Essential Self-Adjointness of Anticommutative Operators

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. 1. Introduction and Main Theorem In this paper, we consider the essential self-adjointness of anticommutative symmetric operators. Let be a symmetric operator on a Hilbert space ; that is, satisfies . It is said that is self-adjoint if and is essentially self-adjoint if its closure is self-adjoint. We are interested in conditions under which a symmetric operator is essentially self-adjoint. The Glimm-Jaffe-Nelson commutator theorem (e.g., [1, Theorem 2.32], [2, Theorem X.36]) is one criterion for the essential self-adjointness of commutative symmetric operators. The commutator theorem shows that if a symmetric operator and a self-adjoint operator obey a commutation relation on a dense subspace , which is a core of , then is essentially self-adjoint on . Historically, Glimm and Jaffe [3] and Nelson [4] investigate the commutator theorem for quantum field models. Faris and Lavine [5] apply it to quantum mechanical models and Fr？hlich [6] considers a generalization of the commutator theorem and proves that a multiple commutator formula follows. Here, we overview the commutator theorem. Let and be linear operators on . Assume the following conditions. is symmetric and is self-adjoint. There exists such that, for all , has a core satisfying , and there exist constants and such that, for all , Theorem A (Glimm-Jaffe-Nelson commutator theorem). Let and be operators satisfying . Suppose (i) or (ii) as follows. (i)There exists a constant such that, for all , (ii)There exists a constant such that, for all , Then, is essentially self-adjoint on . Remark 1. In the commutator theorem, condition (i) is usually supposed. It is also proven under condition (ii) in a similar way to Theorem 2. The idea of the proof of the commutator theorem is as follows. Let and be symmetric operators on a Hilbert space. Then, the real part and the imaginary part of the inner product for are expressed by respectively, where and . In the proof of the commutator theorem, the imaginary part is estimated. In Theorem 2, we prove that an anticommutative symmetric operator is essentially self-adjoint on a dense subspace by estimating the real part. Theorem 2. Assume . In addition, suppose that (I) or (II) holds. (I)There exists a constant such that, for all , (II)There exists a constant such that, for all , Then, is essentially