On self-adjoint operators in krein spaces constructed by clifford algebra cl2

Abstract

Let J and R be anti-commuting fundamental symmetries in a Hilbert space H. The operators J and R can be interpreted as basis (generating) elements of the complex Clifford algebra Cl2(J,R) := span {I, J,R,i,JR}. An arbitrary non-trivial fundamental symmetry from Cl2(J,R) is determined by the formula Jᾱ = α1J +α2R+α3iJR, where alpha;̄ ∈S2. Let S be a symmetric operator that commutes with Cl2(J,R). The purpose of this paper is to study the sets ΣJα(∀α ̄ 2 S2) of self-adjoint extensions of S in Krein spaces generated by fundamental symmetries Jᾱ(Jᾱ-self-adjoint extensions). We show that the setsΣJα and ΣJβ are unitarily equivalent for different ᾱ β̄ ∈ S2 and describe in detail the structure of operators A ∈ Σ Jα with empty resolvent set.