Relationships between Perturbations of a Linear Relation and Its Operator Part

Xu, Guixin; Zhang, Meirong

doi:https://doi.org/10.1155/2022/9134363

Discrete Dynamics in Nature and Society

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Research Article | Open Access

Volume 2022 | Article ID 9134363 | https://doi.org/10.1155/2022/9134363

Relationships between Perturbations of a Linear Relation and Its Operator Part

Guixin Xu¹and Meirong Zhang²

Academic Editor: Filippo Cacace

Received14 Sept 2022

Revised12 Nov 2022

Accepted23 Nov 2022

Published14 Dec 2022

Abstract

In this study, we establish some relationships between perturbations of a linear relation and its operator part by constructing an operator, which is induced by two linear relations including their closedness, hermiticity, self-adjointness, various spectra, defect indices, and perturbation terms.

1. Introduction

Motivated by the study for the adjoint of nondensely defined linear differential operators, the concept of linear relations, a natural generalization of linear operators, was introduced in [1]. Along with the development of operator theory, the spectral theory for linear relations has been extensively studied and has important applications to several problems (cf., [2–14]). It is worth mentioning that the spectra of linear relations may provide some useful tools for the study of certain operators, such as the maximal and minimal operators corresponding to linear continuous Hamiltonian systems or symmetric linear difference equations [12, 15], and the inverse of certain operators in the study of some Cauchy problems associated with parabolic type equations in Banach spaces [16].

To the best of our knowledge, there are still many important fundamental problems of linear relations that have neither been studied nor completed. In 1961, Arens showed that every closed linear relation in a Hilbert space can be decomposed as an operator part and a purely multivalued part [17]; this decomposition provides a bridge between linear relations and operators. In 1985, Dijksma and de Snoo proved that the operator part of a self-adjoint relation is also a self-adjoint in the Hilbert space [18]. Later, Shi et al. established some relationships between the spectra and various spectra of a closed relation and its operator part as well [19]. Enlightened by these works, the main idea of this study was to construct a linear operator , which is induced by two linear relations and such that some perturbations of can be consistent with its operator part and the various spectra of and are identical. Consequently, one can study some perturbation problems about linear relations by using these results and related existing results about operators.

The rest of this study is organized as follows. In Section 2, some preliminary and auxiliary results that will be used in the sequel are introduced. In addition, a new linear operator is introduced, which is induced by two linear relations and , and its properties are studied. In Section 3, the decomposition of a linear relation is given in terms of reducing subspaces, and relationships between a relation and its decomposition parts are established, including their closedness, hermiticity, self-adjointness, defect indices, and spectra. Using these relationships, the corresponding relationships between relation and operator are given in case if is closed. Further, a concept of trace class linear relations is introduced and relationships between various perturbations of a closed relation and its operator part are discussed. It is shown that if relation is a relatively bounded or compact perturbation term of , or belongs to degenerate or trace class linear relations, then is the same perturbation term of , respectively (seeTheorems 9–12).

2. Preliminaries

In this section, we shall recall some basic concepts, give some fundamental results on linear relations, and introduce a new linear operator induced by two relations and study its properties, which will be used in the sequent sections. This section is divided into three subsections.

2.1. Some Basic Concepts and Fundamental Results of Linear Relations

Let denote normed spaces over a number field . The norm of is defined bywhere and are the norms of the spaces and , respectively, still denoted by without any confusion. Similarly, if and are inner product spaces, then the inner product of is defined by

Obviously, if and are complete, then is also complete.

and denote the sets of complex and real numbers, respectively.

Any linear subspace is called a linear relation (briefly, relation or subspace) of . denotes the set of all the linear relations of . In the case where , denotes , briefly.

The domain , range , and null space of are, respectively, defined by

Furthermore, it denotes the following:

is said to be injective if , and surjective if .

It is evident that if and only if can uniquely determine a linear operator from into whose graph is . For convenience, a linear operator (i.e., single-valued operator) from to will always be identified with a relation in via its graph. In addition, if and only if , i.e., is injective if and only if is a linear operator. Further, is said to be closed if , where is the closure of .

Let and . We define

If , we denote

Further, in the case that and are inner product spaces, if and are orthogonal, that is, for all and , then it denotes the following:

The product of linear relations and is defined by (see [17])

Note that if and are operators, then is also an operator.

In the following, we shall briefly recall the concepts of bounded and compact relations, which were introduced in [20, 21].

