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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 Sz.-Nagy 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 i.e. 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 III.2.1 . 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 .