On self-adjoint operators in Krein spaces constructed by Clifford algebra Cl_2

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 ${\mathcal C}l_2(J,R):={span}\{I, J, R, iJR\}$. An arbitrary non-trivial fundamental symmetry from ${\mathcal C}l_2(J,R)$ is determined by the formula $J_{\vecα}=α_{1}J+α_{2}R+α_{3}iJR$, where ${\vecα}\in\mathbb{S}^2$. Let $S$ be a symmetric operator that commutes with ${\mathcal C}l_2(J,R)$. The purpose of this paper is to study the sets $Σ_{J_{\vecα}}$ ($\forall{\vecα}\in\mathbb{S}^2$) of self-adjoint extensions of $S$ in Krein spaces generated by fundamental symmetries ${J_{\vecα}}$ (${J_{\vecα}}$-self-adjoint extensions). We show that the sets $Σ_{J_{\vecα}}$ and $Σ_{J_{\vecβ}}$ are unitarily equivalent for different ${\vecα}, {\vecβ}\in\mathbb{S}^2$ and describe in detail the structure of operators $A\inΣ_{J_{\vecα}}$ with empty resolvent set.

No reviews yet.

No comments yet.