SVEP and Bishop's property for k *-paranormal operators

Abstract

A bounded linear operator T on a complex Hilbert space H is said to be k∗- paranormal ifT ∗ xkT k xfor every unit vector x ∈ H where k is a natural number with 2k. This class of operators is an extension of hyponormal operators and have many interesting properties. We show that k∗-paranormal operators have Bishop's property (β ), i.e., if fn(λ) is an analytic function on some open set D ⊂ C such that (T −z)fn(z) → 0 uniformly on every compact subset K ⊂ D ,t henfn(z) → 0 uniformly on K. In case of k = 2, this means that ∗-paranormal operators have Bishop's property (β ).