Bishop’s property ($\beta $) and essential spectra of quasisimilar operators

Abstract

We analyze the notion of Bishop’s property (β) to obtain some new concepts. We describe some conditions in terms of these concepts for an operator to have its essential spectrum (spectrum) contained in the essential spectrum (spectrum) of every operator quasisimilar to it. A subfamily of such operators is proved to be dense in L(H). 1. Preliminaries and notations The concept of quasisimilarity of linear operators was introduced by B. Sz-Nagy and C. Foias [1] in 1967. Quasisimilarity is an equivalent relation weaker than similarity. Similarity preserves the spectrum and essential spectrum of an operator, but this fails to be true for quasisimilarity. Suppose that S q ∼ T . What condition should be imposed on S and T to insure the equality relation σe(S) = σe(T ) (σ(S) = σ(T ))? A list of results have been announced along this line. We would like to recall that Yang [2] proved that two quasisimilar M -hyponormal operators have equal essential spectra and M. Putinar [3] proved that two densely similar tuples of operators having Bishop’s property (β) ([4]) have equal essential spectra. However, the foregoing works paid more attention to the equality of essential spectra and spectra than the inclusion relations among them, and the methods applied formerly to different families of operators were varied. We are now going to seek some general conditions for a bounded linear operator S ∈ L(H) to have its essential spectrum (spectrum) contained in that of every operator quasisimilar to it. We call such an operator S a (Q) ((P)) operator, denoted as S ∈ (Q) (S ∈ (P )). Of course, if both S and T ∈ (Q) ((P )) and S q ∼ T , then S and T have equal essential spectra (spectra). The results in [2], [3] motivate us to analyze Bishop’s property (β). It is well known that the subdecomposability of an operator T is equivalent to having Bishop’s property (β) [5]. In section 2, we “localize” the property (β) of an operator S to obtain the concepts A(S), E1(S), E2(S), C1(S), C2(S) as defined below and discuss their mutual relations and the relations between them and the spectral structure of S. In section 3, we establish certain sufficient conditions for (Q) ((P), etc.) operators (Theorems 1,2). We then give a number of corollaries and Received by the editors March 27, 1998. 1991 Mathematics Subject Classification. Primary 47B40, 47A10.