Operators without eigenvalues in finite-dimensional vector spaces: Linearization and Spectral Equivalence

Abstract

In this paper $S$ is a closed symmetric linear relation in a Krein space $\mathfrak{H}$ with adjoint $S^{\ast }$, finite and equal defect numbers $d$, and a boundary mapping $\mathsf{b}:S^{\ast }\to {\mathbb{C}}^{2d}$ with Gram matrix $\mathsf{Q}$. We introduce a class ${\mathbb{A}}_{S,\mathsf{b}}$ of self-adjoint extensions of $S$ in a Krein space $\tilde{\mathfrak{H}}$ containing $\mathfrak{H}$ as a Krein subspace of finite codimension, together with a class ${\mathbb{P}}_{\mathsf{Q}}$ of $d\times 2d$ matrix polynomials. Both classes are equipped with natural equivalence relations. Given $\mathcal{P}(z)\in {\mathbb{P}}_{\mathsf{Q}}$, we consider a boundary eigenvalue problem defined by the condition $\mathcal{P}(z)\mathsf{b}(\{f,g\})=0$, $\{f,g\}\in S^{\ast }$, and look for its linearizations. By a linearization we mean a linear relation $\tilde{A}\in {\mathbb{A}}_{S,\mathsf{b}}$ such that the Shtraus extension $T_{\tilde{A}}(z)$ of $S$ determined by $\tilde{A}$ coincides, for all $z\in \overline{\mathbb{C}}$, with the linear relation $\{\{f,g\}\in S^{\ast }:\mathcal{P}(z)\mathsf{b}(\{f,g\})=0\}$. We prove that this correspondence defines a bijection between the equivalence classes in ${\mathbb{A}}_{S,\mathsf{b}}$ 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 $\overline{\mathbb{C}}$ there is a bijection between their Jordan chains.

42 pages