QUASI-FREDHOLM SPECTRUM FOR OPERATOR MATRICES

Abstract

For $A \in \mathcal{L}(X)$, $B \in \mathcal{L}(Y)$ and $C \in \mathcal{L}(Y,X)$ we denote by $M_C$ the operator matrix defined on $X \oplus Y$ by $ M_C= \begin{pmatrix} A & C \\ 0 & B \end{pmatrix}$. In this paper, we prove that $ \sigma_{qF}(A) \cup \sigma_{qF}(B) \subsetneq \bigcup_{C \in \mathcal{L}(Y,X)} \sigma_{qF}(M_C) \cup \sigma_{p}(B) \cup \sigma_{p}(A^*) $, where $\sigma_{qF}(.)$ and $\sigma_{p}(.)$ denote the quasi-Fredholm spectrum and the point spectrum. Furthermore, we consider some sufficient conditions for $M_C$ to be quasi-Fredholm.