Commuting Nonselfadjoint Operators in Hilbert Space: Two Independent Studies: 1272 (Lecture Notes in Mathematics, 1272) - Softcover

Livsic, Moshe S.; Waksman, Leonid L.

Softcover

ISBN 10: 3540183167 ISBN 13: 9783540183167

Publisher: Springer, 1987

Synopsis

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 unitary dilations and characteristic operator functions. It has turned out that the theory has to be based on analytic functions on algebraic manifolds and not on functions of several independent variables as was previously believed. This follows from the generalized Cayley-Hamilton Theorem, due to M.S.Livsic: "Two commuting operators with finite dimensional imaginary parts are connected in the generic case, by a certain algebraic equation whose degree does not exceed the dimension of the sum of the ranges of imaginary parts." Such investigations have been carried out in two directions. One of them, presented by L.L.Waksman, is related to semigroups of projections of multiplication operators on Riemann surfaces. Another direction, which is presented here by M.S.Livsic is based on operator colligations and collective motions of systems. Every given wave equation can be obtained as an external manifestation of collective motions. The algebraic equation mentioned above is the corresponding dispersion law of the input-output waves.

Edition details

• Publisher: Springer
• Publication date: 1987
• Language: English
• ISBN 10: 3540183167
• ISBN 13: 9783540183167
• Binding: Paperback
• Number of pages: 124