TY  - JOUR
ID  - Kuzhel2012
LB  - Kuzhel2012
AU  - Kuzhel, Sergii
AU  - Patsyuck, Olexiy
TI  - On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl<sub>2</sub>
ST  - On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl<sub>2</sub>
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2012
VL  - 32
IS  - 2
SP  - 297
EP  - 316
DO  - http://dx.doi.org/10.7494/OpMath.2012.32.2.297
UR  - http://dx.doi.org/10.7494/OpMath.2012.32.2.297
AB  - 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.
KW  - Krein spaces
KW  - extension theory of symmetric operators
KW  - operators with empty resolvent set
KW  - \(J\)-self-adjoint operators
KW  - Clifford algebra \(Cl_2\)
SN  - 1232-9274
ER  -