Stability of essential spectra of self-adjoint subspaces under compact perturbations

Abstract: This paper studies stability of essential spectra of self-adjoint subspaces (i.e., self-adjoint linear relations) under finite rank and compact perturbations in Hilbert spaces. Relationships between compact perturbation of closed subspaces and relatively compact perturbation of their operator parts are first established. This gives a characterization of compact perturbation in terms of difference between the operator parts of perturbed and unperturbed subspaces. It is shown that a self-adjoint subspace is still self-adjoint under either relatively bounded perturbation with relative bound less than one or relatively compact perturbation or compact perturbation with a certain additional condition. By using these results, invariance of essential spectra of self-adjoint subspaces is proved under relatively compact and compact perturbations, separately. As a special case, finite rank perturbation is discussed. The results obtained in this paper generalize the corresponding results for self-adjoint operators to self-adjoint subspaces.