Triangular and Jordon Representations of Linear Operators

"Triangular and Jordan Representations of Linear Operators" by M. S. Brodskii, published by American Mathematical Society. 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). Book details: 256 pages, published 1972. Order now on BookShops.in with fast, reliable delivery across India.