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We provide necessary and sufﬁcient conditions for a pair S, T of Hilbert space operators in order that they satisfy S ¼ T and T ¼ S. As a main result we establish an improvement of von Neumann’s classical theorem on the positive self-adjoint- ness of S S for two variables. We also give some new characterizations of self- adjointness and skew-adjointness of operators, not requiring their symmetry or skew-symmetry, respectively. Keywords Linear operator Linear relation Adjoint Symmetric operator Self-adjoint operator Operator matrix Mathematics Subject Classiﬁcation 47A5 47B25 1 Introduction The adjoint of an unbounded linear operator was ﬁrst introduced by John von Neumann in  as a profound ingredient for developing a rigorous mathematical framework for quantum mechanics. By deﬁnition, the adjoint of a densely deﬁned linear transformation S, acting between two Hilbert spaces, is an operator T with the largest possible domain such that Dedicated to Professor Franciszek Hugon Szafraniec on the occasion of his 80th birthday. Communicated by Daniel Pellegrino. & Zsigmond Tarcsay email@example.com Zolta´n Sebestye´n firstname.lastname@example.org Department of Applied Analysis and Computational Mathematics, Eo¨tvo¨s L. University, Pa´zma´ny Pe´ter se´ta´ny 1/c., Budapest 1117, Hungary Range-kernel characterizations of operators… 1027 ðSx j yÞ¼ðx j TyÞð1Þ holds for every x from the domain of S. The adjoint operator, denoted by S ,is therefore ‘‘maximal’’ in the sense that it extends every operator T that has property (1). On the other hand, every restriction T of S fulﬁlls that adjoint relation. Thus, in order to decide whether an operator T is identical with the adjoint of S it seems reasonable to restrict ourselves to investigating those operators T that have property (1). This issue was explored in detail in  by means of the operator matrix I T SI cf. also [8, 11, 13, 14]. In the present paper we continue to examine the conditions under which an operator T is equal to the adjoint S of S. Nevertheless, as opposed to the situation treated in the cited papers, we do not assume that S and T are adjoint to each other in the sense of (1). Observe that condition (1) is equivalent to identity S \ T ¼ T : ð2Þ So, still under condition (1), T is equal to the adjoint of S if and only if S \ T ¼ S . In the present paper we are going to guarantee equality S ¼ T by imposing new conditions, weaker than (1), by means of the kernel and range spaces. Roughly speaking, we only require that the intersection of the graphs of S and T be, in a sense, ‘‘large enough’’. We also establish a criterion in terms of the norm of the resolvent of the operator matrix 0 T M ¼ : S;T S 0 As an application we gain some characterizations of self-adjoint, skew-adjoint and unitary operators, thereby generalizing some analogous results by Nieminen  (cf. also ). 2 Preliminaries Throughout this paper, H and K will denote real or complex Hilbert spaces. By an operator S between H and K we mean a linear map S : H! K whose domain dom S is a linear subspace of H. We stress that, unless otherwise indicated, linear operators are not assumed to be densely deﬁned. However, the adjoint of such an operator can only be interpreted as a ‘‘multivalued operator’’, that is, a linear relation. Therefore we are going to collect here some basic notions and facts on linear relations. A linear relation between two Hilbert spaces H