Browder spectra of upper-triangular operator matrices
Let \(M_C=\left[\begin{smallmatrix} A & B\\ O & B\end{smallmatrix}\right]\) be a \(2\times 2\) upper triangular operator matrix acting on the Hilbert space \(H\oplus K\). In this paper, the authors prove, for given operators \(A\) and \(B\), that \[ \bigcup_{C \in L(K,H)}\sigma_b(M_C)=\left(\bigcup_{C\in L(K,H)}\sigma(M_C)\right)\setminus \rho_b(A)\cap\rho_b(B), \] where \(\rho_b(T)=\mathbb{C}\setminus\sigma_b(T)\) denotes the Browder resolvent of an operator \(T\). Moreover, they explore the relations of \(\sigma(A)\cup\sigma(B)\setminus\sigma(M_C)\), \(\sigma_b(A)\cup \sigma_b(B) \setminus\sigma_b(M_C)\) and \(\sigma_w(A)\cup \sigma_w(B)\setminus\sigma_w (M_C)\), where \(\sigma(A)\), \(\sigma_b (A)\) and \(\sigma_w(A)\) denote the spectrum, the Browder spectrum and the Weyl spectrum of \(A\), respectively.
scientific article; zbMATH DE number 5072977