Research Article | Open Access

# Essential Self-Adjointness of Anticommutative Operators

**Academic Editor:**Jen-Chih Yao

#### 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.

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#### Copyright

Copyright © 2014 Toshimitsu Takaesu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.