Let and be normed spaces and . By , or simply when there is no ambiguity about the relation , we denote the natural quotient map from onto . Clearly, is an operator [20]. Furthermore, it denotes that .

For any given , the norms of and are defined byrespectively. If , then is said to be bounded.

is said to be compact if is compact.

It is evident that is compact if and only if for every bounded sequence and has a convergent subsequence in . Moreover, if is compact, then is bounded [20] (Corollary V.2.3).

The following results come from [20], Proposition II.1.4, and [22], Corollary 1.

Lemma 1. Let and be normed spaces and . Then,(i) for every and ;(ii);(iii)For any given , as for some if and only if for each , there exists such that as .

Definition 1. (see [19, 23, 24]). Let be a Banach space over the complex field and .(1)The set is called the resolvent set of .(2)The set is called the spectrum of .(3)For , if there exists for some , then is called an eigenvalue of , while is called an eigenvector of with respect to the eigenvalue . is called the eigensubspace of , and dim is called the multiplicity of . Further, the set of all the eigenvalues of is called the point spectrum of , denoted by .(4)The essential spectrum of is the set of those points of that are either accumulation points of or isolated eigenvalues of infinite multiplicity.(5)The set is called the discrete spectrum of .

2.2. Concepts and Fundamental Properties of Self-Adjoint Linear Relations

In this subsection, we shall briefly recall the concept of self-adjoint relations and another classification of the spectrum of a self-adjoint relation by its spectral family and some fundamental properties of them.

In this part, is always assumed to be a complex Hilbert space.

Let . The adjoint of is defined by

is said to be Hermitian in if and said to be self-adjoint in if .

Arens introduced the following decomposition for a closed linear relation in [17]:where

It can be easily verified that is an operator, and is an operator if and only if . and are called the operator and pure multivalued parts of , respectively. In addition, they satisfy the following properties:

If is an Hermitian, it is evident that

Throughout the present study, the resolvent set and spectrum of mean those of are restricted to .

Lemma 2 (see [Proposition 2.1, Theorems 2.1, 2.2 and 3.4 in [19]). Let be a closed Hermitian relation in . Then,

is a closed Hermitian operator in , and

Further, if is a self-adjoint relation in , then

Remark 1. It follows fromTheorems 2.1 and 2.2 in [19] that (17) also holds ifis a closed Hermitian relation in.

Lemma 3 (see p. 26 in [18]). If is a self-adjoint relation in , then and are self-adjoint relations in and , respectively.

Lemma 4 (see Theorem 2.5 in [19]). Let be an Hermitian relation in . Then, is self-adjoint in if and only if for some .

Next, we shall briefly recall another classification of the spectrum of a self-adjoint relation by its spectral family, including continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum and their some properties (see [19]). The concept of spectral family of a self-adjoint relation was introduced by Coddington and Dijksma in [8].

Let be a self-adjoint relation in . By Lemma 3, is a self-adjoint operator in . Then, has the following spectral resolution:where is the spectral family of in . The spectral family of the relation is defined bywhere is the zero operator defined on . We denote

Let denote the closed linear hull of all the eigenvectors of and . They are called the discontinuous and continuous subspaces in with respect to , respectively. Further, we denote, and , which are called the singular continuous, absolutely continuous, and singular subspaces in with respect to .

The (spectral) discontinuous, continuous, singular continuous, absolutely continuous, and singular parts of are defined byrespectively.

Definition 2. (see Definition 4.1 in [19]). Let be a self-adjoint relation in . The spectra of , , , and are called the continuous spectrum, singular continuous spectrum, absolutely continuous spectrum, and singular spectrum of , respectively, denoted by , , , and , respectively.

Lemma 5 (see Theorem 4.1 in [19]). Let be a self-adjoint relation in . Then,

2.3. An Operator Induced by Two Linear Relations

In this subsection, we shall first introduce a new linear operator induced by two linear relations, which plays an important role in the present study, and then study its some properties. Further, is assumed to be a complex Hilbert space.

Let and be two linear relations in with and . denotes the following orthogonal projection:

It defines that

It follows from that . This means is single-valued. Further, it is evident that and . Consequently, . Note that in the case that is single-valued.

