Browsing by Subject "J-self-adjoint operators"

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On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl₂
(2012) Kužel', Sergìj Oleksandrovič; Pacûk, Oleksìj Mikolajovič
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.