Extended Rakočević's property

Abstract

The purpose of this paper is to introduce and study new extension of Rakočević's property $(w)$ and property $(b)$ introduced by Berkani--Zariouh in \cite{berkani-zariouh1}, in connection with other Weyl type theorems and recent properties. We prove in particular, the two following results: 1. A bounded linear operator $T$ satisfies property $(w_{{\pi }_{00}})$ if and only if $T$ satisfies property $(w)$ and ${\sigma }_{uf}(T)={\sigma }_{uw}(T).$ 2. $T$ satisfies property $(g{w}_{{\pi }_{00}})$ if and only if $T$ satisfies property $(w_{{\pi }_{00}})$ and ${\pi }_0(T)=p_0^a(T).$ Classes of operators are considered as illustrating examples.