Self-Adjoint Extensions of a Symmetric Linear Relation with Finite Defect: Compressions and Straus Subspaces

Abstract

Let S be a symmetric relation with finite and equal defect numbers in the Hilbert space \(\mathfrak H\). If \(\widetilde A\) is a self-adjoint extension of S in some larger Hilbert space \(\widetilde {\mathfrak H}\), the compression of \(\widetilde A\) to \({\mathfrak H}\) is a symmetric extension of S. We study this compression in dependence of the parameter \({\mathcal T}\), which parametrizes the extensions \(\widetilde A\) according to M.G. Krein’s resolvent formula. By means of a fractional transformation, analogous results are proved for the Straus extensions of S at a real point.