GENERALIZED DRAZIN-g-MEROMORPHIC SPECTRUM FOR OPERATOR MATRICES

Abstract

For $C \in \mathcal{L}(K,H)$, $B \in \mathcal{L}(K)$ and $A \in \mathcal{L}(H)$, let $M_C$ be the operator matrix defined on $H \oplus K$ by $ M_C= \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$, whereas $K$ and $H$ are Banach spaces. In this paper, we demonstrate that $\sigma(M_C)=\sigma(B) \cup \sigma(A)$ is equivalent to $$\sigma_{gD(g \mathcal{M})}(M_C)=\sigma_{gD(g \mathcal{M})}(B) \cup \sigma_{gD(g \mathcal{M})}(A)$$
whereas $\sigma_{gD(g \mathcal{M})}(;)$ is the generalized Drazin-$g$-meromorphic spectrum \cite{S}.
Also, we used the local spectral theory to give a sufficient condition to have the last equality.