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, iJR}. An arbitrary non-trivial fundamental symmetry from Cl2(J, R) is determined by the formula J α = α1J + α2R + α3iJR, where α ∈ S 2 . Let S be a symmetric operator that commutes with Cl2(J, R). The purpose of this paper is to study the sets ΣJ α (∀ α ∈ S 2 ) of self-adjoint extensions of S in Krein spaces generated by fundamental symmetries J α (J α -self-adjoint extensions). We show that the sets ΣJ α and ΣJ β are unitarily equivalent for different α, β ∈ S 2 and describe in detail the structure of operators A ∈ ΣJ α with empty resolvent set.
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