A note on Browder spectrum of operator matrices (Q929573)

For \(A\in{\mathcal B}(H)\), \(B\in{\mathcal B}(H)\) and \(C\in{\mathcal B}(K,H)\), define the operator \(M_C\) acting on \(H\oplus K\) by \(M_C= \left(\begin{smallmatrix} A & C\\ 0 & B\end{smallmatrix}\right)\). In this paper the authors, by giving a counterexample, show that some recent results about Browder spectra of upper-triangular operator matrices are not always true and also, under some necessary and sufficient conditions, they determine the set \(\bigcap \sigma_b(M_C)\) over all \(C\in{\mathcal B}(K,H)\), where \(\sigma_b(M_C)\) is the Browder spectrum of \(M_C\).