Abstract

The self-adjoint extensions of symmetric operators satisfying anticommutation relations are considered. It is proven that an anticommutative type of the Glimm-Jaffe-Nelson commutator theorem follows. Its application to an abstract Dirac operator is also considered.

Highlights

  • Introduction and Main TheoremIn this paper, we consider the essential self-adjointness of anticommutative symmetric operators

  • In Theorem 2, we prove that an anticommutative symmetric operator is essentially self-adjoint on a dense subspace by estimating the real part

  • For some z ∈ C \ R, dim ker((H↾D0 )∗ + z♯) = 0 where z♯ = z, z∗

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Summary

Introduction

Introduction and Main TheoremIn this paper, we consider the essential self-adjointness of anticommutative symmetric operators. (C.1) H is symmetric and X is self-adjoint. (C.3) X has a core D0 satisfying D0 ⊂ D(H), and there exist constants a ≥ 0 and b ≥ 0 such that, for all Ψ ∈ D0, Let H and X be operators satisfying (C.1)–(C.3). The idea of the proof of the commutator theorem is as follows.

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