Stability of deficiency indices of Hermitian subspaces under relatively bounded perturbations

This paper is concerned with 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, some of which relax or improve certain conditions of the related results in existing literatures.