (E,

(b) If E is monotone and (XE)F = EO, then E has the matrix Hahn property.

Bibliography: [25], [24], [47]

11.6 Notes on Chapters 9 and 11 We start the notes on Chapters 9 and 11 with some remarks about a further research topic during recent decades which has not yet been treated exhaustively and requires the use of gliding hump arguments. In his monograph [254, 19.5.1 (cf. also 8.6.8, 16.3.10)] A. Wilansky posed the problem `Is co the only FK-AD-space with /3-dual P?' A positive answer has been

independently given by G. Bennett and W. Stadler (cf. [22] and [226]). Both Bennett and Stadler got the answer as a corollary of more general results which lead to a property called the Wilansky property in honour of

A. Wilansky. By definition, an FK-space E has the Wilansky property if for each FK-space F, densely contained in E, the equality E'3 = Fa implies E = F. As in the case of the inclusion theorems, considered in Chapters 9 and 11, gliding hump arguments and certain types of gliding hump properties are important tools for the investigations of the Wilansky property. Besides Bennett and Stadler, D. Noll (cf. [184], [185], [186]), D. Noll and W. Stadler (cf. [187], [188]), and D. J. Fleming and W. H. Ruckle (cf. [84]) contributed to the investigation of the Wilansky property. In the case of Banach spaces a general approach to the Wilansky property is given by A. K. Snyder and G. S. Stoudt (cf. [222]). The next remarks are devoted to recent developments concerning gliding hump properties (including oscillation properties) of sequence spaces. We restrict our interest to `gliding hump properties' whose definitions lead, as in Chapters 9 and 11, to the solution of certain problems in summability and topological sequence spaces by gliding hump arguments. (To get more information on the `gliding hump technique' in functional analysis, in particular in topological sequence spaces, we refer the interested reader to the book of C. Swartz (cf. [235] and the review (MR 98b:46002) of it in the Mathematical Reviews of the AMYMS written by Pedro J. Paul.)

562

Some aspects of topological sequence spaces

By introducing the weak gliding hump property (wGHP) and using the arguments of J. Boos and T. Leiger (cf. [42]), D. Noll (cf. [185]) proved that each sequence space E (containing w) which satisfies the weak gliding hump property has a weakly sequentially complete #-dual. In his thesis C. Stuart (cf. [232], [233] and also [45]) proved that this statement remains true

if one replaces the wGHP by the (weaker) signed weak gliding property (SIGNED WGHP), and, moreover, that (E, rr(E, E13)) has AK if E satisfies the SIGNED WGHP. Independently, J. Boos and D. J. Fleming (cf. [39]) obtained this result for sequence spaces which satisfy the (weaker) SIGNED P_01-oscp (cf. Corollary 11.2.11). A comparison of the methods of proof of Noll and Boos (and his collaborators) on the one hand and that used by Stuart on the other hand is interesting. The method of Noll and Boos arises

from summability-more precisely, from Petersen's proof of the bounded consistency theorem (cf. [194]) or Zeller's theorem on factor sequences (cf. 2.5.3): construction of suitable factor sequences by using gliding hump arguments. In contrast to this, the method used by Stuart (and also, for example, by Swartz in [234]) is based on so-called matrix theorems which

are based-in the end-on the Antosik-Mikusinski matrix theorem (see Antosik and Swartz's Lecture Notes [6] and Swartz's book [235]). The matrix theorems arose from functional analysis as we can conclude from Pedro J. Paul's review, mentioned above: That the gliding-hump techniques can be based on such matrix theorems was first noted by Mikusinski whose original matrix lemma was refined by Antosik and Swartz and used, together with K-convergence and K-space, as a substitute for completeness and barrelledness assumptions in many of the classical uniform boundedness results. At first glance, the application of matrix theorems is designed for 1block sequences whereas the gliding hump arguments apply via factor sequences also to step 1-block sequences. So there arises the question whether there exists a modified matrix theorem which can be applied to step 1-block sequences, for example to prove Theorem 9.1.4 (and Theorem 11.1.4). We should also remark that in the literature gliding hump properties are also studied in the case of spaces of vector-valued sequences and of operator-valued matrices. Concerning this we refer, for example, to [235], [234], [232] and [43].

