Multiparameter Eigenvalue Problems and Expansion Theorems
This book provides a self-contained treatment of two of the main problems of multiparameter spectral theory: the existence of eigenvalues and the expansion in series of eigenfunctions. The results are first obtained in abstract Hilbert spaces and then app
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1356 Hans Volkmer
Multiparameter Eigenvalue Problems and Expansion Theorems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Lecture Notes in Mathematics Edited by A. Dold and 8. Eckmann
1356 Hans Volkmer
Multiparameter Eigenvalue Problems and Expansion Theorems
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo
Author
Hans Volkmer Fachbereich Mathematik, Universitat - Gesamthochschule Essen Universitatsstr. 3, 4300 Essen 1, Federal Republic of Germany
Mathematics Subject Classification (1980): primary: 15A 18, 45F99, 47B25, 34B25 secondary: 15A69, 35P 10, 46C 10
ISBN 3-540-50479-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50479-6 Springer-Verlag New York Berlin Heidelberg
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© Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210
PREFACE Since the pUblication of Atkinson's book "Multiparameter Eigenvalue Problems, Vol. I" in 1972, multiparameter spectral theory has become a subject of growing interest. Several authors have made important contributions to the theory; see the references at the end of this book. There are also two monographs of Sleeman (1978a) and McGhee and Picard (1988) on this subject. The contents of the present book can be called "classical multiparameter to distinguish it from more recent developments in multiparameter theory. The book contains the major results of Atkinson's book (most of them are proved differently), several of the contributions made in the last years and some new results. It is thought as a supplement to the excellent book of Atkinson which has the only drawback that we are waiting for volume II for quite a while. We study two problems: the existence of eigenvalues and the expansion in series of eigenvectors. Each of these problems is treated for multiparameter eigenvalue problems involving (i) Hermitian matrices (ii) compact Hermitian operators, in particular, integral operators (iii) semi bounded selfadjoint operators with compact resolvent, in particular, differential operators. Alltogether this leads to six chapters. It is not assumed that the reader knows already some multiparameter spectral theory but it is supposed that the reader is familiar with the usual one-parameter eigenvalue problems for compact Hermitian operators and their inverses. Brouwer's degree of maps will be used in the first two chapters. Basic properties of the tensor product of linear spaces will be needed in Ch
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