PROGRESS IN MATHEMATICS. Valurne 10. Mathematical Analysis

Transcription

1 PROGRESS IN MATHEMATICS Valurne 10 Mathematical Analysis

2 PROGRESS IN MATHEMATICS Translations of Itogi Nauki- Seriya Matematika 1968: Volume 1- Mathematical Analysis Volume 2 - Mathematical Analysis 1969: Volume 3- Probability Theory, Mathematical Volume 4 - Mathematical Analysis Volume 5 - Algebra 1970: Volume 6- Topology and Geometry Volume 7- Probability Theory, Mathematical Volume 8- Mathematical Analysis 1971: Volume 9 - Algebra and Geometry Volume 10- Mathematical Analysis Volume 11- Probability Theory, Mathematical In preparation: Volume 12- Algebra and Geometry Volume 13- Probability Theory, Mathematical Volume 14 - Algebra, Geometry, and Topology

3 PROGRESS IN MATHEMATICS Volume 10 Mathematical Analysis Edited by R. V. Gamkrelidze V. A. Steklov Mathematics Institute Academy of Seiences of the USSR, Moscow Translated from Russian by J. S. W ood <.:f? SPRINGER SCIENCE+BUSINESS MEDIA LLC 1971

4 The original Russian text was published for the All-Union Institute of Scientific and Technical Information in Moscow in 1969 as a volume of Itogi Nauki- Seriya Maternatika EDITORIAL BOARD R. V. Gamkrelidze, Editor-in-Chief N. M. Ostianu, Secretary P. S. Aleksandrov V. N. Latyshev N. G. Chudakov Yu. V. Linnik M. K. Kerimov M. A. Naimark A. N. Kolmogorov S. M. Nikol'skii L. D. Kudryavtsev N. Kh. Rozov G. F. Laptev V. K. Saul'ev Library of Congress Catalog Card Nurober ISBN ISBN (ebook) DOI / The present translation is published under an agreement with Mezhdunarodnaya Kniga, the Soviet book export agency 1971 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1971 All rights reserved No part of this publication may be reproduced in any form without written permission from the publisher

5 Preface The present book contains three articles: "Systems of Linear Differential Equations," by V. P. Palamodov; "Fredholm Operators and Their Generalizations," by S. N. Krachkovskii and A. S. Dikanskii; and "Representations of Groups and Algebras in Spaces with an Indefinite Metric" by M. A. Naimark and R. S. Ismagilov. In the fi.rst article the accent is on those characteristics of systems of differential equations which distinguish the systems from the scalar case. Considerable space is devoted in particular to "nonquadratic systems," a topic that has very recently stimulated interest. The second article is devoted to the algebraic aspects of the theory of operators (determinant theory in particular) in Banach and linear topological spaces. The third article reflects the present state of the art in the given area of the theory of representations, which has been receiving considerable attention in connection with its applications in physics (particularly in quantum field theory) and in the theory of differential equations. V

6 Contents SYSTEMS OF LINEAR DIFFERENTIAL EQUA TIONS V. P. Palamodov Quadratic Systems of Equations Systems with Constant Coefficients Sperrcer Constructions and the Local Solvability of Systems with Variable Coefficients The Global Solvability Problem and the Generalized Neumann Problem Literature Cited FREDHOLM OPERATORSAND THEIR GENERALIZATIONS S. N. Krachkovskii and A. S. Dikanskii Introduction The <I>-Operators and Their Generalizations in Banach Spaces The <I>-Operators and Their Generalizations in Topological Vector Spaces Abstract Development of the Determinant Theory of Fredholm Literature Cited Supplement to Literature Cited REPRESENTATIONS OF GROUPS AND ALGEBRAS IN SPACES WITH INDEFINITE METRIC M. A. Naimark and R. S. Ismagilov Introduction Theorems on Invariant Subspaces Description of Commutative Symmetrie Algebras of Operators in Spaces of Type IIk vii

7 viii CONTENTS 3. Group Representations Which Are Unitary in Indefinite Metric Dissipative Operators and the Extension of Dual Subspaces Some Unsolved Problems.. Literature Cited