Theory of Commuting Nonselfadjoint Operators

by M.S. Livsic, N. Kravitsky, A.S. Markus, V. Vinnikov

Overview

Considering integral transformations of Volterra type, F. Riesz and B. Sz.-Nagy no­ ticed in 1952 that [49]: "The existence of such a variety of linear transformations, having the same spectrum concentrated at a single point, brings out the difficulties of characterization of linear transformations of general type by means of their spectra." Subsequently, spectral analysis has been developed for different classes of non­ selfadjoint operators [6,7,14,20,21,36,44,46,54]. It was then realized that this analysis forms a natural basis for the theory of systems interacting with the environment. The success of this theory in the single operator case inspired attempts to create a general theory in the much more complicated case of several commuting operators with finite-dimensional imaginary parts. During the past 10-15 years such a theory has been developed, yielding fruitful connections with algebraic geometry and system theory. Our purpose in this book is to formulate the basic problems appearing in this theory and to present its main results. It is worth noting that, in addition to the joint spectrum, the corresponding algebraic variety and its global topological characteristics play an important role in the classification of commuting operators. For the case of a pair of operators these are: 1. The corresponding algebraic curve, and especially its genus. 2. Certain classes of divisors - or certain line bundles - on this curve.

Product Details

ISBN-13:	9789048145850
Publisher:	Springer Netherlands
Publication date:	12/08/2010
Series:	Mathematics and Its Applications , #332
Edition description:	Softcover reprint of hardcover 1st ed. 1995
Pages:	318
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

I Operator Vessels in Hilbert Space.- 1 Preliminary Results.- 2 Colligations and Vessels.- 3 Open Systems and Open Fields.- 4 The Generalized Cayley — Hamilton Theorem.- II Joint Spectrum and Discriminant Varieties of a Commutative Vessel.- 5 Joint Spectrum and the Spectral Mapping Theorem.- 6 Joint Spectrum of Commuting Operators with Compact Imaginary Parts.- 7 Properties of Discriminant Varieties of a Commutative Vessel.- III Operator Vessels in Banach Spaces.- 8 Operator Colligations and Vessels in Banach Space.- 9 Bezoutian Vessels in Banach Space.- IV Spectral Analysis of Two-Operator Vessels.- 10 Characteristic Functions of Two-Operator Vessels in a Hilbert Space.- 11 The Determinantal Representations and the Joint Characteristic Functions in the Case of Real Smooth Cubics.- 12 Triangular Models for Commutative Two Operator Vessels on Real Smooth Cubics.- References.