Closed range and Fredholm properties of upper-triangular operator matrices

Abstract

The closed range and Fredholm properties of the upper-triangular operator matrix $M=(A, C; 0, B) \in\mathcal{B}({\mathcal H}_1\oplus {\mathcal H}_2)$ are studied, where ${\mathcal H}_1$ and ${\mathcal H}_2$ are Hilbert spaces. It is shown that the range $\mathcal{R}(M)$ of $M$ is closed if and only if the following statements hold: (i) $\mathcal{R}(P_{\mathcal{R} (A)^\perp}C|_{\mathcal{N}(B)})$ is closed, (ii) $\mathcal{R}(A)+\mathcal{R}(P_{\overline{\mathcal{R}(A)}} C|_{\mathcal{N}(P_{\mathcal{R}(A)^\perp}C|_{\mathcal{N}(B)})})={\overline{\mathcal{R}(A)}}$, (iii) $\mathcal{R}(B^{*})+\mathcal{R}(P_{\mathcal{N}(B)^\perp}C^* |_{\mathcal{R}(P_{\mathcal{R} (A)^\perp}C|_{\mathcal{N}(B)})^\perp})={\overline{\mathcal{R}(B^*)}}$,\\ where $P_{\mathcal G}$ denotes the orthogonal projection onto ${\mathcal G}$ along ${\mathcal G}^\perp$. Moreover, the analogues for the Fredholmness of $M$ are further presented.