tailieunhanh - Báo cáo toán học: "Calculuses, annihilators and hyperinvariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Calculuses, annihilators and hyperinvariant subspaces . | J. OPERATOR THEORY 9 1983 341 -370 Copyright by INCREST 1983 CALCULUSES ANN1H1LATORS AND HYPERINVARIANT SUBSPACES B. M. SOLOMJAK 0. INTRODUCTION A large amount of recent achievements in operator theory is connected with the problem of invariant subspaces. It remains unknown however whether every bounded operator T in a Banach or even in a Hilbert space B has a nontrivial invariant subspace . a closed subspace X such that TX c X and 0 Xj B. One way to an invariant subspace IS goes through a functional calculus. We shall also follow this words functional calculus for an operator T usually mean a continuous homomorphism p of a function algebra containing z the identity map of C into JỈ Ỡ the algebra of all bounded operators such that p z T. The Riesz-Dunford calculus is defined on the set of functions analytic in a neighbourhood of n T the spectrum of T by the formula R o T - - iyU R L 7 dz 2zr J . def r being a contour surrounding ff T and R Ấ T -- T ẢIỴ1. This calculus enables us to find invariant subspaces corresponding to the isolated parts of the spectrum. Quite an opposite position is occupied by the L ơ T -calculus for normal operators the richest one from the point of view of functions and the poorest one from the point of view of operators. In the common practice intermediate cases are considered. First of all we should like to mention a calculus based on the Cauchy-Green formula developed by E. M. Dyn kyn 11 It is defined on the set of functions analytic in a domain Q containing ơ T in its closure and sufficiently smooth up to SQ. The rate of smoothness depends on the rate of growth of the resolvent near the domain. Some results of this paper can be obtained on the base of this calculus. 342 B. M. SOLOMJAK J. Wermer has constructed a calculus the IFfT j-calculus applicable to operators with spectrum on the unit circumference T CO iy T fe 2 T 5 ỊT ỈỊ oo CO p f - f T - f n T . - 00 We shall also use the algebra ll ỊT W T n H . Throuehout the .