On Compressions of Self-Adjoint Extensions of a Symmetric Linear Relation with Unequal Deficiency Indices

Abstract

Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation \( \tilde{A}\supset A \)in some Hilbert space \( \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} \)is called an (exit space) extension of A. We study the compressions \( C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} \) of extensions \( \tilde{A}=\tilde{A^{\ast }}. \) Our main result is a description of compressions \( C\left(\tilde{A}\right) \) by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of A with the maximal symmetric compression \( C\left(\tilde{A}\right) \) and all extensions \( \tilde{A}=\tilde{A^{\ast }}. \) of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).