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.
Categories: math.FA math-ph math.MP