Relatively Bounded Perturbations of a Closed Linear Relation and Its Adjoint
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.
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  and boundary value problems for differential operators . 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 graph of a linear operator or multivalued linear operator from a linear space to a linear space is a linear subspace in the product space . 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 . 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 , motivated by the need to consider adjoint operators of nondensely defined linear differential operators. The operational calculus of linear relations was developed by Arens . 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 , 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 and 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 Section 2, 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 (, 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).
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.
Let and be Hilbert spaces over the complex field . The norm of is defined bywhere and are the norms of the spaces and , respectively, still denoted by without any confusion. The inner product of is defined by
Let , denotes the orthogonal complement of .
Any linear subspace is called a linear relation (briefly, relation or subspace) of . denotes the set of all linear relations of . In the case that , denotes briefly.
The domain , range , and null space of are, respectively, defined by
A linear relation is said to be injective if and surjective if . Further, denote
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 . Define
If , denote
Further, if and are orthogonal, that is, for all and , then denote
The product of linear relations and is defined as follows (see ):
Note that if and are operators, then is also an operator.
Definition 1. Let . The adjoint of is defined as a relation from to by is said to be Hermitian in if and said to be self-adjoint in if
Let be the flip-flop operator from to defined byIt is clear from Definition 1 thatwhere the orthogonal complements refer to the component wise inner product in and , respectively.
Next, we shall briefly recall the concepts of bounded and relatively bounded relations, which were introduced in [17, 22].
Let . By , or simply , when there is no ambiguity about the relation , denote the natural quotient map from onto . Clearly, is an operator . Further, denote .
Definition 2. Let . For any given , the norms of and are defined byrespectively. If , is said to be bounded.
Definition 3. Let and be two linear relations in with . The linear relation is said to be -bounded if there exist nonnegative numbers and such thatIf is -bounded, then the infimum of all numbers for which (13) holds with some constant is called the -bound of .
Lemma 1 (Propositions II.1.4 and II.1.5 in ). Let and be two linear relations in . Then,(i) for every and (ii) for every (iii) for and Note that the norm is not a real norm since the following inequality may not hold in general (see Exercise II.1.12 in ):However, it holds under some conditions.
Lemma 2 (Theorem 2.3 in ). Let satisfy that and . Then,
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 ). Let be an operator from to and be a subspace of satisfying that . Then,
Lemma 4 (Theorem 4.33 in ). Let and be two orthogonal projections acting on . Then, we havewhere
Theorem 1. Let satisfying and . Assume that there exists a constant such that
For every , let denote the orthogonal projection onto . Then, as .
Proof. Suppose that and . It follows from (Proposition 2.1 in ) thatfor every .
Case 1. (). Let . For any , it follows from (19), (20), and Lemma 1 thatThus,Let . Given any . It can be decomposed as , where and . Let with . There exist such that , , and . By (i) of Lemma 1, we have that for any , there is such thatThis together with Lemma 1, (22), and the fact that implies thatSince . We have . Hence, by the assumption that . Consequently, . Further, noting that , we can get that . Then,It follows from (24) thatTherefore,by the arbitrariness of and Lemma 3.
On the other hand, for any , we can prove in a completely analogous way thatwhich together with (27) and Lemma 4 yields that . Therefore, as .
Case 2. (). In this case, condition (19) turns into . Then, by (23). This together with (25) implies that . Since is arbitrary, we have that . Similarly, we can obtain that for every . Therefore, . This completes the proof.
Remark 1. Theorem 1 is a generalization of Theorem 5.25 in  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 ). Let satisfy that and . If is -bounded with -bound less that 1, then is closed if and only if is closed.
Theorem 2. Let satisfy that and . Assume that is closed and is -bounded with -bound . Further, if , set
If , set
Then,(i)The set is open.(ii)For every , let denote the orthogonal projection from onto . Then, is continuous on (with respect to the norm topology of , where denote the bounded operators on ).