and K is nothing but a linear subspace S of the Cartesian product HK, respectively, a closed linear relation is just a closed subspace of HK. To a linear relation S we associate the following subspaces 1028 Z. Tarcsay and Z. Sebestyén dom S ¼fh 2H : ðh; kÞ2 Sg ker S ¼fh 2H : ðh; 0Þ2 Sg which are referred to as the domain, range, kernel and multivalued part of S, respectively. Every linear operator when identiﬁed with its graph is a linear relation with trivial multivalued part. Conversely, a linear relation whose multivalued part consists only of the vector 0 is (the graph of) an operator. A notable advantage of linear relations, compared to operators, lies in the fact that one might deﬁne the adjoint without any further assumption on the domain. Namely, the adjoint of a linear relation S will be again a linear relation S between K and H, given by S :¼VðSÞ : Here, V : HK ! KH stands for the ‘ﬂip’ operator Vðh; kÞ:¼ðk; hÞ. It is seen immediately that S is automatically a closed linear relation and satisﬁes the useful identity S ¼ S ð¼:ðS Þ Þ: On the other hand, a closed linear relation S entails the following orthogonal decomposition of the product Hilbert space KH: S VðSÞ¼ K H: Note that another equivalent deﬁnition of S is obtained in terms of the inner product as follows: 0 0 0 0 S ¼fðk ; h Þ2K H : ðk j k Þ¼ðh j h Þ for all ðh; kÞ2 Sg: 0 0 In other words, ðk ; h Þ2 S holds if and only if 0 0 ðk j k Þ¼ðh j hÞ8ðh; kÞ2 S: In particular, if S is a densely deﬁned operator, then the relation S coincides with the usual adjoint operator of S. Recall also the dual identities ? ? ker S ¼ðran SÞ ; mul S ¼ðdom SÞ ; where the second equality tells us that the adjoint of a densely deﬁned linear relation is always a (single valued) operator. For further information on linear relations we refer the reader to [1, 2, 4, 10]. 3 Operators which are adjoint of each other Arens  characterized the equality S ¼ T of two linear relations in terms of their kernel and range (see Corollary 2). Below we provide a similar characterization of S T. Observe that the intersection S \ T of the linear relations S and T is again a linear relation, but this is not true for their union S [ T as it is not a linear subspace Range-kernel characterizations of operators… 1029 in general. The linear span of S [ T will be denoted by S _ T, which in turn is a linear relation. Proposition 1 Let S and T be linear relations between two vector spaces. Then the following three statements are equivalent: (i) S T, (ii) ker S ker T and ran S ran ðS \ TÞ, (iii) ran S ran T and ker ðS _ TÞ ker T. Proof It is clear that (i) implies both (ii) and (iii). Suppose now (ii) and let ðh; kÞ2 S then there exists u with ðu; kÞ2 T \ S. Consequently, ðh u; 0Þ2 S, i.e., h u 2 ker S ker T. Hence ðh; kÞ¼ ðh u; 0Þþðu; kÞ2 T þ T T ; which yields S T, so (ii) implies (i). Finally, assume (iii) and take ðh; kÞ2 S. Then ðu; kÞ2 T for some u and hence ðh u; 0Þ2 S _ T, i.e., h u 2 ker T. Consequently, ðh; kÞ¼ðh u; 0Þþðu; kÞ2 T ; which yields S T. h Corollary 1 Let S and T be two linear relations between two vector spaces. The following three statements are equivalent: (i) S ¼ T, (ii) ker S ¼ ker T and ran S þ ran T ran ðS \ TÞ, (iii) ran S ¼ ran T and ker ðS _ TÞ ker ðS \ TÞ. Corollary 2 Let S and T be linear relations between two vector spaces such that S T. Then the following assertions are equivalent: (i) S ¼ T, (ii) ker S ¼ ker T and ran S ¼ ran T. In [17, Theorem 2.9] M. H. Stone established