Generalized Kato Decomposition For Operator Matrices and SVEP

Abstract

In this paper, we show that for a bounded linear operator $T$, the corresponding generalized Kato decomposition spectrum $\sigma_{gK}(T)$ satisfies the equality $\sigma_{gD}(T)=\sigma_{gK}(T)\cup (S(T)\cup S(T^*))$ where $\sigma_{gD} (T ) $ is the generalized Drazin spectrum of $T$ and $S(T )$ (resp., $S(T^*)$ is the set where T (resp., $T^*$) fails to have SVEP. As application, we give sufficient conditions which assure that the generalized Kato decomposition spectrum of an upper triangular operator matrices is the union of its diagonal entries spectra. Moreover, some applications are given.