Finite-dimensional Self-adjoint Extensions of a Symmetric Operator with Finite Defect and their Compressions

Abstract

Let S be a symmetric operator with finite and equal defect numbers d in the Hilbert space \(\mathfrak{H}\), and with a boundary triplet \((\mathbb{C}^d, \Gamma_1, \Gamma_2)\). Following the method of E.A. Coddington, we describe all self-adjoint extensions \(\tilde{A}\) of S in a Hilbert space \(\tilde{\mathfrak{H}}\;=\;\mathfrak{H}\oplus \mathfrak{H}_1\) where \(\mathrm{dim}\;\mathfrak{H}_1\;<\;\infty\). The parameters in this description are matrices \(\mathcal{A,B,U,V}\;\mathrm{and}\;\mathcal{E}\), where \(\mathcal{A}\;\mathrm{and}\;\mathcal{B}\) determine the compression \(A_0\;\mathrm{of}\;\tilde{A}\;\mathrm{to}\;\mathfrak{H}\). According to a result of W. Stenger, this compression \(A_0\) is self-adjoint. Being a canonical self-adjoint extension of S, \(A_0\) can be chosen as the fixed extension in M.G. Krein’s formula for the description of all generalized resolvents of S. Among other results, we describe those parameters which in Krein’s formula correspond to self-adjoint extensions of S having \(A_0\) as their compression to \(\mathfrak{H}\).