Proposition 1. Let with and . If is closed and , then

Proof. We suppose that is closed and . Then, is closed by Proposition II.5.3 in [20]. Consequently, can be decomposed asSince , we have thatHence, is single-valued.
We first show that. For any, there exist and such that by (28). Let , where and . Then, . Note that and . We have . Hence, . Therefore, .
Now, we consider the inverse. For any given , there exists such that , , and since . It follows from (11) that can be decomposed as with and . Let , where and . Then, . Since , we have . Hence, . Note that when , we get . This together with implies that . Since and , we have. Consequently, .
Therefore, (26) holds. This completes the proof.
The following result is a direct consequence of (14) and Proposition 1.

Corollary 1. Let with and . If is closed and Hermitian, then

Now, we give a decomposition of in the case that is closed. This decomposition plays an important role in the study of some properties about linear relations in the present study.

Theorem 1. Let with and . If is closed, then

Proof. We suppose that is closed. It is evident that and are orthogonal.
For any , there exists such that , , and . It follows from (11) that with and . Let , where and . Then, and . Hence, belongs to . This implies that .
On the other hand, for any given , it can be decomposed as with and . Let , where and . Given any , it can be decomposed as with and , and we have that and . Consequently,which yields that .
Therefore, (30) holds and the proof is complete.

Corollary 2. Let with and . If and are closed, then

Proof. Suppose that and are closed. It follows from that . Then, by (11), which together with (30) implies . The proof is complete.

Remark 2. In Theorem 3.1 in [25], Shi showed thatunder some conditions, whereandare orthogonal projections. Note that by the assumption that. Then, is an identity mapping fromonto itself and. Hence, Theorem 3.1 in [25] is consistent with the result in Corollary 2 in the case that is closed. Further, Theorem 3.2 in [25] can be directly derived from Corollary 2.

3. Relationships between Perturbations of and

In this section, we shall investigate the relationships between properties of and and the perturbation terms of and . Note that the relation can be decomposed as (30), we shall consider a general decomposition, which is induced by reducing subspaces and discuss the relationships between a relation and its decomposition parts in Subsection 3.1. Using these results, the corresponding relationships between relation and operator are given in Subsection 3.2. Further, relationships between various perturbations of closed relation and its operator part are studied in Subsection 3.3.

3.1. Decomposition of Relations

The following concept of reducing subspace for a linear relation in Banach spaces can be extended to the corresponding one in Hilbert spaces (cf., [18]).

Let be a Banach space. Suppose that has the decompositionwhere and are closed subspaces of and . Let be the projection on along with and . We denote

It is clear that .

If , then is called a reducing subspace of . We also say that reduces or is reduced by . In this case, one has that

It can be easily verified that reduces if and only if reduces . Further, if reduces , thenwhere

Further, we suppose that is a Hilbert space and . If is reduced by , thenwhere and are defined by (37).

Now, we give a relationship between the closedness of and its decomposition parts.

Proposition 2. Let be a Banach space, be reduced by , and be defined by (37). Then, is a closed relation in if and only if is a closed relation in for each .

Proof. It is evident that is closed in for if is closed.
We suppose that is closed in for . Given any sequence with as , there exist such that by (36). Note that is closed for . Then, is bounded by closed graph theorem. So, there exist such that as for . Consequently, with since is closed in for . Hence, again by (36). Therefore, is a closed relation in . This completes the proof.
In the following, we shall discuss the relationships between the hermiticity and self-adjointness of and its decomposition parts, respectively.

Proposition 3. Let be a Hilbert space, be reduced by , and be defined by (37) with . Then, is an Hermitian relation in if and only if is an Hermitian relation in for each .

Proof. Obviously, is an Hermitian relation in for each if is Hermitian.
We suppose that is an Hermitian relation in for . Given any , there exist and such that and by (38). Since is Hermitian in for , we have thatTherefore, is an Hermitian. The proof is complete.

Lemma 6 (see p. 26 in [18]). Let be a Hilbert space and .(i)Let and be closed subspaces in and . If and , then is a reducing subspace of , .(ii)If is a self-adjoint relation in and reduces with , then defined by (37) is a self-adjoint relation in for .

Proposition 4. Let be a Hilbert space, be reduced by , and be defined by (37) with . Then, is a self-adjoint relation in if and only if is a self-adjoint relation in for each .

Proof. The necessity directly follows from (ii) of Lemma 6. Now, we consider the sufficiency. Since is a self-adjoint relation in , we have for by Lemma 4. So, it follows from (37) and (38) thatwhich implies that is a self-adjoint relation in by Lemma 4 and Proposition 3. This completes the proof.
Now, we discuss the relationships between the defect indices of and its decomposition parts.

