Generalized Kato-meromorphic decomposition, generalized Drazin-meromorphic invertible operators and single-valued extension property

Abstract

In this paper we define and study generalized Kato-meromorphic decomposition and generalized Drazin-meromorphic invertible operators. 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. Among others, we show that T is generalized Drazin-meromorphic invertible if and only if T admits a generalized Kato-meromorphic decomposition and 0 is not an interior point of $$\sigma (T)$$ , and this is also equivalent to the fact that T is a direct sum of a meromorphic operator and an invertible operator. Also we study 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. In particular, we 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, and as a consequence we get several results on cluster points of some distinguished part of the spectrum. Furthermore, we investigate corresponding spectra and prove that these spectra are empty if and only if the operator T is polynomially meromorphic.