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.