Definition 3. (see Definition 2.3 in [26]). Let be a Hilbert space and . The subspace is called the defect space of and , and the number is called the defect index of and .
Let be a Hilbert space. It follows from Theorem 2.3 in [26] that is constant in the upper and lower half-planes if is an Hermitian relation in . In this case, it denotes thatwhich are called the positive and negative defect indexes of , respectively. The pair is called the defect indices of (see [26]).

Proposition 5. Let be a Hilbert space, be reduced by , and be defined by (37) with . Then,where is regarded as a relation in for .

Proof. Given any , it suffices to show that

where is the orthogonal complement of in for .

Given any , there exist and such that . So, for every and , we have and . Hence, for . And consequently, .

On the other hand, for any , it can be decomposed as with . For any , there exists , such that by (37) and (38). Hence,which implies that . Then, . Therefore, (43) holds, which yields that (42) holds and this completes the proof.

The following result can be directly derived from Propositions 3 and 5.

Corollary 3. Let be a Hilbert space and be Hermitian and reduced by . Furthermore, let be defined by (37) with . Then,where is regarded as a relation in for .

In the following, we shall investigate the relationships among the spectra and various spectra of and its decomposition parts, including point spectra, essential spectra, discrete spectra, continuous spectra, singular continuous spectra, absolutely continuous spectra, and singular spectra.

The following result is a generalization of self-adjoint case in Hilbert spaces Proposition 3.2 in [19].

Theorem 2. Let be a Banach space, be reduced by , and be defined by (37). Then,where and are the spectrum and resolvent set of in for , respectively.

Proof. It suffices to show that the second relation in (46) holds, which implies that .
We first show that . For any given , and there exists a constant such that for all ,Obviously, (47) holds for all since . Now, we show that is surjective in . For any given , there exists such that since . It follows from (36) that can be decomposed as with and . Then, . Because and , we have and , and thus, . Consequently, is surjective in . Hence, . With a similar argument, one can show . Then, . Therefore, .
Next, we consider the inverse inclusion. For any given , we have for , and there is a constant such thatFor any , there exist and such that . Then, there is such that for , which implies that . Further, by (36), we can get that . Hence, , and consequently, . Note that is bounded by the closed graph theorem. There exists a constant such thatFor any , it follows from (36) that can be uniquely decomposed as with and . By utilizing (48) and (49), one can get thatwhich yields that is a bounded linear operator defined on , and consequently, . It follows that . Therefore, the second relation in (46) holds. The proof is complete.
Now, we discuss the relationships between the point spectra and essential spectra of and its decomposition parts.

Theorem 3. Let be a Banach space, be reduced by , and be defined by (37). Then,

Proof. It suffices to show that for every . It is evident that since and . For any , we have , which can be decomposed as by (36), where and . Note that and , . One can get that for , which implies that and . This yields that . Therefore, . This completes the proof.
The following result can be easily verified by Theorems 2 and 3. So, its detail proofs are omitted.

Theorem 4. Let be a Banach space, be reduced by , and be defined by (47). Then, To the end of this subsection, we shall discuss the relationships between the continuous spectra, singular continuous spectra, absolutely continuous spectra, and singular spectra of and its decomposition parts, separately.
Suppose that is a Hilbert space and . If is self-adjoint and can be decomposed as (38), then is self-adjoint in for by Lemma 6. Let , where , , and . It is evident thatwhere is the spectral family of in for .

It follows from the second relation in (51) thatand consequently,where and are the discontinuous and continuous subspaces in with respect to for .

By utilizing (53) and (55), we can get thatwhich together with (54) and (55) implies thatwhere , , and are the singular continuous, absolutely continuous, and singular subspaces in with respect to for .

It is derived from (37), (38), and (54)–(57) thatwhere , , , , and are the (spectral) discontinuous, continuous, singular continuous, absolutely continuous, and singular parts of in for , respectively.

Theorem 5. Let be a Hilbert space and be self-adjoint and reduced by . Further, let be defined by (37) with . Then,where is regarded as a relation in for .

Proof. This theorem can be directly derived from (i) of Lemma 6, Theorem 2, and (54)–(58).

3.2. Relationships between and

In this subsection, we shall study the relationships between the properties of and , including their closedness, hermiticity, self-adjointness, various spectra, and defect indices. Further, is always assumed to be a complex Hilbert space in this part.

