# Relatively Bounded Perturbations of a Closed Linear Relation and Its Adjoint

Authors: Guixin Xu
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

This paper is concerned with the relatively bounded perturbations of a closed linear relation and its adjoint in Hilbert spaces. A stability result about orthogonal projections onto the ranges of linear relations is obtained. By using this result, two perturbation theorems for a closed relation and its adjoint are given. These results generalize the corresponding results for single-valued linear operators to linear relations and some of which weaken certain assumptions of the related existing results.


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## Body

## 1. Introduction

Perturbation theory is one of the main topics in both pure and applied mathematics. In particular, the perturbations of linear operators (i.e., single-valued operators) have received lots of attention and many useful results have been obtained (cf. [1–4]).However, when considering the adjoint of a nondensely defined linear operator and the minimal and maximal operators corresponding to a linear discrete Hamiltonian system or a linear symmetric difference equation (cf. [5, 6]), the classical perturbation theory of linear operator is not available in these cases. So, we should apply the perturbation theory of multivalued linear operators to study the above problems. Further, multivalued linear operator theory may provide some useful tools for the study of some Cauchy problems associated with parabolic type equations in Banach spaces [7] and boundary value problems for differential operators [8]. Due to these reasons, it is necessary and urgent to study some topics about multivalued linear operators, which are a necessary foundation of research on those related problems about differential or difference operators.Note that the graphGT of a linear operator or multivalued linear operator T from a linear space X to a linear space Y is a linear subspace in the product space X×Y. Further, it is more convenient to introduce concepts of the inverse, closure, and adjoint for linear subspaces. So, we shall directly study linear subspace (briefly, subspace) in the product space X×Y. A subspace is also called a linear relation (briefly, relation). A linear operator always means a single-valued linear operator for convenience in the present paper.To the best of our knowledge, linear relations were introduced by von Neumann [9], motivated by the need to consider adjoint operators of nondensely defined linear differential operators. The operational calculus of linear relations was developed by Arens [10]. His works were followed by many scholars, and some basic concepts, fundamental properties, self-adjoint extension, resolvent, spectrum, and perturbation for linear relations were studied (cf. [5–8, 11–26]).There are still many important fundamental problems about linear relations that have neither been studied nor completed. It is shown that the closedness and self-adjointness of linear relation are stable under relatively bounded perturbation (cf. [22, 25]). However, they have not been specifically and thoroughly studied. In the present paper, enlightened by the methods used in [4], we shall deeply study the stability of a closed linear relation and its adjoint under more general relatively bounded perturbation in Hilbert spaces. The spaces X and Y are always assumed to be Hilbert spaces throughout the present paper. The results obtained in the present paper not only cover the related existing results about operators but also some of them weaken the conditions of the corresponding existing results (see Remarks 1 and 2).The rest of this paper is organized as follows. In Section2, some notations, basic concepts, and fundamental results about linear relations are introduced. In Section 3, we first give in Theorem 1 a stability result about orthogonal projections onto the ranges of linear relations, which generalize the corresponding result ([4], Theorem 5.25) for linear operators to linear relations. Then, we investigate the relatively bounded perturbations of a closed linear relation and its adjoint. It is shown that the adjoint of the sum of two linear relations is equal to the sum of each adjoint (see Theorem 3).


