On the Stability of Self-Adjointness of Linear Relations

Abstract

This paper focuses on the stability of self-adjointness of linear relations under perturbations in Hilbert spaces. It is shown that a self-adjoint relation is still self-adjoint under bounded and relatively bounded perturbations. The results obtained in the present paper generalize the corresponding results for linear operators to linear relations, and some weaken the conditions of the related existing results.

1. Introduction

Perturbation theory is one of the main topics in both pure and applied mathematics. Since the self-adjoint operators form the most important class of linear operators that appear in applications, the perturbation of self-adjoint operators and the stability of self-adjointness have received lots of attention. In particular, Kato first studied the stability of self-adjointness of closed symmetric operators and showed that the self-adjointness is preserved under relatively bounded perturbations with relative bounds less than 1 [1]. Followed by this work, Devinatz, Zettl, Behncke, Kissin, etc. extended this work about stability of self-adjointness to results about the stability of the deficiency indices [2–5].

With further research of operator theory, more and more multivalued operators and nondensely defined operators have been found. For example, the operators generated by those linear continuous Hamiltonian systems, which do not satisfy the definiteness conditions, and general linear discrete Hamiltonian systems may be multivalued or not densely defined in their corresponding Hilbert spaces (cf. [6–8]). So the classical perturbation theory of linear operators is not available in this case. Motivated by this need, von Neumann [9] first introduced linear relations into functional analysis, and then Arens [10] and many other scholars further studied and developed the fundamental theory of linear relations. A liner relation is also called a linear subspace (briefly, subspace).

Since the theory of linear relations was established, the related perturbation problems have attracted extensive attention of scholars and some good results have been obtained [11–13]. It is well known that the self-adjoint relations are the most important class of linear relations that appear applications. To the best of our knowledge, there seem a few results about the stability of self-adjointness of linear relation under perturbations [11, 13]. But it has not been specifically and thoroughly studied. In the present paper, we shall concentrate on the stability of self-adjointness of linear relations. The results given in the present paper not only weaken the conditions of Theorem 4.1 in [11] but also cover the result obtained in [13, Theorem 5.2].

The rest of the paper is organized as follows. In Section 2, some basic concepts and useful fundamental results about linear relations are introduced. In Section 3, we first show that the deficiency indices of Hermitian relations are invariant under relatively bounded perturbations with relative bounds less than 1. As a consequence, stability of self-adjointness of Hermitian relations under bounded and relatively bounded perturbations is obtained.

2. Preliminaries

In this section, we shall recall some basic concepts and introduce some fundamental results about linear relations.

and denote the sets of the complex numbers and the real numbers, respectively. Let be a complex Hilbert space with inner product . denote the set of all linear relations of . Let . is said to be a closed relation if , where is the closure of . denote the set of all closed relations of .

Let and .

The adjoint of is defined by

is called a Hermitian relation if , and it is called a self-adjoint relation if .

Definition 1 ([14, Definition 2.3]). Let . The subspace is called the deficiency space of and , and the number is called the deficiency index of and .

Let be a Hermitian relation. By [14, Theorem 2.3], is constant in the upper and lower half-planes; that is, for all with and for all with , where . The pair is called the deficiency indices of , and and are called the positive and negative deficiency indices of , respectively.

In the following, we shall recall concepts of the norm of a subspace and relative boundedness of two subspaces, and their fundamental properties.

Let be a closed subspace of . Define the following quotient space [1]:

We define an inner product on the quotient space by

where , with and . It can been easily verified that with this inner product is a Hilbert space. The norm induced by this inner product is the same as that of induced by the norm of .

Now, define the following natural quotient map:

Let . By denote for briefness without confusion. Define

Then is a linear operator with domain [15, Proposition II.1.2]. The norm of at and the norm of are defined by, respectively (see [16, II.1]),

Lemma 2 ([15, Propositions II.1.4–II.1.7]). Let . Then (1) for every and ;(2), for every and ;(3) for , .

Definition 3 ([15, Definition VII.2.1]). Let .(1) is said to be -bounded if and there exists a constant such that(2)If is -bounded, then the infimum of all numbers for which a constant exists such thatis called the -bound of .

Remark 4. Condition (9) is equivalent to the following condition:

where the constants . It can be easily deduced that (10) implies (9) with , whereas (9) implies (10) with and with an arbitrary . Consequently, the -bound of may as well be defined as the infimum of the possible values of .

