Operators Without Eigenvalues in Finite-Dimensional Vector Spaces: Self-Adjoint Couplings and Shtraus Subspaces

In a coupling theorem from 2001 we described a special class of canonical self-adjoint extensions of the direct sum of symmetric linear relations S1 and S2 in Krein spaces H1 and H2 and assigned a unique parameter to each of these extensions. In this paper we assume that dimH2∈N and that S2 is an operator without eigenvalues and construct a model for (H2,S2) based on an essentially unique polynomial matrix P(z). The families of Shtraus subspaces associated with the self-adjoint extensions are characterized as restrictions of S1∗ by polynomial boundary conditions involving P(z) and the parameters. We establish necessary and sufficient conditions on the parameters under which the extensions are similar and the corresponding families of Shtraus subspaces coincide. Related to our results is the equation W(z)P(z)=P(z)V in which the unimodular matrix polynomial W(z) and the invertible matrix V are the unknowns. Explicit examples are given.