Infinite Matrices of Operators
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786 Ivor J. Maddox
Infinite Matrices of Operators
Springer-Verlag Berlin Heidelberg New York 1980
Author Ivor J. Maddox Department of Pure Mathematics Queen's University Belfast BT7 1NN United Kingdom
AMS Subject Classifications (1980): 40-02, 40 C 0 5 , 40 E05, 40 F 05, 4 0 G 0 5 , 4 0 H 0 5 , 46-02 ISBN 3-540-09?64-3 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09?64-3 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in PublicationData.Maddox,Ivor John. Infinite matrices of operators.(Lecture notes in mathematics; 786) Bibliography:p. Includes index. 1. Operator theory. 2. Matrices, Infinite. 3. Summabilitytheory. I. Title. I1. Series: Lecture notes in mathematics(Berlin) ; 786. QA3.L28 no. 786 [QA329] 510s [515.?'24] 80-11702 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
CONTENTS i.
Introduction General remarks Dual sequence spaces
2.
Notation and terminology
3
Standard sequence spaces
3
Group norms
5
Generalized K~the-Toeplitz Theorems of Kojima-Schur,
3.
duals; matrix classes Toeplitz,
8 10
Schur
T h e o r e m of Crone
15
Strong summability
17
Abel, Ces~ro, N ~ r l u n d surmnability
18
Generalized K~the-Toeplitz
19
duals
19
cB(X) o
cB(x)
20
g~ (x)
21
c a (x) o
23
£1(B(X,y))
24
c £~(X)
~B(X), O < p -< 1 P
26
£~(X), 1 < p < P
27
~(X),
29
P
1 < p
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