# Invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations

## TL;DR

This paper investigates how the deficiency indices of Hermitian subspaces in Hilbert spaces remain stable under certain bounded perturbations, extending known results from symmetric operators to more general linear relations.

## Contribution

It establishes invariance results for deficiency indices and self-adjointness of Hermitian subspaces under relatively bounded perturbations, generalizing previous operator theory results.

## Key findings

- Deficiency indices are invariant under certain bounded perturbations.
- Self-adjointness of Hermitian subspaces remains stable under these perturbations.
- In some cases, deficiency indices may decrease when the relative bound equals 1.

## Abstract

This paper is concerned with the stability of deficiency indices of Hermitian subspaces (i.e., linear relations) under relatively bounded perturbations in Hilbert spaces. Several results about invariance of deficiency indices of Hermitian subspaces under relatively bounded perturbations are established. As a consequence, invariance of self-adjointness of Hermitian subspaces under relatively bounded perturbations is obtained. In addition, it is shown that the deficiency indices may shrink in the special case that the relative bound is equal to 1. The results obtained in the present paper generalize the corresponding results for symmetric operators to more general Hermitian subspaces.