Self-adjoint perturbations of spectra for upper triangular operator matrices

This paper is concerned with the self-adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries. A necessary and sufficient condition is given under which such operator matrix admits a Weyl (Fredholm) operator completion by choosing some bounded self-adjoint operator. It is shown that the self-adjoint perturbation of the Weyl (essential) spectrum can be the proper set of the general perturbation. Combining the spectral properties, we further characterize the perturbation of the Weyl (essential) spectrum for Hamiltonian operators.