Opuscula Math. 32, no. 2 (2012), 297-316
http://dx.doi.org/10.7494/OpMath.2012.32.2.297

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
• Revised: 2011-09-15.
• Accepted: 2011-09-19.