## 2. Preliminaries

In this section, we shall introduce some basic concepts and give some fundamental results about linear relations, which will be used in the sequent sections.LetX and Y be Hilbert spaces over the complex field C. The norm of X×Y is defined by(1)x,y=xX2+yY21/2,x∈X,y∈Y,where ⋅X and ⋅Y are the norms of the spaces X and Y, respectively, still denoted by ⋅ without any confusion. The inner product of X×Y is defined by(2)x1,y1,x2,y2=x1,x2+y1,y2,x1,y1,x2,y2∈X×Y.LetE⊂X, E⊥ denotes the orthogonal complement of E.Any linear subspaceT⊂X×Y is called a linear relation (briefly, relation or subspace) of X×Y.LRX,Y denotes the set of all linear relations of X×Y. In the case that X=Y, LRX denotes LRX,X briefly.The domainDT, range RT, and null space NT of T are, respectively, defined by(3)DT≔x∈X:x,y∈Tfor somey∈Y,RT≔y∈Y:x,y∈Tfor somex∈X,NT≔x∈X:x,0∈T.A linear relationT is said to be injective if NT=0 and surjective if RT=Y. Further, denote(4)Tx≔y∈Y:x,y∈T,T−1≔y,x:x,y∈T.It is evident thatT0=0 if and only if T can uniquely determine a linear operator from DT into Y whose graph is T. For convenience, a linear operator (i.e., single-valued operator) from X to Y will always be identified with a relation in X×Y via its graph. In addition, NT=0 if and only if T−10=0, i.e., T is injective if and only if T−1 is a linear operator. Further, T is said to be closed if T=T¯, where T¯ is the closure of T.LetT,A∈LRX,Y and α∈C. Define(5)αT≔x,αy:x,y∈T,T+A≔x,y+z:x,y∈T,x,z∈S,T−α≔x,y−αx:x,y∈Tin the case thatY=X.IfT∩A=0,0, denote(6)T+˙A≔x1+x2,y1+y2:x1,y1∈T,x2,y2∈A.Further, ifT and A are orthogonal, that is, x,y,u,v=0 for all x,y∈T and u,v∈A, then denote(7)T⊕A≔T+˙A.Theproduct of linear relations T∈LRX,Y and A∈LRY,Z is defined as follows (see [10]):(8)AT≔x,z∈X×Z:x,y∈T,y,z∈Afor somey∈Y.Note that ifA and T are operators, then AT is also an operator.Definition 1.
LetT∈LRX,Y. The adjointT∗ of T is defined as a relation from Y to X by(9)T∗≔f,g∈Y×X:g,x=f,yfor allx,y∈T.
T is said to be Hermitian in X2 if T⊂T∗ and said to be self-adjoint in X2 if T=T∗
LetU be the flip-flop operator from X×Y to Y×X defined by(10)Ux,y=y,−x,x,y∈X×Y.
It is clear from Definition1 that(11)T∗=UT=UT,where the orthogonal complements refer to the component wise inner product in Y×X and X×Y, respectively.
Next, we shall briefly recall the concepts of bounded and relatively bounded relations, which were introduced in [17, 22].
LetT∈LRX,Y. By QT, or simply Q, when there is no ambiguity about the relation T, denote the natural quotient map from Y onto Y/T0¯. Clearly, QT is an operator [17]. Further, denote BX≔x∈X:x≤1.Definition 2.
LetT∈LRX,Y. For any given x∈DT, the norms of Tx and T are defined by(12)Tx≔QTx,T≔QT=supQTx:x∈DT∩BX,respectively. If T∞, T is said to be bounded.Definition 3.
LetT and A be two linear relations in X×Y with DT⊂DA. The linear relation A is said to be T-bounded if there exist nonnegative numbers a and b such that(13)Ax≤ax+bTx,x∈DT.
IfA is T-bounded, then the infimum of all numbers b≥0 for which (13) holds with some constant a≥0 is called the T-bound of A .Lemma 1 (Propositions II.1.4 and II.1.5 in [17]).
LetT and S be two linear relations in X×Y. Then,(i)
Tx=dy,T0=d0,Tx for every x∈DT and y∈Tx(ii)
Tx+Sx≤Tx+Sx for every x∈DT+S(iii)
αTx=αTx for α∈C and x∈DT
Note that the normTx is not a real norm since the following inequality may not hold in general (see Exercise II.1.12 in [17]):(14)S−Tx≥Sx−Tx,x∈DS∩DT.
However, it holds under some conditions.Lemma 2 (Theorem 2.3 in [22]).
LetT,S∈LRX,Y satisfy that DS⊂DT and T0¯⊂S0¯. Then,(15)S−Tx≥Sx−Tx,x∈DS.