Lemma 5 ([16, Lemma 2.7]). Let and . If there exists such that for any , then is closed.

Lemma 6 ([12, Propositions 2.1, 3.1, 3.3, Theorem 6.3]). Let .(1) if and only if and .(2)If is -bounded with -bound less than , then is closed if and only if is closed.(3)If is Hermitian, then and

Lemma 7 ([17, Lemma 2.5]). Let be Hermitian. If there is such that , then is a self-adjoint relation.

Lemma 8 ([13, Lemma 5.8]). Let be self-adjoint. If is Hermitian and , then .

3. Main Results

In this section, we shall first study the stability of deficiency indices of Hermitian relations under perturbations. Then, we shall use these results to study the stability of self-adjointness of Hermitian relations.

We shall first prove some useful lemmas:

Lemma 9. Let and satisfy that there exists such thatLet with and , and satisfyfor some constant . Then is closed and satisfies (12) with instead of . Moreover,

Proof. Note that is closed. It follows from (13) and (2) of Lemma 6 that is closed. By Lemmas 2 and 6, (12), and (13), we have that for any and ,This implies that satisfies (12) with instead of .
In addition, by (12), (15), and the closedness of and , and are closed by Lemma 5.
Now we show that (14) holds. Suppose that . Then there exists with . Thus, there exists such that . And there exist and such that . In addition, since is closed, it can be easily verified that is closed. Hence, . For clarity, for every , by denote the element of . By the assumption that , it follows from Lemma 2 thatSince is closed, there exist and such that . Further, by noting that and , it follows thatIt follows thatwhich, together with (13) and (16), yields thatWe claim that . If , then by (13). Hence, . This implies that and . Consequently, . Further with , we have . This is a contradiction with the assumption that . Therefore, . In view of , it follows from (19) thatThis is a contradiction. Hence, .
On the other hand, if , we can similarly find with , and there exists such that . Set . Then , and consequently . By the assumption that , we have that . ThenIn addition, in view of that is closed, there exist and such that . Noting that , we get that . So, . It follows thatwhich, together with (21), yields thatwhich implies thatin which (13) has been used. This is a contradiction with that and by . Therefore, . And consequently, . The whole proof is complete.

Corollary 10. Let and satisfy (12) for some constant . Let with and , and satisfywhere . Then all the conclusions of Lemma 9 hold.

Proof. It follows from (12) and (25) thatHence, (13) is satisfied with . Therefore, the assertion holds by Lemma 9. The proof is complete.

Lemma 11. Let be Hermitian. Then for any and any with and ,

Proof. Fix any and any with . We have thatwhere . Since is Hermitian, it follows from (3) of Lemma 2.3 that andwhich implies thatInserting it into (28), we get thatIn addition, by noting that . Therefore, it follows from (31) that (27) holds. This completes the proof.

Remark 12. Lemma 11 extends the result given in [1, p. 270] for closed symmetric operators to closed Hermitian relations.

Lemma 13. Let be Hermitian relations with and . Suppose that is closed and is -bounded with -bound less than 1. Then is closed and .

Proof. By the assumption that is -bounded with -bound less than 1, there exist and such thatSince , there exists such that . By Remark 2.1 we have thatLet . Then andIn addition, since is Hermitian, by Lemma 11 we get thatThis, together with (3.23), yields thatSince , applying Lemma 3.1 to and we get that is closed and . This completes the proof.

Now, we give the main result of the present paper.

Theorem 14. Let be Hermitian relations with and . If is -bounded with -bound less than 1, then is self-adjoint if and only if is self-adjoint.

Proof. Since is -bounded with -bound less than 1, it follows from (2) of Lemma 6 that is closed if and only if is closed. In this case, by Lemma 13 we get that . This, together with Lemma 7, yields that is self-adjoint if and only if is self-adjoint. The proof is complete.

The following result is a direct consequence of Lemma 8 and Theorem 14. This result is the same as that of [13, Theorem 5.2].

Corollary 15. Let be Hermitian and be self-adjoint with . If is -bounded with -bound less than 1, then is self-adjoint.

Remark 16. By the definition of relative boundedness for subspaces, we shall remark that the results about stability of deficiency indices and self-adjointness obtained in the present paper still hold under bounded perturbations.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

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

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grant 11901153), the Natural Science Foundation of Jiangsu Province, China (Grant BK20170298), and the Chinese Universities Scientific Fund (2018B44014).