Proof. (i)Suppose that is closed and is -bounded with -bound . Set Then, . Note that for every , we have and . Then, by Lemma 2, one has that This implies that Let . Given any . There is such that . By the definition of infimum, there exists such that . Thus, . Further, there is a constant such that , which together with (33) yields that If , for every , there exist and such that and . Again, by (33), we get that which together with (34) yields that is -bounded. Hence, is closed for sufficiently near to by Lemma 5. Therefore, is open.(ii)Define the linear relations by for every . Obviously, , , and . Thus, . Further, it is easy to verify thatandNote that is -bounded. Then, there is a constant such that . Therefore, is continuous on 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 ). Let and be closed subspace of , and let and be the orthogonal projections onto and , respectively. Then, if and only if (or ) and if and only if is an orthogonal projection.
Theorem 3. Let satisfy that and . Assume that is closed, is -bounded with -bound , and is -bounded with -bound . Let , if , setIf , set
Further, denote by the connected component of that contains zero. Then,for all .
Proof. Let and be the orthogonal projections (in ) onto and , respectively, where is the flip-flop operator defined as (10). It follows from Theorem 2 that the operators and depend continuously on . By (11), we get that . Hence, and consequently,By Proposition III.1.5 in , we have that , which together with (11) implies thatThus, for by Lemma 6. Again by Lemma 6, one has that is an orthogonal projection for any . Consequently, the value of can be only 0 or 1. Since depends continuously on , which together with (42) yields thatHence, . Therefore, for all by (11). This completes the proof.
If the relative bounds of with respect to and of with respect to are less than one, then by Lemma 5. Therefore, by Theorem 3, we have the following result.
Corollary 1. Let satisfy that and . Assume that is closed, is -bounded with -bound less than 1, and is -bounded with -bound less than 1. Then, .
Remark 2. Let be self-adjoint and be Hermitian with . Then, by Lemma 5.8 in . If is -bounded with -bound less than one, then is -bounded with -bound less than one by Corollary III.1.13 in , and thus by Corollary 1, which means is self-adjoint. So, we shall remark that Corollary 1 generalizes Theorem 5.2 in  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.
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
This research was supported by the National Natural Science Foundation of China (Grant no. 11790273).
S. Goldberg, Unbounded Linear Operators, Theory and Applications, McGraw-Hill, New York, NY, USA, 1966.
T. Kato, Perturbation Theory for Linear Operators, Spinger-Verlag, New York, NY, USA, 2nd edition, 1984.
M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, Cambridge, CA, USA, 1972.
J. Weidmann, Linear Operators in Hilbert Spaces Graduate Texts in Math, Springer-Verlag, New York, NY, USA, vol. 68, 1980.
