Triangular and Jordan Representations of Linear Operators

In this book we present the foundations of the theory of triangular and Jordan representations of bounded linear operators in Hilbert space, a subject which has arisen in the last 10-15 years. It is well known that for every selfadjoint matrix of finite order there eXists a unitary transformation which carries it into diagonal form. Geometrically this means that a finite-dimensional Hilbert space, in which there is given a selfad- joint operator A, is representable in the form of the orthogonal sum of one-dimen- sional subspaces invariant relative to A. More than 60 years ago David Hilbert formulated the infinite-dimensional analog of this fact. Any square matrix, according to Schur's theorem, can be reduced by means of a certain unitary transformation to triangular form. The first step in the theory of triangular representations of nonselfadjoint operators operating in infinite-dimensional spaces was taken by M. S. Livsic [1] in 1954. U sing the theory of characteristic functions created by him, he con- structed a triangular functional model of a bounded linear operator with nuclear imaginary component. Later on, thanks to the investigations of L. A. Sahnovic [1,2], A. V. Kuzel' [1,2], V. T. PoljackiT[l] and others, triangular functional models of operators belonging to other classes were found. Simultaneously, in the work of the present author [1- 4], 1. C. Gohberg and M. G. KreIn, [1--6], Ju.1. Ljubic and V. 1. Macaev [1,2,3], V. 1. Macaev [1,2], V. M. BrodskiT [1], and V. M. Brod- skiT and the present author [1], the theory of abstract triangular representations was formulated. It was proved in particular that every completely continuous operator, and also every bounded operator with a completely continuous imaginary component, whose eigenvalues tend to zero sufficiently rapidly, is representable in an integral form which is the natural analog of the ri£ht side of formula (1). An- alogously, invertible operators, close in a certain sense to unItary operators, turned out to be connected with formula (2).