Stability of Adjointable Mappings in Hilbert

Transcription

1 Stability of Adjointable Mappings in Hilbert arxiv:math/ v2 [math.fa] 1 Aug 2005 C -Modules M. S. Moslehian Abstract The generalized Hyers Ulam Rassias stability of adjointable mappings on Hilbert C -modules is investigated. As a corollary, we establish the stability of the equation f(x) y = xg(y) in the context of C -algebras. 1 Introduction In 1940, S. M. Ulam [22] proposed the following problem: Given a group G 1, a metric group (G 2,d) and a positive number ǫ, does there exist a δ > 0 such that if a function f : G 1 G 2 satisfies the inequality d(f(xy),f(x)f(y)) < δ for all x,y G 1 then there exists a homomorphism T : G 1 G 2 such that d(f(x),t(x)) < ǫ for all x G 1? If this problem has a solution, we say that the homomorphisms are stable or say the functional equation f(xy) = f(x)f(y) is stable. This terminology can be applied to various functional equations and mappings; see [1], [2], [12], [13], [14] and references therein. In the next year, D. H. Hyers [7] gave a partial solution of Ulam s problem for the case that G 1 and G 2 are Banach spaces Mathematics Subject Classification. Primary 39B82, secondary 46L08, 47B48, 39B52 46L05, 16Wxx Key words and phrases. generalized Hyers Ulam Rassias stability; Hilbert C -module; adjointable mapping; C -algebra. 1

2 In 1978, Th. M. Rassias [16] extended the theorem of Hyers by considering an interesting stability problem with an unbounded Cauchy difference as follows: Assume that E 1 and E 2 are real normed spaces with E 2 complete, f : E 1 E 2 is a mapping such that for each fixed x E 1 the mapping t f(tx) is continuous on R, and let there exist ε 0 and p < 1 such that f(x + y) f(x) f(y) ε( x p + y p ) for all x,y E 1. Then there exists a unique linear mapping T : E 1 E 2 such that for all x E 1. f(x) T(x) 2ǫ 2 2 p x p In 1991, Z. Gajda [5] following the same approach as in Th.M. Rassias [16] provided an affirmative solution to this quastion for p > 1. This phenomenon is called Hyers-Ulam- Rassias stability. It is shown that there is no analogue of Rassias result for p = 1 (see [5] and [20]). A number of Rassias type results related to the stability of various functional equations are presented in [17][18], [19]. In 1992, a generalization of Rassias theorem was obtained by Găvruta [6] as follows: Suppose (G, +) is an abelian group, E is a Banach space and the so-called admissible control function ϕ : G G [0, ) satisfies ϕ(x,y) := 1 2 n ϕ(2 n x,2 n y) < 2 n=0 for all x,y G. If f : G E is a mapping with f(x + y) f(x) f(y) ϕ(x,y) for all x,y G, then there exists a unique mapping T : G E such that T(x + y) = T(x) + T(y) and f(x) T(x) ϕ(x,x) for all x,y G. This phenomenon of stability that essentially was introduced by Th. M. Rassias [16] is called the generalized Hyers Ulam Rassias stability. During the last decades several stability problems of functional equations have been investigated in the sprit of Hyers Ulam Rassias Găvrurta. See [3] and [8] for more detailed information on stability of functional equations. 2