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265. K. Zeller. Merkwiirdigkeiten bei Matrixverfahren; Einfolgenverfahren. Arch. Math. 4, 1-5 (1953). 266. K. Zeller. Ober die Darstellbarkeit von Limitierungsverfahren mittels Matrixtransformationen. Math. Z. 59, 271-277 (1953). 267. K. Zeller and W. Beekmann. Theorie der Limitierungsverfahren (2. Au$.). Springer, Berlin - Heidelberg - New York, 1970. 268. A. Zygmund. Trigonometric series. Vol. I, II. Cambridge University Press, Cambridge, 1988. Reprint of the 1979 edition.

Index List of symbols Miscellaneous *, 129

o
<M>,6 <,39

Dn(v), 246 G(T), 331 HK (A), 492

Kn (v), 247 Ln, 252

DL, 168 DR, 168

0,168 (Sp), 44

(Spo), 45, 160 (Zn), 44 (Zr), 63 (Zs), 46

(Zst), 47 KG, 93 KP, 93 Distinguished subspaces of K-spaces

LA , 250

Bx, 356

Mn, 236 An, 186

Fx, 356

r, 105 E-, 103, 232 ERF, 229 XM, 95 215 235

Kern f , 286 Sgn, 474 Sol Q, 560 conv A, 470 sgn, 40

Ia , 454

®,6

zr1 , 296, 349

-'109

Bx , 386

Fx , 386 Sx, 356 Wx, 356

Distinguished subspaces of domains BA, 407 FA, 407 IA, 78, 407 IA t, 416 LA, 407 PA, 407 SA, 407 TA, 407 WA, 407 AA, 78, 407 Domains of summability methods EA, 397

Wt, 254

CAI, 10

ak(f), 244 ix, 350 px(t), 158

Cs 15

vB(z), 235 Conditions (EI), 519 MK (A), 87

S1,504 S2, 511 53, 513 S4, 513 0 , 168

CPP, 158-165, 186-191 WA, 13 id& , 229 CA, 13 COA, 13 6s.@, 229 MA ; 42

Dual spaces

X-, 284 X', 284

Xf 356

576

Index

X° 341 X'6d , 39, 394 X13, 39, 341 X- f, 341 XCC, 342

Function spaces

B(X), 284 B(X, Y), 284 BV([0,1]), 144 C(R), 294 C21,244 M(D), 269 C, 10, 158 C, 15 Cp, 158

Abb(Z, X), 267 Hom(X, Y), 284 Limit functionals 229 AA L, 78, 407

limA, 13

limsup, 51 Lim, 16

Al-Iim, 10 A-Iim, 13 Bj- lim, 15 F-Iim, 17 Pp Iim, 158 Iim, 6 Metrics, semi-metrics

dy, 271

d,,, 270 d,, 273 dp, 269 Neighbourhood (systems)

K,(x),270 U,, 282

U,(x),

270

U, (x) , 270 UP, 282, 293 B, 293

B(x), 263, 293

8P,

293

U, 282, 294 U(x), 263, 264

u,(x),263 up , 282

u,p,(x), 270 u p,(x), 270 Norms, semi-norms

Ply, 294 11,20 1

lbv , 286, 378, 554

II

11,36

523

11

II

IlL,t476 T

II

Il

111,6,283 11., 6, 249, 282, 283

II

Il,, 476

II

Ili, 483

II

114 ,

II II

ilp, 282, 283 llx"" , 288

II

Ilba, 6

II

Ilbv, 6, 286

483

h(A), 51 4i , 349 Sequence spaces

C, 16 M(E, F), 394 I, 16 II, , 339 X k7 X , 557

8,340

£, 6, 283 476 £1, 6, 283

t-, 5, 270 tP, 283

n, 339 FA, 537 T, 7, 20, 40, 41, 47, 60, 93, 125, 149, 558

w, 5, 273 iv , 228 gyp, 6

bs, 6, 292

by, 6 bvo, 6

c, 6 co, 6 co; , 476 cs, 6, 292

a, 229 d, 339

d,, 340 f , 17, 19, 20, 55, 56, 69, 95, 359, 374, 558

fo, 20, 56, 61, 97, 463

m,5

mo, 7, 40, 41, 52, 94, 95, 121, 343, 368, 542, 548

m,, 476 Sequences/coefficients

(fj), 67

index

577

Sn(x),103

Bi , 14, 24, 49, 51, 59, 153, 222,

[A]nk, 13 [Z]k, 5

B., 162, 196, 197

Onxk, 136 (f, t), 247 e, 7 en, 7

s(f,t), 247 sf(z), 216 8f (z), 216 sn(f,t), 245 X[n]