a simple yet effective sufﬁcient condition for an operator to be self-adjoint: a densely deﬁned symmetric operator S is necessarily self-adjoint provided it is surjective. In that case, it is invertible with bounded and self-adjoint inverse due to the Hellinger–Toeplitz theorem. Here, density of the domain can be dropped from the hypotheses: a surjective symmetric operator is automatically densely deﬁned (see also [16, Corollary 6.7] and [15, Lemma 2.1]). Below we establish a generalization of Stone’s result for a pair of operators. Proposition 2 Let H; K be real or complex Hilbert spaces and let S : H!K and T : K!H be (not necessarily densely deﬁned or closed) linear operators such that 1030 Z. Tarcsay and Z. Sebestyén ran ðS \ T Þ¼K and ran ðT \ S Þ¼ H: Then S and T are both densely deﬁned operators such that S ¼ T and T ¼ S. Proof For brevity, introduce the following notations :¼S \ T ; T :¼T \ S : 0 0 Observe that S and T are adjoint to each other in the sense that 0 0 ðS x j yÞ¼ðx j T yÞ; x 2 dom S ; y 2 dom T : 0 0 0 0 We claim that S and T are densely deﬁned: let z 2ðdom S Þ , then by surjectivity, 0 0 0 z ¼ T v for some v 2 dom T . Hence 0 0 0 ¼ðx j zÞ¼ðx j T vÞ¼ðS x j vÞ; x 2 dom S ; 0 0 0 which implies v ¼ 0 and also z ¼ 0. The same argument shows that T is densely deﬁned too. We see now that S and T are densely deﬁned operators such that ? ? ker S ðran S Þ ¼f0g; ker T ¼ðran TÞ ¼f0g; and ran ðS \ T Þ¼K. Corollary 1 applied to S and T implies that S ¼ T . The same argument yields equality S ¼ T. h Corollary 3 Let S : H!K and T : K!H be (not necessarily densely deﬁned) surjective operators such that ðSx j yÞ¼ ðx j TyÞ; x 2 dom S; y 2 dom T : Then S and T are both densely deﬁned operators such that S ¼ T and T ¼ S. From Proposition 2 we gain a sufﬁcient condition of self-adjointness without the assumptions of being symmetric or densely deﬁned: Corollary 4 Let H be a Hilbert space and let S : H!H be a linear operator such that ran ðS \ S Þ¼ H. Then S is densely deﬁned and self-adjoint. Proof Apply Proposition 2 with T :¼S. h Clearly, if S is a symmetric operator, then S \ S ¼ S. Hence we retrieve [17, Theorem 2.9] by M. H. Stone as an immediate consequence (cf. also [16, Corollary 6.7]): Corollary 5 Every surjective symmetric operator is densely deﬁned and self-adjoint. In the next result we give a necessary and sufﬁcient condition for an operator S to be identical with the adjoint of a given operator T. Theorem 1 Let H; K be real or complex Hilbert spaces and let S : H!K and T : K!H be (not necessarily densely deﬁned or closed) linear operators. The following two statements are equivalent: Range-kernel characterizations of operators… 1031 (i) T is densely deﬁned and S ¼ T , (ii) (a) ðran TÞ ¼ ker S, (b) ran S þ ran T ran ðS \ T Þ. Proof It is obvious that (i) implies (ii). Assume now (ii) and for sake of brevity introduce the operator S :¼S \ T : We start by establishing that T is densely deﬁned. Let g 2ðdom T Þ , then ð0; gÞ2 T , i.e., g 2 ran T . By (ii) (a), T g ¼ S h ¼ Sh for some h 2 dom S . Then it follows that ðh; ShÞ2 T and therefore ðTk j hÞ¼ðk j ShÞ¼ðk j gÞ¼ 0; k 2 dom T ; whence we infer that h 2ðran TÞ . Again by (ii) (a) we have h 2 ker S and thus g ¼ Sh ¼ 0. This proves that T is densely deﬁned and as a consequence, T is an operator. Next we prove that T S: ð3Þ To see this consider g 2 dom T . By (ii) (b), T g ¼ S h ¼ Sh ¼ T h for some h 2 dom S . Then it follows that T ðg hÞ¼ 0, i.e., g h 2 ker