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

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.