Lemma 7. Let satisfy that and . If is closed and , then is reduced by .

Proof. We suppose that is closed and . It follows from the assumption and thatNote that (15) holds if , which together with Proposition 1 and Proposition 2.1 in [21] implies thatTherefore, is reduced by by Theorem 1 and (i) of Lemma 6. This completes the proof.
The following result comes from [6], and we shall give its proof for completeness.

Lemma 8. Let be closed. Then, is a self-adjoint relation in .

Proof. We suppose that is closed. Obviously, is closed in for any and . Then, is Hermitian in . Note that . Therefore, is a self-adjoint relation in by Lemma 4. The proof is complete.
By Propositions 2 ‐ 4, Lemmas 7 and 8, (61) and (62), one can easily get the following results.

Theorem 6. Let satisfy that and . If is closed and , then(i) is closed if and only if is closed in ;(ii) is an Hermitian relation in if and only if is an Hermitian operator in .(iii) is a self-adjoint relation in if and only if is a self-adjoint operator in .Now, we give a relationship between the spectra and various spectra of and .

Theorem 7. Let satisfy that and .(i)If is closed and , then(ii)If is closed and is self-adjoint, then where is regarded as a relation in .

Proof. The first assertion of Theorem 7 can be easily verified by (16), Theorems 1‐4, and Lemma 7. Suppose that is self-adjoint. Then, . And consequently, reduces by Lemma 7. It follows from (16), and Theorems 1, 5, and 6 that (63) holds. This completes the proof.

Remark 3. By Corollary 2and Lemmas 3 and 5, (63) holds.
The following results can be easily directly derived from (14) and Theorem 7.

Corollary 4. Let satisfy that and . If is closed and Hermitian, thenwhere is regarded as a relation in .
To the end of this subsection, we shall give a relationship between the defect indices of and .

Theorem 8. Let satisfy that and . If is closed and is Hermitian, thenwhere is regarded as a relation in .

Proof. We suppose that is closed and is Hermitian. Then, . And consequently, reduces by Lemma 7. It is derived from Theorem 1, Corollary 3, and (ii) of Theorem 6 that (65) holds. Thus, the proof is complete.

3.3. Relationships between Perturbation Terms of and

In this subsection, we shall discuss the relationships between the perturbation terms of and if is closed. Including relatively bounded and relatively compact perturbation terms, finite rank perturbation term, and trace class perturbation term.

We shall first recall the concepts of relatively bounded and compact relations, which were introduced by Cross [20].

Let and be normed spaces, , and denote the space , where

We then define by for . is called the graph operator of .

Definition 4. (see Definition VII.2.1 in [20]). Let , , and be normed spaces, , and with .(1)The linear relation is said to be -bounded if there exist nonnegative numbers and such that If is -bounded, then the infimum of all numbers for which (67) holds with a constant , is called the -bound of .(2)The linear relation is said to be -compact (or relatively compact to) if is compact, i.e., is compact.

Lemma 9 (see Lemma 2.7 in [22]). Let be a Hilbert space and can be decomposed as (11). Then, (11) holds and

Lemma 10. Let be a Hilbert space and satisfy that and . Then,

Proof. It suffice to show that the first relation in (69) holds. Let . It follows from (i) of Lemma 1 that for any , there exists such that . Let , where and . Then, by (25). Consequently, . Hence, by the arbitrariness of . Therefore, (69) holds and this completes the proof.
By Lemmas 9 and 10, one can easily get the following result.

Theorem 9. Let be a Hilbert space and satisfy that is closed, , and . If is -bounded with -bound less that , then is -bounded with -bound less than , .
Now, we give a relationship between the relatively compact perturbation of and .

Theorem 10. Let be a Hilbert space and satisfy that is closed, , and . If is -compact, then is -compact.

Proof. It follows from (13) and Lemma 9 that . We suppose that is -compact. Then, for any given bounded sequence in , and has a convergent subsequence in . So, for each , there is such that is convergent in by (iii) of Lemma 1. Let with and . Hence, , and is convergent in . This means has a convergent subsequence in . Therefore, is -compact. Thus, the proof is complete.
With a similar argument to that used in the proof of Theorem 10, one can easily show the following results hold.

Corollary 5. Let and be Hilbert spaces and . Then, is a compact relation if and only if is a compact operator, where is an orthogonal projection.

