Extended Rakočević's property

Abstract: The purpose of this paper is to introduce and study new extension of Rako\v{c}evi\'{c}'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 $(gw_{\pi_{00}})$ 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.