Perturbation of spectra of operator matrices and local spectral theory (Q852817)

Let \(X\) and \(Y\) be Banach spaces and let \(L(X,Y)\) denote the space of all bounded linear operators from \(X\) to \(Y\) (\(L(X):=L(X,X)\)). For \(A\in L(X)\), \(B\in L(Y)\) and \(C\in L(X,Y)\), denote by \(M_{C}\) the operator defined on \(X\oplus Y\) by \(\left[\begin{smallmatrix} A & C\\ 0 & B \end{smallmatrix} \right]\). In this article, the defect set \(D^{\Sigma }=(\Sigma (A)\cup \Sigma (B))\setminus \Sigma (M_{C})\) is studied for different spectra including the spectrum, the essential spectrum, the Weyl spectrum and the approximate point spectrum. The obtained results are applied to the stability of such spectra (\(D^{\Sigma }=\varnothing\)) and the classes of operators \(C\) for which stability holds of \(M_{C}\) using local spectral theory.