T ¼ðran TÞ ¼ ker S: Consequently, g ¼ðg hÞþ h 2 dom S and Sg ¼ Sh ¼ T g, which proves (3). It only remains to show that the converse inclusion S T ð4Þ holds also true. For let g 2 dom S and choose h 2 dom S such that Sg ¼ S h ¼ T h ¼ Sh: Then g h 2 ker S ¼ðran TÞ ¼ ker T whence we get g ¼ðg hÞþ h 2 dom T and T g ¼ T h ¼ Sg, which proves (4). h A celebrated theorem by von Neumann  states that S S and SS are positive and selfadjoint operators provided that S is a densely deﬁned and closed operator between H and K: In that case, I þ S S and I þ SS are both surjective. In  it has 1032 Z. Tarcsay and Z. Sebestyén been proved that the converse is also true: If I þ S S and I þ SS are both surjective operators, then S is necessarily closed (cf. also ). Below, as the main result of the paper, we establish an improvement of Neumann’s theorem: Theorem 2 Let H; K be real or complex Hilbert spaces and let S : H!K and T : K!H be linear operators and introduce the operators S :¼S \ T and T :¼T \ S . The following statements are equivalent: (i) S, T are both densely deﬁned and they are adjoint of each other: S ¼ T and T ¼ S, (ii) ran ðI þ T S Þ¼H and ran ðI þ S T Þ¼K. 0 0 0 0 Proof It is clear that (i) implies (ii). To prove the converse implication observe ﬁrst that ðS u j vÞ¼ ðu j T vÞ; u 2 dom S ; v 2 dom T : 0 0 0 0 We start by showing that S is densely deﬁned. Take a vector g 2ðdom S Þ , then there is u 2 dom S such that g ¼ u þ T S u. Consequently, 0 0 0 0 ¼ðu j gÞ¼ðu j uÞþðT S u j uÞ¼ kuk þkS uk; 0 0 0 whence u ¼ 0, and therefore also g ¼ 0. It is proved analogously that T is densely deﬁned too, and therefore the adjoint relations S and T are operators such that 0 0 S T and T S . 0 0 0 0 We are going to prove now that S and T are adjoint of each other, i.e., 0 0 S ¼ T ; T ¼ S : ð5Þ 0 0 0 0 Consider a vector g 2 dom T and take u 2 dom S and v 2 dom T such that 0 0 0 g ¼ u þ T S u and T g ¼ v þ S T v: 0 0 0 0 0 Since u is in dom T we infer that T S u 2 dom T and hence 0 0 0 0 T g ¼ T u þ T T S u: 0 0 0 0 0 It follows then that 0 ¼ v T u þ S T v T T S u ¼ðI þ T T Þðv S uÞ 0 0 0 0 0 0 0 0 0 which yields v ¼ S u 2 dom T : As a consequence we obtain that 0 0 g ¼ u þ T S u ¼ v þ T v; 0 0 0 and therefore that g 2 dom S . This proves the ﬁrst equality of (5). The second one is proved in a similar way. Now we can complete the proof easily: since S T and T T it follows that 0 0 T ¼ S T T ; 0 0 0 whence T ¼ T ¼ S , and therefore T S. On the other hand, T S implies 0 0 Range-kernel characterizations of operators… 1033 S S T ¼ T ; whence we conclude that S ¼ T . It can be proved in a similar way that T ¼ S . h As an immediate consequence we conclude the following result: Corollary 6 Let H and K be real or complex Hilbert spaces and let S : H!K be a densely deﬁned operator. The following statements are equivalent: (i) S is closed, (ii) S S and SS are self-adjoint operators, (iii) ran ðI þ S SÞ¼H and ran ðI þ SS Þ¼K. Proof Apply Theorem 2 with T :¼S . h In the ensuing theorem we provide a renge-kernel characterization of operators T that are identical with the adjoint S of a densely deﬁned symmetric operator S.We stress that no condition on the closedness of the operator or density of the domain is imposed. On the contrary: we get those properties from the other conditions. Theorem 3 Let H be a real or complex Hilbert space and let T : H!H be a (not necessarily densely deﬁned or closed) linear operator and let T :¼T \ T . The following two