Opuscula Math. 32, no. 2 (2012), 297-316

Opuscula Mathematica

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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  • Sergii Kuzhel
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. Mickiewicza 30, 30-059 Krakow, Poland
  • Olexiy Patsyuck
  • National Academy of Sciences of Ukraine, Institute of Mathematics, 3 Tereshchenkivska Street, 01601, Kiev-4, Ukraine
  • Received: 2011-03-17.
  • Revised: 2011-09-15.
  • Accepted: 2011-09-19.
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Cite this article as:
Sergii Kuzhel, Olexiy Patsyuck, On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl2, Opuscula Math. 32, no. 2 (2012), 297-316, http://dx.doi.org/10.7494/OpMath.2012.32.2.297

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