The Bishop’s property (β) for class A operators

Abstract

We say that an operator T on a Hilbert space H has the Bishop?s property (?) if for an arbitrary open set U ? C and analytic functions fn : U ? H with ?(T ? z) fn(z)? converges to 0 uniformly on every compact subset of U as n ? ?then ? fn? converges to 0 uniformly on every compact subset ofU as n ? ?. An operator T on H is called to be hyponormal if T*T ? TT*, and T is called to be class A if T*T ? |T2|. In this papaer, we give an elementary proof of the assertion that every hyponormal operator has the Bishop?s property (?). And we show that every class A operator has the Bishop?s property (?). Moreover, we also show a class A operator T is similar to a hyponormal operator if T is invertible, and hence T has the growth condition (G1).