248 Bs, , 159, 160

C1, 8, 9, 13, 14, 20, 21, 23, 49, 51, 55, 58, 82, 86, 87, 91, 104,

109, 112, 113, 116, 127, 134, 168, 170-172, 178, 194, 195, 248, 423, 444 CQ , 104-112, 126-128, 140, 148,

149, 151, 162, 163, 172, 175-177, 194, 196, 233,

318, 355

Subsets/members of R, C D1, 10

255, 256, 406, 479, 536

E., 140, 141, 147, 151, 153, 155, 157, 163, 202, 203, 226,

Dr , 217

235, 406

GS[f], 217

F, 17

GQ , 224

K1, 226

H-, 100, 101, 106, 111, 139, 148, 149, 151, 176

S[f], 217

H,, 137-152, 223

W,901 207

1,14

[a] ,117

Jp, 158

K, 5 N, 5 N°, 5

K[a, Q] , 154

Non, 28

Ps, , 158-165, 186-191 PAtn f 159 Rs, , 112-126, 178-186, 558 Ss, 153

L, 162 Np, 126-135, 149, 152

Nn , 134

3a , 53 Ra, 53 X(A), 46 X(f), 379, 411 bnk, 7 S, 462 U, 241

p(T),

Sf, 152 T. , 153 Z0 , 127 Zj , 8, 12, 14, 20, 21, 23, 39, 49, 51, 55, 424

236

a(T), 236 Summability methods, matrices (H,Pn), 137 (N, p) , 126 (N, pn) , 126

A, 137 E, 38

E-', 38

A, 19 diag, 60 Topologies

(R,p), 112 (R,pn), 112

j3(X,Y), 320 rl(X,Y), 351

A, 228

o(X,Y), 299, 315 ary(X,Y), 354 -r(X,Y), 320 rp , 293

A1, 10, 15, 21, 23, 24, 55, 158, 162, 190, 192, 194-196,

210, 405, 406 AQ , 162 Aln, 249

B(a, ji) , 162 Bo, 15

B6, 15 B1, 15, 24, 159, 160, 163, 165,

190, 197, 203, 405, 406

;7, 517

-r., 273, 294, 349

General Index AB, 356, 357 AB-space, 357

578

Index

Abel's convergence test, 27

Abel's partial summation formula, 27

Abel's test, 28 Abel's theorem, 10 Abel-type method, 158 Abel method A,, 15 Abel method, generalized, 162 Abschnittsdichte, 356 absolutely Ia-summable sequences, 476

absolutely A-bounded domain, 81 absolutely A-bounded sequences, 81

absolutely p-summable sequence, 283, 304

absolutely convex set, 283 absolutely equivalent matrices, 482 absolutely summable sequence, 6 ABSOLUTE SP_GHP, 542, 543, 546

ABSOLUTE sP_oscP, 472, 474, 542, 543

absolute strong pointwise gliding hump property, 542 absolute strong pointwise oscillation property, 472 absolute summability, 25 absorbing set, 283 AD-space, 356 adherent point, 265, 300 adherent point of a net, 550 adjoint map, 292 AK, 355, 357 AK-space, 357 algebraic dual, 284 almost convergent, 17 almost coregular matrix, 421, 423 a-dual, 341 a-space, 342 analytic function, 340 sequence, 339 analytic continuation, 214 along a sequence of circles, 214 Antosik-Mikusinski matrix theorem, 562 application domain, 13, 229, 398 associative matrix, 417 associative part, 407 asymptotic equal, 109 Bo-invariant, 554 B-invariant, 554 b-comparison, 530

b-consistent summability methods, 23

b-equivalent summability methods, 22

b-stronger summability method, 22 b-weaker summability method, 22 Baire space, 279 balanced set, 283 Banach, theorem of, 333 Banach-Steinhaus theorem, 290, 291, 333, 336, 337 Banach space, 280 barrel, 336 barrelled space, 336 Bernstein's basis polynomials, 150 ,0-dual, 39, 341 #3-space, 342

bi-inverse matrix, 37 binomial coefficients, 101 BK-space, 359 BK-topology, 359 block sequence, 461 block sequence, 1-, 542 Borel kernel, 248

matrix Bi , 14 method B1, 14, 15 method, generalized, 162 Borel-type method, 159 boundary behaviour of a power series, 206 bounded domain, 13 linear operators, 285 maps, set of, 270 partial sums, 6 sequence, 6, 279 set, 279, 281, 289, 301 set, norm, 289

