Partially fundamentally reducible operators in Krein spaces

TL;DR

This paper studies a special class of self-adjoint operators in Krein spaces called partially fundamentally reducible operators, providing conditions under which they are similar to Hilbert space self-adjoint operators, based on boundary triples and Weyl functions.

Contribution

It introduces the concept of partial fundamental reducibility in Krein spaces and establishes criteria for similarity to Hilbert space self-adjoint operators using Weyl functions.

Findings

Sufficient conditions for similarity to Hilbert space self-adjoint operators.

Characterization of positive self-adjoint extensions via asymptotic behavior of Weyl functions.

Extension of boundary triple techniques to partially fundamentally reducible operators.

Abstract

A self-adjoint operator $A$ in a Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$ is called partially fundamentally reducible if there exist a fundamental decomposition ${\mathcal K} = {\mathcal K}_+ [\dot{+}] {\mathcal K}_-$ (which does not reduce $A$) and densely defined symmetric operators $S_+$ and $S_-$ in the Hilbert spaces $\bigl({\mathcal K}_+,[\,\cdot\,,\cdot\,]\bigr)$ and $\bigl({\mathcal K}_-,-[\,\cdot\,,\cdot\,]\bigr)$, respectively, such that each $S_+$ and $S_-$ has defect numbers $(1,1)$ and the operator $A$ is a self-adjoint extension of $S =S_+ \oplus (-S_-)$ in the Krein space $\bigl({\mathcal K},[\,\cdot\,,\cdot\,]\bigr)$. The operator $A$ is interpreted as a coupling of operators $S_+$ and $-S_-$ relative to some boundary triples $\bigl({\mathbb C},\Gamma_0^+,\Gamma_1^+\bigr)$ and $\bigl({\mathbb C},\Gamma_0^-,\Gamma_1^-\bigr)$. Sufficient conditions for a nonnegative partially fundamentally reducible operator $A$ to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions $m_+$ and $m_-$ of $S_+$ and $S_-$ relative to the boundary triples $\bigl({\mathbb C},\Gamma_0^+,\Gamma_1^+\bigr)$ and $\bigl({\mathbb C},\Gamma_0^-,\Gamma_1^-\bigr)$. Moreover, it is shown that under some asymptotic assumptions on $m_+$ and $m_-$ all positive self-adjoint extensions of the operator $S$ are similar to self-adjoint operators in a Hilbert space.