Spectra for upper triangular linear relation matrices through local spectral theory

Abstract

Let X and Y be Banach spaces. When A and B are linear relations in X and Y, respectively, we denote by \(M_{C}\) the linear relation in \(X\times Y\) of the form \(\left( \begin{array}{cc} A &{} C \\ 0 &{} B \\ \end{array} \right) \), where 0 is the zero operator from X to Y and C is a bounded operator from Y to X. In this paper, by using properties of the SVEP, we study the defect set \((\Sigma (A)\cup \Sigma (B))\backslash \Sigma (M_{C})\), where \(\Sigma \) is the spectrum, the approximate point spectrum, the surjective spectrum, the Fredholm spectrum, the Weyl spectrum, the Browder spectrum, the generalized Drazin spectrum and the Drazin spectrum.