set, pointwise, 289 set, uniformly, 289 variation, 6

bounded consistency theorem, 81, 528, 537 bs-norm, 6 by-norm, 6

Ci-matrix, 14 C1 -method, 13

C,-matrix, 104 C0-method, 104 canonical neighbourhood basis of zero, 293

Cartesian product, 264

Index

Cauchy's limit theorem, 9 Cauchy's theorem for double series, 33

Cauchy integral formulae, 214 Cauchy product, 31 Cauchy sequence, 275, 327 Cesaro

matrix C1, 14 matrix Ca , 104 method C1, 13 method CQ , 104, 162 sections, 393 characteristic function, 95

characteristic of a matrix, 46 closed balls, 270 closed graph, 331 closed graph lemma, 332 closed graph theorem, 332, 335, 337,

convergence factor sequence, 39 convergent

net, 550 sequence, 6, 264 sequence, pointwise, 290 series, 229 convergent double sequence, 16 convex hull, 470 convexity theorem, 176 convex set, 283 convolution, 129 coordinatewise convergence, 301 product of sequences, 39 sum, 461 coregular matrix, 49 decomposition of linear functionals,

521

closed graph theorem, Kalton's, 335 closed map, 331 closed subset, 265 closure of a set, 265

cluster point, 68, 265, 300

coarser topology, 267 coercive method (matrix), 21 cofinal subnet, 550 cofinal subset, 550 column-finite matrix, 82 column condition, 44 compact convergence, 215 compactly summable series, 235 compact set, 268, 551 compact sets in K-spaces, 551 compact space, 268 compact space, sequentially, 279 comparison of domains, 22 compatible with the dual pair, 318 complete semi-metric space, 275 semi-metrizable locally convex space, 329 semi-normed space, 280 completion of a normed space, 281 conservative for null sequences, 21 conservative method (matrix), 21 consistency theorem, 464 consistent summability methods, 23 continuation, analytic, 214 continuous, sequentially, 267 continuous map, 266, 300 continuous summability method, 158

conull matrix, 49, 213

579

309

Dedekind's test, 28 definiteness, 269, 280 dense subset, 93, 265 diagonal matrix, 60 diameter, 488 difference matrix, 137 differences of order k, 136 direct sum, 6 Dirichlet's test, 28

Dirichlet kernel, 246 Dirichlet series, 34

discrete

power series method, 159 summability method, 158 discrete Borel method Bi , 14 discrete metric, 271 discrete topology, 262 disjointly supported, 94 distance, 269

distinguished subspace, 356, 386393

distributive laws, 35 domain (in C), 214 domain of a matrix method, 13, 97, 397 summability method, 12 double sequences, 16 dual, 284, 307 dual map, 292 dual pair, 314 dual space, 284, 307 Du Bois-Reymond's test, 28 Einfolgeverfahren, 73

entire function, 341

580

Index

entire sequence, 340 e-d-criterion, 274 e-neighbourhood, 270 e-no-criterion, 273 equicontinuous, 289, 519 equicontinuous, pointwise, 289 equicontinuous, uniformly, 289 equicontinuous family, 312 equiconvergent, 22, 91 equipotent, 22 equivalent semi-metric, 274 equivalent semi-norms, 285 equivalent summability methods, 22 Euclidean distance, 269 metric, 269 Euclidean distance, 269 Euler matrix EQ , 140 matrix, generalized, 233 method EQ , 140, 163 Euler-Knopp method EQ , 140 eventually constant sequence, 265 F-space, 329

-y-convergence, 517 -f-dual, 341 -y-space, 342

gamma function, 105, 148 gap sequence, 172

relative to p, 182 gap Tauberian condition, 172 theorem for Ci , 172

theorem for Rp, 182 generalized Abel method, 162 Borel method, 162 Euler matrix, 233 Euler method, 233 generated topology, 293

generation of FK-spaces by linear maps, 365 gliding hump method, 42 property, 463 property, absolute strong, 542 property, signed weak, 562 property, weak, 562 graph, 331

factor algebra of a regular matrix, 537

factor sequence, 61 factor sequence of a regular matrix, 537

FAK, 355, 357 FAK-space, 357 family (of semi-norms), 293 FC-effective matrix, 249 Fejer kernel, 248 Fejer's theorem, 4, 255 FF-form, 229 FF-method, 228, 229 finer topology, 267 finitely non-zero sequences, 6 finite subcovering, 268 first category, subset of, 279 fixed point form, 236 FK-product, 394 FK-space, 359, 361 FK-topology, 359, 361 Fourier-effective matrix, 249 Fourier coefficients, 244 Fourier series, 244 Frechet combination, 272, 303 Frechet space, 329 functional sectional convergence, 355