G. Ren and Y. Shi, “Self-adjoint extensions for discrete linear Hamiltonian systems,” Linear Algebra and Its Applications, vol. 454, pp. 1–48, 2014.View at: Publisher Site | Google Scholar
Y. Shi and H. Sun, “Self-adjoint extensions for second-order symmetric linear difference equations,” Linear Algebra and Its Applications, vol. 434, no. 4, pp. 903–930, 2011.View at: Publisher Site | Google Scholar
A. Favini and A. Yagi, “Multivalued linear operators and degenerate evolution equations,” Annali di Matematica Pura ed Applicata, vol. 163, no. 1, pp. 353–384, 1993.View at: Publisher Site | Google Scholar
J. Behrndt, S. Hassi, and H. de Snoo, “Boundary value problems, wely functions, and differential operators,” Monographs in Mathematics, vol. 108, 2020.View at: Google Scholar
J. von Neumann, “Functional operator II: the geometry of orthogonal spaces,” Annals of Mathematics Studies, vol. 22, 1950.View at: Google Scholar
R. Arens, “Operational calculus of linear relations,” Pacific Journal of Mathematics, vol. 11, no. 1, pp. 9–23, 1961.View at: Publisher Site | Google Scholar
T. Álvarez, A. Ammar, and A. Jeribi, “On the essential spectra of some matrix of linear relations,” Mathematical Methods in the Applied Sciences, vol. 37, no. 5, pp. 620–644, 2014.View at: Publisher Site | Google Scholar
T. Y. Azizov, J. Behrndt, P. Jonas, and C. Trunk, “Compact and finite rank perturbations of closed linear operators and relations in Hilbert spaces,” Integral Equations and Operator Theory, vol. 63, no. 2, pp. 151–163, 2009.View at: Publisher Site | Google Scholar
A. G. Baskakov and A. S. Zagorskii, “Spectral theory of linear relations on real Banach spaces,” Mathematical Notes, vol. 81, no. 1-2, pp. 15–27, 2007.View at: Publisher Site | Google Scholar
E. Chafai and M. Mnif, “Perturbation of normally solvable linear relations in paracomplete spaces,” Linear Algebra and Its Applications, vol. 439, no. 7, pp. 1875–1885, 2013.View at: Publisher Site | Google Scholar
E. A. Coddington, “Extension theory of formally normal and symmetric subspaces,” Memoirs of the American Mathematical Society, vol. 134, 1973.View at: Publisher Site | Google Scholar
E. A. Coddington and A. Dijksma, “Self-adjoint subspaces and Eigen function expansions for ordinary differential subspaces,” Journal of Differential Equations, vol. 20, no. 2, pp. 473–526, 1976.View at: Publisher Site | Google Scholar
R. Cross, “Multivalued linear operators,” Monographs and Textbooks in Pure and Applied Mathematics, vol. 213, 1998.View at: Google Scholar
R. Cross, A. Favini, and Y. Yakubov, “Perturbation results for multivalued linear operators,” Parabolic Problems, vol. 80, pp. 111–130, 2011.View at: Publisher Site | Google Scholar
A. Dijksma and H. de Snoo, “Self-adjoint extensions of symmetric subspaces,” Pacific Journal of Mathematics, vol. 54, no. 1, pp. 71–100, 1974.View at: Publisher Site | Google Scholar
Y. Shi, C. Shao, and Y. Liu, “Resolvent convergence and spectral approximations of sequences of self-adjoint subspaces,” Journal of Mathematical Analysis and Applications, vol. 409, no. 2, pp. 1005–1020, 2014.View at: Publisher Site | Google Scholar
Y. Shi, C. Shao, and G. Ren, “Spectral properties of self-adjoint subspaces,” Linear Algebra and Its Applications, vol. 438, no. 1, pp. 191–218, 2013.View at: Publisher Site | Google Scholar
Y. Shi, G. Xu, and G. Ren, “Boundedness and closedness of linear relations,” Linear and Multilinear Algebra, vol. 66, no. 2, pp. 309–333, 2018.View at: Publisher Site | Google Scholar
H. Sun, Q. Kong, and Y. Shi, “Essential spectrum of singular discrete linear Hamiltonian systems,” Mathematische Nachrichten, vol. 289, no. 2-3, pp. 343–359, 2016.View at: Publisher Site | Google Scholar
H. Sun and Y. Shi, “Spectral properties of singular discrete linear Hamiltonian systems,” Journal of Difference Equations and Applications, vol. 20, no. 3, pp. 379–405, 2014.View at: Publisher Site | Google Scholar
G. Xu and Y. Shi, “Perturbations of spectra of closed subspaces in Banach spaces,” Linear Algebra and Its Applications, vol. 531, pp. 547–574, 2017.View at: Publisher Site | Google Scholar
G. Xu and Y. Shi, “Essential spectra of self-adjoint relations under relatively compact perturbations,” Linear and Multilinear Algebra, vol. 66, no. 12, pp. 2438–2467, 2018.View at: Publisher Site | Google Scholar