Fredholm consistency of upper-triangular operator matrices
Summary: 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 \(M_C=\begin{bmatrix} A&C\\0&B\end{bmatrix}\) 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 \(\bigcap_{C\in\mathcal{B}(\mathcal{K},\mathcal{H})}\sigma_{\mathrm{FC}}(M_C)=\sigma_{\mathrm{FC}}(M_0)\).
scientific article; zbMATH DE number 6830234
scientific article