fundamental set, 313

Hahn-Banach extension principle, 287

Hahn-Banach theorem, 287, 308 Hahn property, 558 Hahn property, matrix, 94 HQ-matrix, 100 HQ-method, 100 Hausdorff matrix Hp, 137 method Hp, 137 methods, potent, 149 Hausdorff space, 268 Holder matrix HQ , 100, 139 method HQ , 100, 139 hump, 42 hyperplane, 312

i-semi-norm, 483

identity matrix, 14 IFK-space, 367 inclusion map, 266, 268 inclusion theorem, 21, 539, 552 index sequence, 40 indiscrete semi-metric, 271 induced semi-metric, 271 induced topology, 262 inset, 78, 407

Index

internal inset, 416 intersection method, 25 intersection of FK-spaces, 362 invariance of PA, 453 p-uniqueness, 404, 453 invariant statement, 452 inverse matrix, 37 isometric isomorphism, 291 K-space, 350 K-topology, 350 Kalton's closed graph theorem, 335

Karamata's Tauberian theorem for Laplace transforms, 204 Karamata matrix K[a, /3J , 154 kernel, 286 corresponding to a matrix, 247 kernel, Borel, 248 Dirichlet, 246 Fejer, 248 KG, 93 Knopp, theorem of, 109 Knopp and Schnee, theorem of, 111 Kbthe space, 342 Kolmogoroff, theorem of, 302 KP, 93 Kronecker symbol, 90 PI-norm, 6 Landau order symbols, 168

581

lower triangular matrix, 37 Pp-norm, 283, 304 Mackey space, 320, 336, 521 topology, 320

Mackey's theorem, 322 Mackey-Arens, theorem of, 322 major rearrangement theorem, 32 map, continuous, 266 map, linear, 284 mapping theorem, 394 matrix, 13 Hahn property, 94, 557 map, 13, 361 method, 13 method in RR-form, 229 theorem, 562 theorem, Antosik-Mikusinski, 562

matrix, column-finite, 82 maximal inset, 417 Mazur-Orlicz, bounded consistency theorem, 81 Mazur-Orlicz type, theorem of, 462, 463, 540 meagre subset, 279 mean value condition, 87 property, 87, 109, 128, 418 Mercer's theorem, 91

Laurent matrix Se, 153

metric, 269 metric, discrete, 271

lcs, 293 Lebesgue constant, 250

metric, generated by a norm, 280 metrizable locally convex space, 303

left inverse matrix, 37 LFK-space, 367, 393 limit formula, 44, 46, 51, 55, 80, 114,

Minkowski functional, 283, 295 Mittag-Leffier star, 217 mixed topology, 460, 517 moment sequence, 144 monotone sequence space, 542 monotonicity of FK-topologies, 360 p-bounded domain, 478 p-bounded sequence, 476 p-comparison, 478 p-consistency, 478 p-consistent matrices, 478 p-continuous matrix, 455 la-equivalent matrices, 478 p-space, 455 p-stronger, 478

127, 371, 373

limit of a net, 550 a sequence, 265 limit superior, 51 linear form, 284 functional, 284 map, 284 operator, 284 locally convex

space, 293 space, sequentially complete, 329 topology, 293 logarithmic method, 162 logarithmic weighted mean, 116

p-stronger matrix, 478 p-unique matrix, 421, 447, 448, 451, 453, 454

p-weaker matrix, 478

582

index

multiplicative matrix, 419 multiplier, 394, 537 space, 394 neighbourhood, 262 basis, 263, 264 collection, 263 system, 264 system of zero, 294 neighbourhoods of zero, 294 net, 300, 550 Neumann series, 237 Norlund

matrix Np, 126 method Np, 126 norm, 280 normable metric space, 281 normal matrix, 37 normal topology, 351 norm bounded set, 289 normed space, 280 nowhere dense, 279 null domain, 13 null sequence, 6

ID R (1)-Al -i C,-theorem, 195 oscillation property, 472

P-perfect matrix, 444 para-norm, 305 para-norm, total, 305 partial summation, 27 perfect matrix, 78, 444 perfect sequence space, 342 V-topology, 368

