On spectral stability for self-adjoint extensions

Abstract

We prove that given a symmetric completely non-selfadjoint operator $B$ with finite deficiency indices $(n,n)$ on a Hilbert space and a boundary triplet $\left({\mathbb{C}}^n,{\Gamma }_1,{\Gamma }_2\right)$ for $B^{\ast }$, the set of points in the spectrum of $A_1$ (the self-adjoint extension with domain $Ker{\Gamma }_1$) which are not eigenvalues of maximum multiplicity for any self-adjoint extension of $B$ disjoint of $A_1$, is a dense ${\text{G}}_{\delta }$ set in $\sigma (A_1)$. Furthermore, a proof of a Malamud's theorem that generalizes a well-known result of the Aronszajn-Donoghue theory on the characterization of eigenvalues is offered.