statements are equivalent: (i) there exists a densely deﬁned symmetric operator S such that S ¼ T, (ii) (a) ker T ¼ðran T Þ , (b) ran T ¼ ran T ¼ ran T . In particular, if any of the equivalent conditions (i), (ii) is satisﬁed, then T is a densely deﬁned and closed operator such that T T. Proof It is straightforward that (i) implies (ii) so we only prove the converse. We start by proving that T is densely deﬁned. Take g 2ðdom TÞ , then g 2 mul T ran T . By (ii) (b), there exists h 2 dom T such that g ¼ T h ¼ Th. 0 0 Consequently, ðh; gÞ2 T and for every f 2 dom T, ðTf j hÞ¼ ðf j gÞ¼ 0; which yields h 2ðran TÞ . Observe that (ii) (a) and (b) together imply that ðran TÞ ¼ ker T ; ð6Þ whence we infer that h 2 ker T and therefore that g ¼ Th ¼ 0. This means that T is a (single valued) operator. Our next claim is to show that T T : ð7Þ 1034 Z. Tarcsay and Z. Sebestyén To this end, let g 2 dom T , then T g ¼ T h for some h 2 dom T . From inclusion 0 0 0 T T we conclude that g h 2 ker T ¼ðran TÞ , thus g ¼ðg hÞþ h 2 dom T and Tg ¼ Th ¼ T h ¼ T g; which proves (7). Next we show that T is densely deﬁned too, i.e., T is closable. To this end consider a vector g 2ðdom T Þ ¼ mul T . Since mul T ran T ,we can ﬁnd a vector h 2 dom T such that g ¼ T h. For every k 2 dom T , 0 0 ðh j T kÞ¼ ðTh j kÞ¼ðg j kÞ¼ 0; thus h 2ðran T Þ . By (ii) (a) we infer that h 2 ker T and hence g ¼ Th ¼ 0 and thus ðdom T Þ ¼f0g, as it is claimed. Finally we show that T is closed. Take g 2 dom T , then T g ¼ Th for some h 2 dom T, according to assumption (ii) (b). Hence g h 2 ker T ¼ðran T Þ , thus g h 2 ker T because of (ii) (a). Con- sequently, g ¼ðg hÞþ h 2 dom T which proves identity T ¼ T . Summing up, S:¼T is a densely deﬁned operator such that S T ¼ S . In other words, T is identical with the adjoint S of the symmetric operator S. h 4 Characterizations involving resolvent norm estimations Let H and K be real or complex Hilbert spaces. For given two linear operators S : H!K and T : K!H, let us consider the operator matrix 0 T M :¼ ; S;T S 0 acting on the product Hilbert space H K. More precisely, M is an operator S;T acting on its domain dom M :¼dom S dom T by S;T M ðh; kÞ:¼ðTk; ShÞðh 2 dom S; k 2 dom TÞ: S;T Assume that a real or complex number k 2 K belongs to the resolvent set qðM Þ, S;T which means that k T M k ¼ S;T S k has an everywhere deﬁned bounded inverse. In that case, for brevity’s sake, we introduce the notation R ðkÞ:¼ðM kÞ S;T S;T for the corresponding resolvent operator. In the present section, we are going to establish some criteria, by means of norms of the resolvent operator R ðkÞ, under which the operators S and T are adjoint of S;T each other. Our approach is motivated by the classical paper of T. Nieminen  (cf. Range-kernel characterizations of operators… 1035 also ). We emphasize that our framework is more general than that of  for many ways: we do not assume that the operators under consideration are densely deﬁned or closed, and also the underlying space may be real or complex. Theorem 4 Let S : H!K and T : K!H be linear operators between the real or complex Hilbert spaces H and K. The following assertions are equivalent: (i) S and T are densely deﬁned such that S ¼ T and T ¼ S, (ii) every non-zero real number t belongs to the resolvent set of M and S;T kR ðtÞk ; 8t 2 R; t 6¼ 0: ð8Þ S;T jtj Proof Let us start by proving that (i) implies (ii). Assume therefore that S is densely deﬁned and closed and that T ¼ S . Consider a non-zero