PMI, 417 pointwise bounded family, 333 bounded set, 289 Cauchy sequence, 290 convergence, 333 convergent sequence, 290 equicontinuous, 289 limit, 290 polar topology, 319 positive homogeneity, 280 positive Norlund method, 129 potent Hausdorff methods, 149

matrix (method), 93, 111, 117, 149, 557

o-Tauberian theorem for power series methods, 188 for the Abel method, 190 for the Borel method, 190 JD-Tauberian theorem for Riesz methods, 180 for the Abel method, 192 for the Borel method, 197 for the Euler methods, 202 i7-O-theorem for power series methods, 190

0(1)-Al -* Ci-theorem, 194 Okada's theorem, 220 1-block sequence, 542 one-sequence method, 73

one-sided oscillation Tauberian theorem for Rp, 178, 185 open ball, 270 open covering, 268 open map, 331 open mapping theorem, 331 open set, 262 operator norm, 286 semi-norm, 286 operator, linear, 284 oR-Tauberian theorem for the Abel method, 192

Riesz methods, 117 power series method, 158 power series method Pp, 158 power series method, discrete, 159 power series methods, ID-l)-theorem for, 190 prime number theorem, 256 product metric, 272 semi-metric, 272 semi-norm, 281 space, 264, 272 topology, 264, 296 product of a matrix and a sequence, 34 FK-spaces, 394 locally convex spaces, 296 matrices, 35 semi-normed spaces, 281 series, 31 projection, 267, 296 map, 349 quotient representation, 448, 459,

484, 499, 531 quotient space, 282

R-F-equivalent matrices, 230

Index

rearrangement, 28 regular for null sequences, 23 regular summability method, 23 relatively compact, 268 relative topology, 262 replaceable matrix, 412, 420-422, 454

Riemann-Stieltjes integral, 144 Riesz

matrix RP , 112 method Rp, 112 methods, potent, 117 right inverse matrix, 37 row-finite matrix, 36 row norm condition, 44 row sum condition, 46 RR-form, 229 RR-method, 228, 229 RR-regular summation matrix, 230 SAK, 355, 357 SAK-space, 357 Saks space, 460, 516-522, 524, 526, 529

scalar product of sequences, 34 scaling, 94 Schauder basis, 521 Schur theorem, 51, 326, 479 second category, subset of, 279 second dual, 288 sectional boundedness, 92, 356 convergence, 355 section density, 356 section of a sequence, 355 semi-definiteness, 269, 280 semi-metric, 268 space, 269

space, complete, 275

semi-metric, generated by a seminorm, 280 semi-metric, indiscrete, 271 semi-metrizable locally convex space, 303 semi-norm, 280 semi-normable locally convex space, 302 semi-metric space, 281 semi-normed space, 280

space, complete, 280 spaces, product of, 281 separable space, 288, 334

583

separating family of semi-norms, 295 separation

of sets, 312 of sets, strict, 312 separation theorem, 312, 493 sequences of zeros and ones, 7 sequence space, 5 sequence with bounded variation, 6 sequential continuity, 274 sequential dual, 356 sequentially complete, 329, 521, 526, 528

sequentially continuous, 267 series, 228

set of all double sequences, 16 set of all sequences, 5 SIGNED P_01-oscP, 542, 543, 546, 557 SIGNED P_GHP, 463, 464, 474, 475, 477, 542, 546 SIGNED P_oSCP, 462, 463, 472, 474,

477,540-542 signed pointwise 01-oscillating property, 542 gliding hump property, 463 oscillating property, 462 signed weak gliding hump property, 562

simultaneously A-consistent matrices, 500 b-consistent matrices, 500 consistent matrices, 500 singularity S1, 504 singularity S2, 511 singularity 53 , 513 singularity S4, 513 slowly decreasing sequence, 169 slowly increasing sequence, 169 slowly oscillating relative to 179

slowly oscillating sequence, 62, 169 SM-method, 19 solid hull, 560 solid sequence space, 342 somewhere dense, 279 Sonnenschein

matrix Sf , 152 method Sf , 152 space, barrelled, 336 spectral radius, 236 spectrum, 236 star, 217 star-like set, 217

584

Index

statistical convergence, 25, 537 step 1-block sequence, 462 with respect to an index sequence, 461

