Commuting Nonselfadjoint Operators in Hilbert Space Two Independent
Classification of commuting non-selfadjoint operators is one of the most challenging problems in operator theory even in the finite-dimensional case. The spectral analysis of dissipative operators has led to a series of deep results in the framework of un
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Moshe S. Livsic Leonid L. Waksman
Commuting Nonselfadjoint Operators in Hilbert Space Two Independent Studies
Spri nger-Verlag Berlin Heidelberg New York London Paris Tokyo
Authors
Moshe S. Livsic Dept. of Mathematics Beer Sheva Ben-Gurion University of the Negev, Israel Leonid L. Waksman ul. 1 konnoi Armii d. 18/9 kv. 6 344029 Rostov-na-Donu, USSR
Mathematics Subject Classification (1980): 15, 35A07, 46, 47, 47005, 47H06, 73,81,93 ISBN 3-540-18316-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-18316-7 Springer-Verlag New York Berlin Heidelberg
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Commuting Nonselfadjoint Operators in Hilbert Space
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M.S. LJ.vsJ.c: Commuting Nonselfadjoint Operators and Collective Motions of Systems •.•••.••••••.•• 1 L. Waksman: Harmonic Analysis of Mult1-Parameter Semigroups of Contractions •••••••••••••••... 40
COMMUTING NONSELFADJOINT OPERATORS AND COLLECTIVE MOTIONS OF SYSTEMS
Moshe S.
TABLE OF CONTENTS Introduction §l.
Single-Operator Colligations: Review of Basic Results
5
§2.
Commutative Colligations and Collective Motions
9
§3.
Characteristic (Transfer) Functions
14
§4.
The Output Realization of Colligations
17
§5.
Composit Colligations
22
§6.
Couplings and Resolutions of Commutative Colligations
23
§7.
A One Dimensional Wave Equation
27
§ 8.
Symmetries
32
COMMUTING NONSELFADJOINT OPERATORS AND COLLECTIVE MOTIONS OF SYSTEMS M.S. Department of Mathematics Ben Gurion University of the Negev Beer Sheva, Israel Introduction The spectral analysis of nonselfadjoint operators in the single operator case was developed during the fifties and sixties thanks to the efforts of many mathematicians. Then it was realized that this analysis forms a mathematical basis for the theory of open systems, interacting with the environment. In the light of the success of this theory, attempts have been made to create an analogous theory for several commuting operators, based on a generalization of the Nagy-Foias functional model to the case of several variables. It turned out, however, that this approach was ineffective. Moreover, the following generalization of the classical Cayley-Hamilton Theorem was obtained: Two
with by a The degree of this equation does not exceed the dimension of the sum of ranges of imaginary parts.
This result shows that the genuine theory of commuting nonselfadjoint ope
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