Stability of Self-Adjoint Extensions of Symmetric Linear Relations under Relatively Bounded Perturbations on Hilbert Spaces

Abstract

We investigate the stability of self-adjoint extensions of a closed symmetric linear relation \(S\) when subjected to a relatively bounded perturbation by another symmetric linear relation \(T\). We demonstrate that the deficiency indices of \(S\) are stable under such perturbations, provided the relative bound is small enough. This result is fundamental, as it ensures that the perturbed relation \(S+T\) possesses the same number of self-adjoint extensions as the original relation \(S\). It is established that if \(T\) is relatively bounded with respect to \(S\) with a relative bound less than \(\frac{1}{2}\), then \(S\) admits self-adjoint extensions if and only if \(S+T\) does.