2026/07/23 by Branko Ćurgus, Aad Dijksma
Mathematics · #math.SP
In this paper S is a closed symmetric linear relation in a Krein space \mathfrak H with adjoint S^*, finite and equal defect numbers d, and a boundary mapping b: S^* → ℂ2d with Gram matrix \mathsf Q. We introduce a class \mathbb AS,b of self-adjoint extensions of S in a Krein space \widetilde\mathfrak H containing \mathfrak H as a Krein subspace of finite codimension, together with a class ℙ\mathsf Q of d × 2d matrix polynomials. Both classes are equipped with natural equivalence relations. Given P(z) ∈ ℙ\mathsf Q, we consider a boundary eigenvalue problem defined by the condition P(z)b(\f,g\)=0, \f,g\∈ S^*, and look for its linearizations. By a linearization we mean a linear relation \widetildeA ∈ \mathbbAS, \mathsf b such that the Shtraus extension T\widetilde A(z) of S determined by \widetilde A coincides, for all z ∈ \mathbb C, with the linear relation \\f,g\∈ S^* : \mathcal P(z)\mathsf b(\f,g\)=0\. We prove that this correspondence defines a bijection between the equivalence classes in \mathbb AS, \mathsf b and those in \mathbb P\mathsf Q. Moreover, we provide a condition under which the resulting linearizations are spectrally equivalent to the boundary eigenvalue problem: their regular and spectral points coincide, and for each eigenvalue in \mathbb C there is a bijection between their Jordan chains.