Fredholm consistency of upper-triangular operator matrices

Abstract

In this paper, for given operators $A\in \mathcal{B}(\mathcal{H})$ and $B\in \mathcal{B}(\mathcal{K})$, we characterize the set of all $C\in \mathcal{B}(\mathcal{K},\mathcal{H})$ such that the operator matrix $M_C=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]$ is Fredholm consistent. We completely describe the sets ${\bigcap }_{C\in \mathcal{B}(\mathcal{K},\mathcal{H})}{\sigma }_{\mathrm{FC}}(M_C)$ and ${\bigcup }_{C\in \mathcal{B}(\mathcal{K},\mathcal{H})}{\sigma }_{\mathrm{FC}}(M_C)$. Also, we prove that ${\bigcap }_{C\in \mathcal{B}(\mathcal{K},\mathcal{H})}{\sigma }_{\mathrm{FC}}(M_C)={\sigma }_{\mathrm{FC}}(M_0)$.