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رای دهید ♥Triangular and Jordan Representations of Linear Operators
by M. S. Brodskii

Persian Title: مثلثی و اردن نمایندگی از اپراتورهای خطی

Summary and Info
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 1015 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 finitedimensional Hilbert space, in which there is given a selfad joint operator A, is representable in the form of the orthogonal sum of onedimen sional subspaces invariant relative to A. More than 60 years ago David Hilbert formulated the infinitedimensional 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 infinitedimensional 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, [16], 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).
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