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Let A be a symmetric linear relation in the Hilbert space $${\mathfrak {H}}$$H with equal deficiency indices $$n_\pm (A)\le \infty $$n±(A)≤∞. A self-adjoint linear relation $$\widetilde{A}\supset A$$A~⊃A in some Hilbert space $$\widetilde{\mathfrak {H}}\supset {\mathfrak {H}}$$H~⊃H is called an exit space extension of A; such an extension is called finite-codimensional if $$\dim (\widetilde{\mathfrak {H}}\ominus {\mathfrak {H}})< \infty $$dim(H~⊖H)<∞. We study the compressions $$C (\widetilde{A})=P_{\mathfrak {H}}\widetilde{A}\upharpoonright {\mathfrak {H}}$$C(A~)=PHA~↾H of exit space extensions $$\widetilde{A}=\widetilde{A}^*$$A~=A~∗. For a certain class of extensions $$\widetilde{A}$$A~ we parameterize the compressions $$C (\widetilde{A})$$C(A~) by means of abstract boundary conditions. This enables us to characterize various properties of $$C (\widetilde{A})$$C(A~) (in particular, self-adjointness) in terms of the parameter for $$\widetilde{A}$$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 and equal deficiency indices.

On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation