Self-adjoint perturbations of left (right)Weyl spectrum for upper triangular operator matrices

Abstract

Let H be a separable infinite-dimensional Hilbert space. Given the operators A ? B(H) and B ? B(H), we define MX := [A X 0 B] where X ? S(H) is a self-adjoint operator. In this paper, a necessary and sufficient condition is given for MX to be a left (right) Weyl operator for some X ? S(H). Moreover, it is shown that ? X?S(H) ?*(MX) = ? X?S(H)?Inv(H) ?*(MX) = ? X?B(H) ?*(MX) ? ?, where ?* is the left (right)Weyl spectrum. Finally, we further characterize the perturbation of the left (right) Weyl spectrum for Hamiltonian operators.