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

Sergii Kuzhel, Oleksii Patsiuk

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.

http://arxiv.org/abs/1105.2969
Functional Analysis (math.FA); Mathematical Physics (math-ph)