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Day July 27, 2011

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.

Functional Analysis (math.FA); Mathematical Physics (math-ph)