3 The notion of Hilbert C -module is a generalization of the notion of Hilbert space. This object was first used by I. Kaplansky [10]. The research on this subject began by M. A. Rieffel [21] and W. L. Paschke [15]. It plays a key role in operator algebras, KK-Theory [9], operator spaces [4], quantum group theory [23], Morita equivalence [21] and so on. Interacting with the theory of operator algebras and including ideas from non-commutative geometry it progresses and produces results and new problems attracting attention. Let A be a C -algebra and E be a linear space which is a right A-module with a scalar multiplication satisfying λ(xa) = x(λa) = (λx)a for x E, a A, and λ C. The space E is called a pre-hilbert A-module if there exists an inner product.,. : E A with the following properties: (i) x, x 0 and x, x = 0 iff x = 0 (ii) x, y + λz = x, y + λ x, z (iii) x, ya = x, y a (iv) x, y = y, x E is called a (right) Hilbert A-module if it is complete with respect to the norm x = x, x 1/2. Left Hilbert A-modules can be defined in a similar way. (i) Every inner product space is a left Hilbert C-module. (ii) Let A be a C -algebra Then every right ideal I of A is a Hilbert A-module if one defines a, b = a b, a, b I. Assume that F is another Hilbert A-module. A mapping T : E F is called adjointable if there exists a mapping S : F E such that Tx, y = x, Sy for all x E, y F. The mapping S is denoted by T and is called the adjoint of T. If T is adjointable, then it is a bounded A-linear mapping [11]. Indeed, for each x in the unit ball of E, define τ x : F A by τ x (y) = Tx, y = x, Sy. Then τ x (y) x Sy Sy. Hence { τ x : x E, x = 1} is bounded. Due to Tx = sup Tx, y = sup τ x (y) = τ x y =1 y =1 and by applying the Banach-Steinhaus theorem we conclude that T is bounded. The reader is referred to [11] for more details on Hilbert C -modules. 3

4 In this paper we deal with the following equation in the context of Hilbert C -modules: When is it true that the solution of an equation differing slightly from a given one, must of necessity be close to the solution of the given equation. Our main purpose is to prove the generalized Hyers Ulam Rassias stability of adjointable mappings on Hilbert C -modules. This significant result provides us a solution for stability of the equation f(x) y = xg(y) in the context of C -algebras. In particular, the linear mapping Tx = λx for certain λ C is the unique solution of functional equation f(x)y = xg(y), x, y C in spirit of Hyers Ulam Rassias stability. 2 Main Results. Throughout this section, A denotes a unital C -algebra and E and F denote Hilbert A- modules. We are going to establish the stability of adjointable mappings on Hilbert C - modules. Theorem 2.1 Suppose f : E F and g : F E are mappings for which there exists a function ϕ : E F [0, ) such that 2 (n+m) ϕ(2 n x, 2 m y) < (1) m=1 f(x), y x, g(y) ϕ(x, y) (2) for all x E, y F. Then there exists a unique adjointable linear mapping T : E F such that T(x) f(x), y 2 2 n ϕ(2 n x, y) + ϕ(x, y) < for all x E, y F. Moreover, if S is the adjoint of T then x, Sy g(y) 2 2 n ϕ(x, 2 n y) + ϕ(x, y) <. 4

5 Proof. Let y F be an arbitrary fixed element. Using f(2x) 2f(x), y f(2x), y 2x, g(y) + 2f(x), y 2x, g(y) and the inequality (2), we obtain f(2x) 2f(x), y ϕ(2x, y) + 2ϕ(x, y), x E. (3) Assume that for some positive integer n, the inequality 2 n f(2 n n 1 x) f(x), y 2 2 k ϕ(2 k x, y) + 2 n ϕ(2 n x, y) + ϕ(x, y) holds for all x E. By applying (3) with x replaced by 2 n x, we get k=1 2 (n+1) f(2 n+1 x) f(x), y 2 (n+1) f(2 n+1 x) 2 n f(2 n x), y Using induction we infer that = n f(2 n x) f(x), y 2 (n+1) (ϕ(2 n+1 x, y) + 2ϕ(2 n x, y)) n 1 +2 k=1 n k=1 2 k ϕ(2 k x, y) + 2 n ϕ(2 n x, y) + ϕ(x, y) 2 k ϕ(2 k x, y) + 2 (n+1) ϕ(2 n+1 x, y) + ϕ(x, y). 2 n f(2 n n 1 x) f(x), y 2 2 k ϕ(2 k x, y) + 2 n ϕ(2 n x, y) + ϕ(x, y) (4) for all x E and for all n. k=1 By induction on p, one can verify that 2 p f(2 p x) 2 q f(2 q x), y for all q p and for all x E. p k=q+1 (2 k ϕ(2 k x, y) + 2 (k 1) ϕ(2 k 1 x, y)) Since the norm of each element z of a Hilbert C -module is equal to sup{ z w : w = 1} and since by (1) for each x E, y F the series (2 n ϕ(2 n x, y)+2 (n 1) ϕ(2 n 1 x, y)) is 5

