TAILIEUCHUNG - Báo cáo toán học: "On M-spectral sets and rationally invariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một bộ M-quang phổ và subspaces bất biến hợp lý. | J. OPERATOR THEORY 7 1982 247- 266 Copyright by INCREST 1982 ON Af-SPECTRAL SETS AND RATIONALLY INVARIANT SUBSPACES c. APOSTOL and B. CHEVREAU 1. INTRODUCTION Let J be a separable infinite dimensional complex Hilbert space and let denote the algebra of all bounded linear operators on Jf. If .J is a subalgebra of a nontrivial -invariant subspace is a subspace Ji of such that 0 Ji JC and such that A JI cz Ji for each A in J. If is the algebra of all polynomials in a fixed operator A an invariant subspace is exactly an invariant subspace for A and if ja is the commutant of A an j -in variant subspace is exactly an hyperinvariant subspace for A. In 1978 s. Brown proved that every subnormal operator has a nontrivial invariant subspace see 8 . The method he used was immediately adapted by various authors to obtain invariant subspaces for other classes of operators see 1 2 3 4 10 15 . The present paper is another one in the same area and can be regarded as a sequel to 10 In that paper the authors first established a theorem that gives sufficient conditions for a representation of 2f G into to have a nontrivial invariant subspace. If G is a bounded open set in c Jf G denotes the Banach algebra of functions bounded and analytic in G equipped with the supremum norm. This result paved the way for a systematic investigation of the existence of invariant subspaces for operators A for which there exists a bounded open set G such that 1 G is an M-spectral set for A . IIr A II M sup_ r 2 for every rational ASG function r with poles off G_ and 2 the intersection of the spectrum of A ơ A with G is dominating in G . A oo sup J iG l for any function h in 2Z G . Eventually it was 2eơM nG proved that if the boundary of G consists of finitely many disjoint Jordan loops then the operator A has a nontrivial subspace invariant under the rational functions of A with poles off G . 248 c. APOSTOL and B. CHEVREAU Our main result Theorem says that the same conclusion holds if in .

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