On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

TL;DR

This paper investigates self-adjoint extensions of symmetric operators in Krein spaces constructed via Clifford algebra elements, revealing their unitary equivalence and characterizing operators with empty resolvent set.

Contribution

It introduces a framework for analyzing J-self-adjoint extensions using Clifford algebra and describes their structure and equivalence properties.

Findings

Sets of extensions are unitarily equivalent for different symmetries.

Characterization of operators with empty resolvent set.

Structural description of self-adjoint extensions.

Abstract

Let $J$ and $R$ be anti-commuting fundamental symmetries in a Hilbert space $\mathfrak{H}$. The operators $J$ and $R$ can be interpreted as basis (generating) elements of the complex Clifford algebra ${\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}$. An arbitrary non-trivial fundamental symmetry from ${\mathcal C}l_2(J,R)$ is determined by the formula $J_{\vec{\alpha}}=\alpha_{1}J+\alpha_{2}R+\alpha_{3}iJR$, where ${\vec{\alpha}}\in\mathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${\mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $\Sigma_{{J_{\vec{\alpha}}}}$ ($\forall{\vec{\alpha}}\in\mathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${{J_{\vec{\alpha}}}}$ (${{J_{\vec{\alpha}}}}$-self-adjoint extensions). We show that the sets $\Sigma_{{J_{\vec{\alpha}}}}$ and $\Sigma_{{J_{\vec{\beta}}}}$ are unitarily equivalent for different ${\vec{\alpha}}, {\vec{\beta}}\in\mathbb{S}^2$ and describe in detail the structure of operators $A\in\Sigma_{{J_{\vec{\alpha}}}}$ with empty resolvent set.