6 convergent, we deduce that {2 n f(2 n x)} is Cauchy for each x E. Due to the completeness of F, the sequence {2 n f(2 n x)} is convergent. Set lim n T(x) = lim n 2 n f(2 n x), x E. Similarly one can show that there exists a mapping S : F E defined by S(y) = g(2 n y). Replacing x by 2 n x and y by 2 n y in (2), we obtain 2 n 2 n f(2 n x), y x, 2 n g(2 n y) 2 2n ϕ(2 n x, 2 n y). (5) As n the right hand side of (5) tends to zero. Therefore Tx, y = x, Sy for all x E, y F. Hence T and S are the adjoint of each other. In particular, they are bounded A-linear mappings. It follows from (4) that T(x) f(x), y 2 2 n ϕ(2 n x, y) + ϕ(x, y) <. If T is another adjointable mapping satisfying for all x E, y F, then T (x) f(x), y 2 2 n ϕ(2 n x, y) + ϕ(x, y) < T x Tx, y = lim T x 2 p f(2 p x), y p lim (2 (n+p)+1 ϕ(2 n+p x, y)) + 2 p ϕ(2 p x, y)) ( p (2 p n=p = lim = 0. This implies the uniqueness of T. (2 n ϕ(2 n+p x, y)) + 2 p ϕ(2 p x, y)) Remark 2.2 (i) In Theorem 2.1, for establishing the continuity of T or S we do not need to have any regular assumption on f or g. (ii) Replacing x with 0 and y with 2 m f(0) in (2) of Theorem 2.1, we get f(0), f(0) 2 m ϕ(0, 2 m f(0)). By the convergence of the series (1) with x = 0, we conclude that f(0), f(0) = 0 and so f(0) = 0 and similarly g(0) = 0. 6

7 f(2 n x) 2 n f(x) (iii) If f(2x) = 2f(x), x E then T(x) = lim = lim = f(x). A similar n 2 n n 2 n statement is true for g and S. Corollary 2.3 The equation f(x) y = xg(y), x A, y B is stable where f : A B and g : B A are mappings between right ideals A, B of a given C -algebra C. Corollary 2.4 Suppose that f : E F and g : F E are mappings and δ > 0. Let f(x), y x, g(y) δ for all x E, y F. Then there exists a unique adjointable linear mapping T : E F such that T(x) f(x) 3δ for all x E. Furthermore, assuming E = F, (i) if (f g)(x) x δ, then T is unitary; (ii) if f(x) g(x) δ, then T is self-adjoint; (iii) if (f g)(x) (g f)x δ, then T is normal. Proof. Put ϕ(x, y) = δ. Then 2 (n+m) ϕ(2 n x, 2 m y) = 4δ. Applying Theorem 2.1, m=1 there is an adjointable mapping satisfying Tx f(x), y 3δ, x E, y F. Hence Tx f(x) sup 3δ = 3δ. y =1 If T is another mapping with T x f(x) 3δ, x E, then Tx T x Tx f(x) + T x f(x) 6δ, x E replacing x by 2 n x in the later inequality, we obtain Tx T x 62 n δ, x E which gives the uniqueness. Next suppose that E = F. We are going to prove (i). The rest is similar. For each fixed y E with y = 1, we have g(x), g(y) x, y g(x), g(y) f(g(x)), y + f(g(x)), y x, y δ + f(g(x)) x 2δ for all x E. Hence 2 n g(2 n x), 2 n g(2 n y) x, y 2 2n+1 δ. It follows from Sx = lim n 2 n g(2 n x) that Sx, Sy = x, y. Therefore TSx, y = x, y and so TS = I. Similarly one can show that ST = I. 7

