Generalized Kato Decomposition For Operator Matrices and SVEP

TL;DR

This paper explores the relationship between the generalized Kato decomposition spectrum and other spectral sets of bounded linear operators, providing conditions for spectra of operator matrices and applications in spectral theory.

Contribution

It establishes a new spectral equality involving the generalized Kato decomposition spectrum and SVEP failure sets, and applies this to operator matrices.

Findings

The generalized Kato decomposition spectrum equals the generalized Drazin spectrum union SVEP failure sets.

Conditions are provided under which the spectrum of an operator matrix is the union of its diagonal entries' spectra.

Applications demonstrate the utility of these spectral relations in operator theory.

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.