#### 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.

#### 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 Fröhlich [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 self-adjoint on .*

*Proof of Theorem 2. *We show that, for some , where . Let and let . Since , we have
First, we assume that (I) holds. Let satisfy . Since is a core of , it follows from and (I) that and for all ,
By (8) and (9), we have
Since , we have from (10). Then, we have . Next, we suppose that (II) follows. Let satisfy . Since is a core of , it also follows from and (II) that and for all ,
Then from (8) and (11), we have
Since and , we have from (12). Thus, the proof is obtained.

#### 2. Application of Theorem 2

We apply Theorem 2 to abstract Dirac operator theory [1, 7]. Let be a Hilbert space. Let and be self-adjoint operators on . Assume that is bounded, and, . Then is called an abstract Dirac operator on with unitary involution . We construct an abstract Dirac operator by weakly commuting operators. Let and be densely defined linear operators on a Hilbert space. The weak commutator of and is defined for and for by Let , , be self-adjoint operators on a Hilbert space . Set . Assume that satisfy the following condition. is dense in . For all , , .

Let be a bounded self-adjoint operator satisfying the following condition.For all , , .

Let be a Hilbert space. Let and be bounded self-adjoint operators on satisfying the following anticommutation relations:.

Then, the next assertion holds.

Theorem 3. *Let . Assume . Then,
**
is self-adjoint on .*

*Remark 4. *In the case where strongly commute, Theorem 3 has been proven ([8, Theorem 4.3], [9, Lemma 6.7]) by strongly anticommuting methods [10, 11].

It is seen that and . Then, from Theorem 3, is an abstract Dirac operator on with the unitary involution .

To prove Theorem 3, we show some lemmas.

Lemma 5. *Let , , be closed operators on a Hilbert space on . Suppose that is dense in and for , , . Then is closed.*

*Proof. *We see that , . Then, . Then from a closedness criterion (e.g., [1, Theorem B1], [12, Proposition 1]), is closed.

From an argument of quadratic forms, there exists a self-adjoint operator on such that , and for all ,

Lemma 6. *Assume . Then, for all ,
*

*Proof. *Since is positive and self-adjoint, it follows that , . Then, for all ,
By and (17), we have for all . Note that is a core of , since is self-adjoint. In addition, for all , , . Hence, it follows that for all . Thus, the proof is obtained.

*Proof of Theorem 3. *Since is bounded, it is enough to show that is self-adjoint. Let . We show that and satisfy and (I) in Theorem 2. Since is symmetric and is self-adjoint, is satisfied. Since and , we see that, for all ,
Then, is satisfied. Since , it follows that . Then, by , we see that, for all ,
Then, for all , and hence is satisfied. By Lemma 6, it is seen that, for all ,
Then by and (20), we have . Then, from (18), it follows that for all . Then (I) is satisfied, and hence is self-adjoint from Theorem 2. In addition, by and , we see that, for ,
Then, from Lemma 5, , and hence the proof is obtained.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

It is a pleasure to thank Professor Akito Suzuki and Professor Fumio Hiroshima for their comments. This work is supported by JSPS Grant 24·1671.