## 3. Main Results

In this section, we shall investigate the relatively bounded perturbations of a closed linear relation and its adjoint. For this purpose, we need to discuss the stability of orthogonal projections onto the ranges of linear relations.We first give the following auxiliary results about linear operators.Lemma 3 (Theorem 4.3 in [4]).
LetT be an operator from X to Y and M be a subspace of Y satisfying that RT⊂M¯. Then,(16)Tx=supTx,y:y∈Mwithy=1,x∈DT.Lemma 4 (Theorem 4.33 in [4]).
LetP1 and P2 be two orthogonal projections acting on X. Then, we have(17)P1−P2=maxρ12ρ21,where(18)ρj,k=supPjh:h∈RPk,h≤1.Theorem 1.
LetA,B∈LRX,Y satisfying DA⊂DB and B0⊂A0. Assume that there exists a constant c≥0 such that(19)Bx≤cAxforx∈DA.For everyk∈C, let Pk denote the orthogonal projection onto RA+kB¯. Then, Pk−P0⟶0 as k⟶0.Proof.
Suppose thatDA⊂DB and B0⊂A0. It follows from (Proposition 2.1 in [22]) that(20)A=A+kB−kB,for every k∈C.Case 1.
(c〉0). Let 0k1/2c. For any x∈DA, it follows from (19), (20), and Lemma 1 that(21)Bx≤cA+kB−kBx≤cA+kBx+ckBx≤cA+kBx+12Bx.
Thus,(22)Bx≤2cA+kBx.
LetY=RA+kB¯⊕RA+kB¯. Given any h∈RP0=RA. It can be decomposed as h=h1+h2, where h1∈RA+kB¯ and h2∈RA+kB¯. Let x,g∈A+kB with g=1. There exist g1,g2∈Y such that x,g1∈A, x,g2∈B, and g=g1+kg2. By (i) of Lemma 1, we have that for any ϵ〉0, there is w∈Bx such that(23)wBx+ϵ.
This together with Lemma1, (22), and the fact that g∈RA+kB implies that(24)w<2cA+kBx+ϵ≤2cg+ϵ=2c+ϵ.
Sincex,w,x,g2∈B. We have 0,g2−w∈B. Hence, 0,g2−w∈A by the assumption that B0⊂A0. Consequently, 0,kg2−w∈A. Further, noting that x,g1∈A, we can get that x,g1+kg2−w∈A. Then,(25)Pkh,g=h1,g=h,g=h,g1+kg2​​​​=h,g1+kg2−w+kw=h,kw=k¯h,w.
It follows from (24) that(26)Pkh,g≤khw2c+ϵkh.
Therefore,(27)Pkh≤2ckh,by the arbitrariness of ϵ and Lemma 3.
On the other hand, for anyh′∈RPk=RA+kB, we can prove in a completely analogous way that(28)P0h′≤ckh′,which together with (27) and Lemma 4 yields that Pk−P0≤2ck. Therefore, Pk−P0⟶0 as k⟶0.Case 2.
(c=0). In this case, condition (19) turns into Bx=0. Then, wϵ by (23). This together with (25) implies that Pkh,g≤khwϵkh. Since ϵ is arbitrary, we have that Pkh=0. Similarly, we can obtain that P0h′=0 for every h′∈RA+kB. Therefore, Pk−P0=0. This completes the proof.Remark 1.
Theorem1 is a generalization of Theorem 5.25 in [24] for operators to linear relations.
Next, we shall discuss the perturbations of a closed linear relation and its adjoint. We first recall a stability result of the closedness for linear relations.Lemma 5 (Theorem 6.3 in [22]).
LetT,S∈LRX,Y satisfy that DT⊂DS and S0⊂T0. If S is T-bounded with T-bound less that 1, then T+S is closed if and only if T is closed.Theorem 2.
LetT,S∈LRX,Y satisfy that DT⊂DS and S0⊂T0. Assume that T is closed and S is T-bounded with T-bound b. Further, if b〉0, set(29)Ω=z1b:T+zSis closed.Ifb=0, set(30)Ω=z∈C:T+zSis closed.Then,(i)
The setΩ is open.(ii)
For everyz∈Ω, let Qz denote the orthogonal projection from X×Y onto T+zS. Then, Qz is continuous on Ω (with respect to the norm topology of BX×Y, where BX×Y denote the bounded operators on X×Y).Proof.
(i)
Suppose thatT is closed and S is T-bounded with T-bound b. Set(31)Φ≔r≥0:there existsa≥0such thatSx≤ax+rTx,x∈DT.Then,b=infr:r∈Φ. Note that for every z∈C, we have DT+zS=DT⊂DS and S0⊂T+zS0=T0. Then, by Lemma 2, one has that(32)T+zSx≥Tx−zSx=Tx−zSx.This implies that(33)−zSx≤T+zSx−Tx.Letb〉0. Given any z0∈Ω. There is ϵ>0 such that z01/b+ϵ. By the definition of infimum, there exists r1∈Φ such that r1b+ϵ1/z0. Thus, r1z01. Further, there is a constant a1≥0 such that Sx≤a1x+r1Tx, which together with (33) yields that(34)1−r1z0Sx≤a1x+r1T+z0Sx.Ifb=0, for every z0∈Ω, there exist r2∈Φ and a2≥0 such that r21/z0 and Sx≤a2x+r2Tx. Again, by (33), we get that(35)1−r2z0Sx≤a2x+r2T+z0Sx,which together with (34) yields that S is T+z0S-bounded. Hence, T+zS is closed for z sufficiently near to z0 by Lemma 5. Therefore, Ω is open.(ii)
