On compressions of self-adjoint extensions of a symmetric linear relation

Abstract

Let $A$ be a symmetric linear relation in the Hilbert space $\gH$ with equal deficiency indices $n_\pm (A)\leq\infty$. A self-adjoint linear relation $\wt A\supset A$ in some Hilbert space $\wt\gH\supset \gH$ is called an exit space extension of $A$; such an extension is called finite-codimensional if $\dim (\wt\gH\ominus\gH)< \infty$. We study the compressions $C (\wt A)=P_\gH\wt A\up\gH$ of exit space extensions $\wt A=\wt A^*$. For a certain class of extensions $\wt A$ we parameterize the compressions $C (\wt A)$ by means of abstract boundary conditions. This enables us to characterize various properties of $C (\wt A)$ (in particular, self-adjointness) in terms of the parameter for $\wt A$ in the Krein formula for resolvents. We describe also the compressions of a certain class of finite-codimensional extensions. These results develop the results by A. Dijksma and H. Langer obtained for a densely defined symmetric operator $A$ with finite deficiency indices.