Finite-dimensional self-adjoint extensions of a symmetric operator with finite defect and their compressions

Let S be a symmetric operator with finite and equal defect numbers d in the Hilbert space (Symbol Presented), and with a boundary triplet (Cd, Γ1, Γ2). Following the method of E.A. Coddington, we describe all self-adjoint extensions à of S in a Hilbert space (Formula Presented) where dim (Formula Presented). The parameters in this description are matrices A, B, U, V and E, whereA and B determine the compression A0 of à to H. According to a result of W. Stenger, this compression A0 is self-adjoint. Being a canonical self-adjoint extension of S, A0 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 A0 as their compression to (Symbol Presented)