Weyl’s theorem for upper triangular operator matrix and perturbations

Abstract

Let and be infinite dimensional separable complex Hilbert spaces and T a bounded linear operator on . ‘Weyl’s theorem’ holds for operator T when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. Let . In this paper, we characterize the invariant subset and the diagonal operator matrix satisfying upper triangular operator matrix , where .