Stirling's formula, 59 Stolz angle, 207 Stone-Cech compactification, 537 strong subsequence, 544 summability, 25 topology, 320 stronger semi-metric, 274 summability method, 22 topology, 267 strongly conservative method (matrix), 21, 55, 59, 94, 149, 374, 549, 558

regular method (matrix), 55, 58, 59, 77, 116, 125, 149-151 strongly A-summable, 353 subset of first category, 279 subset of second category, 279 subspace of a (semi-)normed space, 280 of a locally convex space, 294 of an FK-space, closed, 361 of a semi-metric space, 271 of a topological space, 262 substar, 217 summability function, 177 functional, 12 method, 12 order, 86, 113 summable matrix series, 237 sequence, 6, 12 series, 12, 229, 235 summation domain, 229 summation matrix, 38 summation method, 229 sum of FK-spaces, 362 supremum metric, 270 supremum norm, 6, 282 symmetry, 269

T-statistical convergence, 25, 537 Tauberian condition, 22, 168, 169 Tauberian condition, gap, 172 local, 171 one-sided local, 171

Tauberian theorem, 22 for Laplace transforms, 204 Tauberian theorem, gap, 172 Tauberian theorem for A with MK(A), one-sided oscillation, 183 Ci, one-sided oscillation, 170 Ci, oscillation, 170 R? , gap, 182 power series methods, o-, 188 the Abel method, OR-, 192 the Abel method, o-, 190 the Abel method, JD-, 192 the Borel method, o-, 190 the Borel method, i?-, 197, 202 Taylor matrix Ta , 153 test function, 414 theorem of Abel and Stolz, 207 Banach-Steinhaus type, 520 Fejer, 4, 255 Landau, 209 Mazur-Orlicz type, 459, 463, 484, 529

Mercer, 91 theorem on the inverse operator, 331

thin sequence, 7 Toeplitz matrix, 393 sections, 393

Toeplitz, Silverman, Kojima and Schur, theorem of, 46 Toeplitz-Silverman-type theorems, 98, 368-375, 546-549

topological dual, 341 isomorphism, 311 product, 264, 296 space, 262 subspace, 262 topologically isomorphic, 311 topology, 262 topology, strong, 320 topology generated by a semi-metric, 270 a semi-norm, 280 topology of coordinatewise convergence , 301

the dual pair, 318 uniform convergence on equicontinuous subsets of X, 320 uniform convergence on the elements of M, 319 total family of semi-norms, 295

Index

totally monotone sequence, 142 transition matrix, 75 translation, 94, 293 translation invariance, 281 triangle, 37 triangle inequality, 269, 280 triangle inequality, second, 269 triangular matrix, lower, 37 two-norm convergence, 517 two-norm space, 516, 517 type M, matrix of, 82-88, 101, 109,

113, 128, 141, 445, 447, 448, 536

ultimately constant sequence, 265, 339

uniform boundedness principle, 290 uniformly bounded set, 289 uniformly equicontinuous, 289 union method, 25 union of FK-spaces, 365-367, 393, 406, 556 uniqueness of FK-topologies, 360 unit vector, 7 V-summable, 12

very conull matrix, 423, 433, 434, 444, 454

weak y-dual topology, 354 weaker semi-metric, 274 summability method, 22 topology, 267 weak gliding hump property, 562 weakly bounded subset, 319 weak sectional convergence, 355 weak topology, 299, 315, 316 weighted mean, logarithmic, 116 weighted mean method, 112 WGHP, 562

Wiener's Tauberian theorem, 204, 256

Wiener theory, 204 Wilansky property, 393, 561

zero matrix, 14 C-dual, 341 C-space, 342 Zweier

matrix Z., 127 matrix Z1 , 14, 424, 541 method Z,,, 127

585

method Zi, 13

List of Names Abel, N. H., 4, 207 Agnew, R. P., 67, 77, 78, 92, 140 Alexiewicz, A., 515, 516, 522 Andersen, A. F., 109, 176 Antosik, P., 562 Atalla, R. E., 537 Bajsanski, B. M., 154 Baker, J. W., 513 Balser, W., 257 Banach, S., 84, 338 Baumann, H., 484 Beekmann, W., ix, 25, 157, 204, 214, 452-454

Benholz, M., 367, 394 Bennett, G., 72, 94, 395, 439, 460, 522, 539, 561

Bingham, N. H., 204 Boos, J., 81, 367, 453, 455, 474, 484, 513, 529, 536, 562

Borwein, D., 395 Brauer, G., 537 Brown, A., 166 Brudno, A. L., 26, 78, 81, 522, 537 Buntinas, M., 393-395 Connor, J., 25, 395, 537 Cooper, J. B., 516 Copping, J., 72, 484, 498, 513 Cramer, H., 256 Darevsky, V., 73 Eiermann, M., 243 Fast, H., 25 Faulstich, K., 226 Fekete, M., 25 Fleming, D. J., 81, 394, 474, 561, 562 Ford, W. B., 126 Fridy, J. A., 25 Gao, X., 395

