Perturbation of spectra of operator matrices and local spectral theory

For A is an element of L(X),B is an element of L(Y) and C is an element of L(Y,X)we denote by M-C the operator defined on X circle plus Y by [(A)(0) (C)(B)]. In this article, we study defect set D-Sigma = (Sigma(A) boolean OR Sigma(B)) \ Sigma(M-C) for different spectra including the spectrum, the essential spectrum, Weyl spectrum and the approximate point spectrum. We then apply the obtained results to the stability of such spectra (D-Sigma = 0) and the classes of operators C for which stability holds of MC using local spectral theory. (c) 2005 Published by Elsevier Inc.