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Báo cáo nghiên cứu khoa học: "Contractions with rich spectrum have invariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các cơn co thắt có phổ giàu có subspaces bất biến. | Copyright by INCREST 1979 J. OPERATOR THEORY 1 1979 123-136 CONTRACTIONS WITH RICH SPECTRUM HAVE INVARIANT SUBSPACES s. BROWN B. CHEVREAU and c. PEARCY 1. INTRODUCTION Let be a separable infinite dimensional complex Hilbert space and let denote the algebra of all bounded linear operators on . As usual a subspace Ji of is said to be a nontrivial invariant subspace for an operator A r JU if 0 70 Ji X Jf and AJÍ .JÍ. In 4 the first author solved the invariant subspace problem for subnormal operators in XiX and in so doing originated a technique for constructing invariant subspaces that was amenable to wider application. In this paper we use the techniques and results of 4 to show Theorem 4.1 that all contraction operators in with sufficiently rich spectrum have nontrivial invariant subspaces. The main new contributions to the ideas of 4 contained in the present paper are the use of the Sz.-Nagy-Foias functional calculus for contractions and Lemmas 4.5 and 4.6. For completeness however we have chosen to begin at the beginning and thus we have included some preliminary material of a general nature 2 as well as some material on the Sz.-Nagy-Foias functional calculus 3 . In addition because of the different setting many of the results from 4 appear in a slightly different form. Nevertheless it must be said that the credit for Theorem 4.1 largely belongs to the first author. Additional interesting results based on 4 and the present paper have been obtained by Agler 1 and Stampfli 12 . 2. GENERAL PRELIMINARIES For purposes of completeness we include some preparatory material of a general nature. Proposition 2.1. Let X be a complex Banach space with dual X and let J be a weak closed subspace of X . If J denotes the preannihilator of J in X then the annihilator fj a of J in X is equal to J and the mapping a from X aJ onto J defined by setting off f o 71 e XfJ where 7Ĩ is the quotient map of X onto Xi J is an isomorphism of XfJ onto J. 124 s. BROWN B. CHEVREAU and c. PEARCY .