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

Abstract

We define and characterize generalized Drazin-g-meromorphic invertible operators, as well as
generalized Kato-g-meromorphic decomposition.
We say that a bounded linear operator $T$ acting on a Banach space $X$ is g-meromorphic if every non-zero point of its spectrum is an isolated point.
For $T$ is said to be generalized Drazin-g-meromorphic
invertible if there exists a bounded linear operator $S $ acting on $X$ such that
$TS=ST$, $STS=S$, $ TST-T$ is g-meromorphic. We shall say that a bounded linear operator $T$ acting on a Banach space $X$ admits a generalized Kato-g-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 $T_N$ is g-meromorphic.
Among others, we show that $T$ is generalized Drazin-g-meromorphic invertible if and only if $0$ is not an accumulation point of its Koliha-Drazin spectrum and this is also equivalent to the fact that $T$ is a direct sum of a g-meromorphic operator and an invertible operator, as well as to the fact that $T$ admits a generalized Kato-meromorphic
decomposition and $0$ is not an interior point of $\sigma (T)$.
Also we study bounded linear operators which can be expressed as a direct sum of a g-meromorphic operator and a bounded below (resp. surjective, upper (lower) semi-Fredholm, Fredholm, upper (lower) semi-Weyl, Weyl) operator.