Generalized Drazin-meromorphic invertible operators and generalized Kato-meromorphic decomposition

Abstract

A bounded linear operator $T$ on a Banach space $X$ is said to be generalized Drazin-meromorphic invertible if there exists a bounded linear operator $S$ acting on $X$ such that $TS=ST$, $STS=S$, $ TST-T$ is meromorphic. We shall say that $T$ admits a generalized Kato-meromorphic decomposition if there exists a pair of $T$-invariant closed subspaces $(M,N)$ such that $X=M\oplus N$, the reduction $T_M$ is Kato and the reduction $T_N$ is meromorphic. In this paper we shall investigate such kind of operators and corresponding spectra, the generalized Drazin-meromorphic spectrum and the generalized Kato-meromorphic spectrum, and prove that these spectra are empty if and only if the operator $T$ is polynomially meromorphic. Also we obtain that the generalized Kato-meromorphic spectrum differs from the Kato type spectrum on at most countably many points. Among others, bounded linear operators which can be expressed as a direct sum of a meromorphic operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator are studied. In particular, we shall characterize the single-valued extension property at a point $\lambda_0\in\mathbb{C}$ in the case that $\lambda_0-T$ admits a generalized Kato-meromorphic decomposition. As a consequence we get several results on cluster points of some distinguished parts of the spectrum.