Garling, D. J. H., 554 Gawronski, W., 227 Goes, G., 393-395 Goldie, C. M., 204 Grofle-Erdmann, K.-G., 227, 367, 394, 455

Grothendieck, A., 518, 519 Halmos, P. R., 166

Hardy, G. H., 3, 25, 112, 144, 145; 166, 176, 191, 196

Hausdorff, F., 101, 136, 140, 142, 144, 166

586

Index

Henriksen, M., 537 Hille, E., 221 Hurwitz, W. A., 140 Jakimovski, A., 227, 395 Kothe, G., 339, 341 Kalton, N. J., 72, 334, 460, 522, 539 Kangro, G. F., viii, 25 Karamata, J., 154, 192, 204 King, J. P., 227 Kline, J., 25, 537 Knopp, K., 109, 140 Kojima, T., 46 Kolk, E., 25 Kratz, W., 188, 196 Kublanowskaja, W. N., 228 Kuttner, B., 41, 92, 94-96, 117, 149 Landau, E., 175, 208 Leibniz, G., 3 Leibowitz, G., 166 Leiger, T., 81, 393, 474, 562 Le Roy, E., 226 Littlewood, J. E., 25, 169, 176, 191, 196

Lorch, L., 256 Lorentz, G. G., 17, 19, 20, 51, 55, 136, 149, 177, 505, 513

Luh, W., 226 Luh, Y., 98 Miiller, J., 227 Maddox, J., 25, 41, 92, 94, 95 Magee, J. C., 394, 455 Mahowald, M., 337 Mazur, S., 26, 67, 69, 73, 78, 81, 338, 522

Meyer-Konig, W., 203, 211, 212, 395, 439, 479 Meyers, G., 393 Mikusinski, J., 562 Miller, H. I., 25

Mittag-Lefer, G., 226 Norlund, N. E., 126

Petersen, G. M., 19, 81, 484, 500,

505, 512, 513, 537, 562

Peyerimhoff, A., 87 Pringsheim, A., 25 Ramanujan, M. S., 155 Rhoades, B. E., 166 Riesz, M., 112, 176, 211 Ruckle, W. H., 368, 394, 455, 561 Rudolf, 0., 166 Saxon, S. A., 394 Schempp, W., 243 Schur, I., 46, 51, 479 Semadeni, Z., 515, 522 Sharma, N. K., 166 Shawyer, B., 25 Shields, A. L., 166 Silverman, L. L., 46, 136 Siskakis, A. G., 166 Sledd, W. T., 154, 156, 256 Snyder, A. K., 395, 474, 561 Sonnenschein, J., 100, 152 Srinivasan, V. K., 256 Stadler, W., 561 Stadtmiiller, K., 226 Stadtmiiller, U., 188, 197 Stieglitz, M., 19, 98

Stolz, 0., 207 Stoudt, G. S., 561 Stuart, C. E., 562 Swartz, C., 561, 562 Tali, A., 536 Tauber, A., 169 Teugels, J. L., 204 Tietz, H., 98, 192, 395 Toeplitz, 0., 339, 341 Tomm, L., 226 Trautner, R., 226, 227 Varga, R. S., 243 Watson, B., 25 Wielandt, H., 192 Wiener, N., 204

Neuser, R., 536 Newman, D. J., 256 Newton, Sir I., 3 Niethammer, W., 227, 228, 242, 243 Noll, D., 394, 561, 562 Ohlenroth, W., 537 Orhan, C., 25

Wilansky, A., 67, 72, 81, 144, 145,

Orlicz, W., 26, 67, 69, 78, 81, 338,

157, 192, 204, 214, 259, 338, 355, 359, 393, 395, 401, 425, 439, 453, 479,

460, 515, 516, 522

Parameswaran, M. R., 92, 94, 96, 117, 149

Paul, P. J., 561, 562 Persson, A., 515, 516

259, 338, 386, 393, 395, 396, 399, 420, 439, 454,

455,56-1

Wiweger, A., 515, 516 Woronoj, G. Th., 126 Zeller, K., ix, 25, 61-63, 67, 72, 73,

505, 513, 529, 537, 562 Zygmund, A., 255