Define the linear relationsA,B∈LRX,X×Y by(36)Ax=x,f+z0g:x,f∈T,x,g∈S,Bx=0,h:x,h∈S,for everyx∈DT. Obviously, DA=DB=DT, A0=0×T0, and B0=0×S0. Thus, B0⊂A0. Further, it is easy to verify that(37)Ax=T+z0Sx+x,Bx=Sx,and(38)RA=T+z0S,RA+kB=T+z0+kS.
Note thatS is T+z0S-bounded. Then, there is a constant c≥0 such that Bx≤cAx. Therefore, Qz is continuous on Ω by Theorem 1. The proof is complete.
In the following, we shall give a general perturbation result about a closed linear relation and its adjoint.Lemma 6 (Theorem 4.30 in [4]).
LetM and N be closed subspace of X, and let PM and PN be the orthogonal projections onto M and N, respectively. Then, M⊥N if and only if PMPN=0 (or PNPM=0) and if and only if PM+PN is an orthogonal projection.Theorem 3.
LetT,S∈LRX,Y satisfy that DT⊂DS and S0⊂T0. Assume that T is closed, S is T-bounded with T-bound b, and S∗ is T∗-bounded with T∗-bound b′. Let r=maxb,b′, if r〉0, set(39)Ω=z1r:T+zS,T∗+z∗S∗is closed.
Ifr=0, set(40)Ω=z∈C:T+zS,T∗+z∗S∗is closed.Further, denote byΩ0 the connected component of Ω that contains zero. Then,(41)T+zS∗=T∗+z∗S∗,for all z∈Ω0.Proof.
LetQz and Qz′ be the orthogonal projections (in X×Y) onto T+zS and U−1T∗+z∗S∗, respectively, where U is the flip-flop operator defined as (10). It follows from Theorem 2 that the operators Qz and Qz′ depend continuously on z∈Ω. By (11), we get that T⊥=U−1T∗. Hence, X×Y=T⊕U−1T∗ and consequently,(42)Q0+Q0′=I.
By Proposition III.1.5 in [9], we have that T∗+z∗S∗⊂T+zS∗, which together with (11) implies that(43)U−1T∗+z∗S∗⊂U−1T+zS∗=T+zS,z∈Ω.
Thus,QzQz′=Qz′Qz=0 for z∈Ω by Lemma 6. Again by Lemma 6, one has that I−Qz−Qz′ is an orthogonal projection for any z∈Ω. Consequently, the value of I−Qz−Qz′ can be only 0 or 1. Since I−Qz−Qz′ depends continuously on z∈Ω, which together with (42) yields that(44)I−Qz−Qz′=I−Q0−Q0′=0,z∈Ω0.
Hence,X×Y=T+zS⊕U−1T∗+z∗S∗. Therefore, T∗+z∗S∗=UT+zS=T+zS∗ for all z∈Ω0 by (11). This completes the proof.
If the relative bounds ofS with respect to T and of S∗ with respect to T∗ are less than one, then z∈C:z≤1⊂Ω0 by Lemma 5. Therefore, by Theorem 3, we have the following result.Corollary 1.
LetT,S∈LRX,Y satisfy that DT⊂DS and S0⊂T0. Assume that T is closed, S is T-bounded with T-bound less than 1, and S∗ is T∗-bounded with T∗-bound less than 1. Then, T+S∗=T∗+S∗.Remark 2.
LetT∈LRX be self-adjoint and S∈LRX be Hermitian with DT⊂DS. Then, S0⊂T0 by Lemma 5.8 in [25]. If S is T-bounded with T-bound less than one, then S∗ is T∗-bounded with T∗-bound less than one by Corollary III.1.13 in [17], and thus T+S∗=T∗+S∗=T+S by Corollary 1, which means T+S is self-adjoint. So, we shall remark that Corollary 1 generalizes Theorem 5.2 in [25] for self-adjoint relations to general linear relations.Remark 3.
For studying stabilities of spectra of self-adjoint linear relations, we need to study stabilities of self-adjointness of linear relations. The results obtained in the present paper can be available in this case. Further, we shall apply these results to study stabilities of essential spectra of self-adjoint linear relations under some perturbations in our forthcoming paper.



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