real number t and a pair of vectors h 2 dom S and k 2 dom S , then we have t S h 2 2 ¼kth S kk þkSh þ tkk St k 2 2 2 2 2 ¼ t ½khk þkkk þ kShk þkS kk which implies that M þ t is bounded from below and the norm of its inverse S;T R ðtÞ satisﬁes (8). However it is not yet clear that R ðtÞ is everywhere S;T S;T deﬁned. Since we have t S h h S k ¼ t þ ; St k Sh k it follows that 1 1 ran ðM þ tÞ¼ S W S ; ð9Þ S;T t t where W is the ‘ﬂip’ operator Wðk; hÞ:¼ðh; kÞ. Since S is densely deﬁned and closed according to our hypotheses, the subspace on the right hand side of (9)is equal to H K. This proves statement (ii). For the converse direction, observe that (8) implies 2 2 t T h h t ; h 2 dom S; k 2 dom T : St k k Hence from (8) we conclude that 2 2 0kSxk þkTyk þ tfðSx j yÞðx j TyÞþðy j SxÞðTy j xÞg 2 2 ¼kSxk þkTyk þ 2t RefðSx j yÞðx j TyÞg 1036 Z. Tarcsay and Z. Sebestyén for every t 2 R. Consequently, ReðSx j yÞ¼ Reðx j TyÞ; x 2 dom S; y 2 dom T : In the real Hilbert space case it is straightforward that S and T are adjoint to each other. In the complex case, replace x by ix to get ImðSx j yÞ¼ Imðx j TyÞ; x 2 dom S; y 2 dom T : So, in both real and complex cases, we obtain that S T and T S . With notation of Theorem 2 this means that S ¼ S and T ¼ T. Since we have 0 0 I T IT ¼ M þ 1; ¼ 1 M ; S;T S;T SI SI we conclude that I þ TS 0 ¼½M þ 1 ½1 M S;T S;T 0 I þ ST is a surjective operator onto H K, which entails ran ðI þ TSÞ¼H and ran ðI þ STÞ¼K. An immediate application of Theorem 2 completes the proof. h As an immediate consequence of Theorem 4 we can establish the following characterizations of self-adjoint, skew-adjoint and unitary operators. Corollary 7 Let H be a real or complex Hilbert space. For a linear operator S : H!H the following assertions are equivalent: (i) S is densely deﬁned and self-adjoint, (ii) Every non-zero real number t is in the resolvent set of M and S;S kR ðtÞk ; 8t 2 R; t 6¼ 0: ð10Þ S;S jtj Proof Apply Theorem 4 with T :¼S to conclude the desired equivalence. h Corollary 8 Let H be a real or complex Hilbert space. For a linear operator S : H!H the following assertions are equivalent: (i) S is densely deﬁned and skew-adjoint, (ii) every non-zero real number t is in the resolvent set of M and S;S kR ðtÞk ; 8t 2 R; t 6¼ 0: ð11Þ S;S jtj Proof Apply Theorem 4 with T :¼ S. h Corollary 9 Let H and K be a real or complex Hilbert spaces. For a linear operator U : H!K the following assertions are equivalent: Range-kernel characterizations of operators… 1037 (i) U is a unitary operator, (ii) ker U ¼f0g, every non-zero real number t is in the resolvent set of M 1 U;U and kR 1ðtÞk ; 8t 2 R; t 6¼ 0: ð12Þ U;U jtj Proof An application of Theorem 4 with S:¼U and T :¼U shows that U is densely deﬁned and closed such that U ¼ U . Hence, ran U dom U. Since we have ran U þ dom U ¼H for every densely deﬁned closed operator U, we infer that dom U ¼H and therefore U is a unitary operator. h Acknowledgements Open access funding provided by Eo¨tvo¨s Lora´nd University (ELTE). The corresponding author Zs. Tarcsay was supported by DAAD-TEMPUS Cooperation Project ‘‘Harmonic Analysis and Extremal Problems’’ (grant no. 308015). Project no. ED 18-1-2019-0030 (Application- speciﬁc highly reliable IT solutions) has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, ﬁnanced under the Thematic Excellence Programme funding scheme. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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