Weyl type theorem and its perturbations for bounded linear operators

Abstract

As two variations of Weyl's theorem, a-Weyl's theorem and property $(\omega)$ are introduced by Rako$\check{c}$evi$\acute{c}$.
In this paper, we study a-Weyl's theorem and property $(\omega)$ for functions of bounded linear operators.
And concrete examples are given to show that the two properties are independent of each other.
We give the necessary and sufficient condition for a bounded linear operator with both a-Weyl's theorem and property $(\omega)$ utilizing the induced spectrum of topological uniform descent.
Also, we investigate the perturbations of operator functions satisfying both a-Weyl's theorem and property $(\omega)$.