TAILIEUCHUNG - Báo cáo toán học: "(BCP)-operators and enrichment of invariant subspaces lattices "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: (BCP) nhà khai thác và làm giàu của lattices subspaces bất biến. | 1. OPERATOR THEORY 9 1983 187 - 202 Copyright by INCREST 198Ĩ BCP -OPERATORS AND ENRICHMENT OF INVARIANT SUBSPACE LATTICES c. FOIAS and c. M. PEARCY This paper is dedicated with warm affection to Professor Béla on the occasion of his. seventieth birthday. 1. INTRODUCTION Let be a separable infinite dimensional complex Hilbert space and let denote the algebra of all bounded linear operators on If A e we denote by ơ A the spectrum of A by re A the essential . Calkin spectrum of A and by Ơ1CG4 and aIe A the left ancl right essential spectra of A respectively. Moreover we write r A for the spectral radius of A and w A for the numerical radius of A. Recall that an operator A in is a completely nonunitary contraction if p4 Ị 1 and there exists no nonzero reducing subspace Ji for A such that A Jl is a unitary operator. In this paper the Banach algebra F H00 D of bounded holomorphic functions h on the open unit disc D Ằ e c 2 1 with supremum norm ll illoo sup A Z will be useful. In particular there is an -functional calculus for any AeD completely nonunitary contraction A so that the operator h A is defined for every h in and has various properties reflecting those of A and h cf. 20 Theorem . Recall that a subset s of D is said to be dominating for the unit circle c ỔD if sup A A IMloo heH les and that these subsets of D can be characterized by the property that almost every point of Cis a nontangential limit point of 5 cf. 5 . In analogy with this characterization we say that a subset s of D is dominating for a subset s of c if almost every point of J is a nontangential limit point of s. Let BCP denote the class of all completely nonunitary contractions A in. ẫ for which ơe A n D is dominating for c. We permit ourselves the indulgence of referring to such operators A as BCP -operators. 188 c. FOIaS and c. M. PfeARCY The class BCP was first studied in 6 where the existence of nontrivial invariant subspaces for BCP -operators was proved and this study .

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