Linear and Complex Analysis Problem Book 199 Research Problems
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1043
Linear and Complex Analysis Problem Book 199 Research Problems
Edited by V. P. Havin, S. V. Hruscev and N. K. Nikol'skii
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Editors
Victor P. Havin Leningrad State University Stary Peterhof, 198904 Leningrad, USSR Sergei V. Hruscev Nikolai K. Nikol'skii Leningrad Branch of the VA Steklov Mathematical Institute Fontanka 27,191011 Leningrad, USSR Scientific Secretary to the Editorial Board V.l. Vasyunin
AMS Subject Classifications (1980): 30,31,32,41,42,43,46,47,60,81 ISBN 3-540-12869-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12869-7 Springer-Verlag New York Heidelberg Berlin Tokyo Library of Congress Cataloging in Publication Data. Main entry under title: Linear and complex analysis problem book. (Lecture notes in mathematics; 1043) 1. Mathematical analysis-Problems, exercises, etc. I. Khavin, Viktor Petrovich. II. Krushchev, S. V. Ill. Nikol'skii, N. K. (Nikolai Kapitonovich) IV. Series: Lecture notes in mathematics (Springer-Verlag; 1043) OA3.L28 no.l043 [OA301] 510s [515'.0761 83-20344 ISBN 0-387-12869-7 (U.S.) 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 'Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
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x List of Participants • • • • • • • • • • • • • • • • Acknowledgements • • • • • • .XIII • • • • • Preface. • • • • • • • • • • • • • • • • • • • • • • • • • • XVI
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PROBLEMS Chapter 1. ANALYSIS IN FUNCTION SPACES. • • • • • • • • •• 1.1. 1.2. 1.3.
1.4c. 1.5c•. 1.6. 1.7. 1.8. 1.9. 1.10c. 1.11. 1.12. 1.13c. 1.14.
Uniformly convergent Fourier series • • • • • • • • • Compactness of absolutely summing operators • • • • • When is ••••••••••• Local theory of spaces of analytic functions. • • • • oo Complemented subspaces of A, 1-\1 and H • • • • • • • Spaces of Hardy type. • • • • • • • • • • • • • • • • Bases in HP spaces on the ball • • • • • • • • • • • Spaces with the approximation property? • • • • • • • Operator blocks in Banach lattices. • • • • • • • • • Isomorphisms and bases. • • • • • • • • • • • • • Isomorphic classification of F-spaces • • • • • • • • Weighted spaces of entire functions • • • • • • • Linear functionals and linear convexity. • • • • Supports of analytic functionals. • • • • • • • • • •
· ..... • • • • • • • ·. The spectral radius in quotient algebra • • • • · • · Extremum problems • • • · • · • · · • • •H• • •• ·• •• ·• Maximum principles for quotient norms Open semigroups in Banach algebras. • ·• · •• • ·• · • •• Homomorphisms from C*-algebras • • · · · ·· · • Analyticity in the Gelfand space of mUltipl
