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

TL;DR

This paper investigates how the essential spectra of self-adjoint subspaces in Hilbert spaces remain stable under various compact and finite rank perturbations, extending known results from operators to subspaces.

Contribution

It provides new characterizations of compact perturbations and proves the invariance of essential spectra for self-adjoint subspaces under these perturbations.

Findings

Essential spectra are invariant under relatively compact perturbations.

Self-adjoint subspaces remain self-adjoint under certain bounded and compact perturbations.

Finite rank perturbations are a special case of the general results.

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.