Weyl's type theorems and perturbations

Weyl’s theorem for a bounded linear operator T on complex Banach spaces, as well as its variants, a-Weyl’s theorem and property (w), in general is not transmitted to the perturbation T + K, even when K is a ”good” operator, as a commuting finite rank operator or compact operator. Weyl’s theorems do not survive also if K is a commuting quasi-nilpotent operator. In this paper we discuss some sufficient conditions for which Weyl’s theorem, a-Weyl’s theorem as well as property (w) is transmitted under such kinds of perturbations.