8 Corollary 2.5 Let : (0, ) (0, ) be a function satisfying (2) < 2 and (ts) (t) (s), t, s (0, ). Let f : E F and g : F E be mappings satisfying f(x), y x, g(y) ( x ) + ( y ) for all x E, y F. Then there exists a unique adjointable linear mapping T : E F such that T(x) f(x), y (2+ (2)) ( x ) 2 (2) + 3 ( y ). Proof. Setting ϕ(x, y) = ( x ) + ( y ), we get 2 (n+m) ϕ(2 n x, 2 m y) = (2 m ( (2) m=1 m=1 2 )n ( x ) + 2 n ( (2) 2 )m ( y )) = (2) ( x ) + (2) ( y ) 2 (2) 2 (2) (2)( ( x ) + ( y )) = <. 2 (2) Applying Theorem 2.1, we obtain a unique adjointable mapping satisfying T(x) f(x), y (2 + (2)) ( x ) 2 (2) + 3 ( y ). Remark 2.6 For given p < 1, the function (t) = t p satisfies the conditions of the preceding corollary. Therefore one may consider the unbounded control function ϕ(x, y) = x p + y p. References [1] C. Baak and M. S. Moslehian, On the stability of J*-homomorphisms, Nonlinear Analysis, to appear. [2] C. Baak, H. Y. Chu and M. S. Moslehian, On linear n-inner product preserving mappings, Math. Inequalities and Appl., to appear. [3] S. Czerwik (ed.), Stability of Functional Equations of Ulam Hyers Rassias Type, Hadronic Press,

9 [4] E. G. Effros, Z. J. Ruan, Operator spaces. London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York, [5] Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14 (1991), [6] P. Găvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), [7] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), [8] D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston, Basel, Berlin, [9] K. K. Jensen and K. Tomsen, Elements of KK-Theory, Birkhauser, Boston-Basel-Berlin, [10] I. Kaplansky, Modules over operator algebras, Amer J. Math., 75(1953), [11] E. C. Lance, Hilbert C -modules, LMS Lecture Note Series 210, Cambridge Univ. Press, [12] M. S. Moslehian, Orthogonal stability of the Pexiderized quadratic equation, J. Differ. Equations. Appl., to appear. [13] M. S. Moslehian, Approximately vanishing of topological cohomology groups, J. Math. Anal. Appl., to appear. [14] M. S. Moslehian, Approximate (σ, τ)-contractibility, Nonlinear Functional Analysis and Applications, to appear. [15] W. L. Paschke, Inner product modules over B*-algebras, Trans Amer. Math. Soc. 182(1973),

10 [16] Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), [17] Th. M. Rassias, On the stability of functional equations in Banach spaces, Journal of Mathematical Analysis and Applications 251(2000), [18] Th. M. Rassias (ed.), Functional Equations and Inequalities, Kluwer Academic Publishers, Dordrecht, Boston, London, [19] Th. M. Rassias(ed.), Functional Equations, Inequalities and Applications Kluwer Academic Publishers, Dordrecht,Boston,Londonm, [20] Th. M. Rassias and P. Šemrl, On the behavior of mappings which do not satisfy Hyers- Ulam stability, Proc. Amer. Math. Soc. 114 (1992), [21] M. A. Rieffel, Morita equivalence representations of C -algebras, Adv. in Math., 13(1974), [22] S. M. Ulam, Problems in modern mathematics, Chapter VI, Science Editions, Wiley, New York, [23] S. L. Woronowicz, compact quantum groups, Symetrics quantiques(les Houches, 1995), , North Holland, Amsterdum, Mohammad Sal Moslehian Dept. of Math., Ferdowsi Univ., P. O. Box 1159, Mashhad 91775, Iran msalm@math.um.ac.ir moslehian/ 10