On the $g_{z}$-Kato decomposition and generalization of Koliha Drazin invertibility

Abstract

In \cite{koliha}, Koliha proved that $T\in L(X)$ ($X$ is a complex Banach space) is generalized Drazin invertible operator equivalent to there exists an operator $S$ commuting with $T$ such that $STS = S$ and $\sigma(T^{2}S - T)\subset\{0\}$ which is equivalent to say that $0\not\in \mbox{acc}\,\sigma(T).$ Later, in \cite{rwassa,rwassa1} the authors extended the class of generalized Drazin invertible operators and they also extended the class of pseudo-Fredholm operators introduced by Mbekhta \cite{mbekhta} and other classes of semi-Fredholm operators. As a continuation of these works, we introduce and study the class of $g_{z}$-invertible (resp., $g_{z}$-Kato) operators which generalizes the class of generalized Drazin invertible operators (resp., the class of generalized Kato-meromorphic operators introduced by \v{Z}ivkovi\'{c}-Zlatanovi\'{c} and Duggal in \cite{rwassa2}). Among other results, we prove that $T$ is $g_{z}$-invertible if and only if $T$ is $g_{z}$-Kato with $\tilde{p}(T)=\tilde{q}(T)<\infty$ which is equivalent to there exists an operator $S$ commuting with $T$ such that $STS = S$ and $\mbox{acc}\,\sigma(T^{2}S - T)\subset\{0\}$ which in turn is equivalent to say that $0\not\in \mbox{acc}\,(\mbox{acc}\,\sigma(T)).$ As application and using the concept of the Weak SVEP introduced at the end of this paper, we give new characterizations of Browder-type theorems.