Corollary 6. Let be a Hilbert space and satisfy that and . If is compact, then is compact.

Proposition 6. Let be a Hilbert space and satisfy that and . Then, is an Hermitian relation in if and only if is an Hermitian operator in .

Proof. It follows from the assumption that , which means that is a linear operator in the Hilbert space .
Suppose that is an Hermitian relation in . For any and , there exists such that and , where . Note that and the assumption that is Hermitian, we getHence, is an Hermitian operator in .
Now, we consider the inverse. We assume that is an Hermitian operator in . We fixed . There exist and such that and . Hence, . By the assumption that is Hermitian and the fact that , we getTherefore, is an Hermitian relation in . This completes the proof.
Let be a Hilbert space and . If is self-adjoint, then by Proposition III.1.4 in [20], which together with Proposition 6, can easily achieve the following result.

Corollary 7. Let be an Hilbert space and , satisfying and . If is self-adjoint, then is an Hermitian relation in if and only if is a symmetric operator in , that is, is a densely defined Hermitian operator in .
Now, we introduce the concept of degenerate linear relation, which is a generation of single-valued case.

Definition 5. Letandbe normed spaces and .is said to be degenerate ifis bounded and .
Let be degenerate. It is evident that is degenerate. Hence, is compact (see p.160 in [27]), that is, is compact.
Motivated by the definition of the norm of a linear relation, we introduce the concept of trace class linear relations.

Definition 6. Letandbe Hilbert spaces andwith. We say thatbelongs to trace class relations ifbelongs to trace class operators, whereis the orthogonal projection.

Theorem 11. Let be a Hilbert space and satisfy that and . If is degenerate, then is also degenerate.

Proof. We suppose that is degenerate, that is, and is bounded. Then, is bounded by Lemma 10. Let be a base of . They can be decomposed as with and , . Now, we show that for any given , it can be expressed as a linear combination of , . We then set with , where . There exist and such thatNote that and , we can get that , which yields that . Therefore, is degenerate and the proof is complete.
Now, we recall a necessary and sufficient condition about trace operators (cf., Theorem 7.12 in [28]).

Lemma 11. Let be an operator from Hilbert space into Hilbert space with . Then, belongs to trace class operators if and only if there exist sequences from and from such that for each , and there is a constant sequence for which and can be expressed as

Remark 4. In the case thatin Lemma 11, belongs to trace class operators if and only ifcan be expressed aswhere form an orthonormal family of eigenvectors of and the are the associated (repeated) eigenvalues with (cf., [27], p.543).

Theorem 12. Let be a Hilbert space and with and , and . If belongs to trace class relations, then belongs to trace class operators in .

Proof. The assumptions and implies that .
We suppose that belongs to trace class relations, that is, belongs to trace class operators. It follows from Lemma 11 that there exist sequences and from and from satisfying , , and such thatLet and with and for . We shall show that , where . Given any , there is such that . Since , can be decomposed as with , , and . Then, and . By (75) and the fact that , one can get thatNote that and , we have that . Consequently, . Hence, . Let . By setting and for each , we can get where . Therefore, belongs to trace class in by Lemma 11. This completes the whole proof.

Remark 5. In the present study, we construct a linear operator, which is induced by two linear relations, and then establish the relationships between the perturbation terms of a closed relation and the perturbation terms of its operator part (see Theorems 9–12), and give the relationships between spectrum of a perturbed relation and spectrum of a perturbed operator (see Theorem 7). By using the results obtained in the present study, we shall deeply study stabilities of the spectra of linear relations under some perturbations in our forthcoming study, especially the invariance of the absolutely continuous spectrum of a self-adjoint linear relation under trace class perturbation.

Remark 6. Note that the constructing technique in the present study can also be applied in our previous works. For example, Theorem 5.2 in [29] can be followed by Lemma 3, Corollary 7, Theorems 6 and 9, ([29], Lemma 5.8), and ([27], Theorem V.4.3); Theorem 5.3 in [29] can be followed by Lemmas 2, 3, and 10, Corollaries 4 and 7, Theorem 10, ([29], Lemma 5.8); and ([27], Theorem V.4.10); Theorem 4.1 in [22] can be followed by Lemmas 2 and 3, Corollaries 4 and 7, Theorems 6 and 10, ([29], Lemma 5.8), and ([28], Theorem 9.9), respectively.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Grants no. 11790273).

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Copyright © 2022 Guixin Xu and Meirong Zhang. 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.

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