On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl₂

Sergii Kuzhel
Olexiy Patsyuck

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 \(Cl_2(J,R):=\text{span}\{I,J,R,iJR\}\). An arbitrary non-trivial fundamental symmetry from \(Cl_2(J,R)\) 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 \(Cl_2(J,R)\). 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.

Keywords: Krein spaces, extension theory of symmetric operators, operators with empty resolvent set, \(J\)-self-adjoint operators, Clifford algebra \(Cl_2\).

Mathematics Subject Classification: 47A55, 47B25, 47A57, 81Q15.

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