The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

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1 Vol. 45 No. 4 SCIENCE IN CHINA Series A April 2002 The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space DING Guanggui Department of Mathematics, Nankai University, Tianjin , China ding gg@nankai.edu.cn Received April 18, 2001 Abstract Let E and F be Hilbert spaces with unit spheres S 1E and S 1F. Suppose that V 0 S 1E S 1F is a Lipschitz mapping with Lipschitz constant k =1such that V 0[S 1E] V 0[S 1E]. ThenV 0 can be extended to a real linear isometric mapping V from E into F. In particular, every isometric mapping from S 1E onto S 1F can be extended to a real linear isometric mapping from E onto F. Keywords: isometric mapping, strictly convex, smooth point. 1 Preliminaries Tingley proposed the following problem in ref. [1]: Let E and F be real normed spaces with unit spheres S 1 E ands 1 F. Suppose that V 0 : S 1 E S 1 F is a surjective isometry i.e. V 0 x 1 V 0 x 2 = x 1 x 2 for all x 1, x 2 S 1 E. Is V 0 necessarily the restriction to S 1 E of a linear or affine isometry on E? in the complex spaces, it is evident that the answer is negative. For example, we take E = F = C complex plane and V 0 x = x. In Tingley s paper, he only obtaind the affirmative answer on the assertion V 0 x = V 0 x x S 1 E when the spaces are finite dimensional. In ref. [2], the above conclusion can also be obtained when the spaces are strictly convex Banach spaces. Recently, many papers have been published and the newest result has been obtained when E is some Banach space and F is CΩ cf. ref. [3]. Concerning isometry there is another probleml: What conditions are to be added to a Dopp-mapping T : E F so that T becomes an isometry? where E and F are real normed spaces and Dopp is the distance one preserving property ; that is, for all x 1, x 2 E with x 1 x 2 =1, T x 1 T x 2 = 1. There are also many papers available concerning the above problem [4] and the newest result is as follows: Let E and F be normed spaces with dime 2 such that one of them is strictly convex. Suppose that T : E F is a homeomorphism satisfying the Dopp. Then, T is a linear isometry up to translation [5]. Combining the above two problems, we can propose the following problem: What conditions are to be added to a mapping V 0 : S 1 E S 1 F such that V 0 becomes a real linear isometry on E? where E and F are real or complex normed spaces.

2 480 SCIENCE IN CHINA Series A Vol. 45 In this paper, we shall demonstrate that if V 0 satisfies the Lipschitz condition with k =1and V 0 [S 1 E] V 0 [S 1 E], then the answer of the above extension problem is affirmative when E and F are Hilbert spaces. 2 Main results We shall first give two lemmas. Lemma 2.1. Let E and F be normed spaces and E be strictly convex, and V 0 a mapping from the unit sphere S 1 E intos 1 F. If V 0 [S 1 E] V 0 [S 1 E] and V 0 x 1 V 0 x 2 x 1 x 2, x 1,x 2 S 1 E, then V 0 is one-to-one, and V 0 x = V 0 x for all x S 1 E. For each x S 1 E, let y = V 0 x S 1 F. Then, for every x 1 V0 1 y and x 2 V0 1 y, by the hypothesis of V 0 we have 2 x 1 + x 2 x 1 x 2 V 0 x 1 V 0 x 2 = y y =2. From the strictly convex condition of E, we get x 1 = x 2, and hence V 0 is one-to-one and x = x 1 = x 2. Moreover, from x = x 2 we know That is, we obtain V 0 x =V 0 x 2 =V 0 [V 1 0 y] = y = V x. V 0 x = V 0 x, x S 1 E. Note. The condition V 0 [S 1 E] V 0 [S 1 E] in the above lemma is not removable, since a counterexample can be given as follows: Take V 0 x y 0 y 0 is some element of S 1 F for all x S 1 E. Lemma 2.2. In Lemma 2.1, if both E and F are inner-product spaces, we have V 0 x 1 λv 0 x 2 = x 1 λx 2, λ R, x 1,x 2 S 1 E. Firstly, we prove that V 0 is isometric on S 1 E. In fact, x 0 1, x 0 2 S 1 E such that V 0 x 0 1 V 0 x 0 2 x 0 1 x 0 2, and then V 0 x 0 1 V 0 x 0 2 < x 0 1 x By the parallelogram law of norm in the inner-product space we have From Lemma 2.1, we have 4 = 2 V 0 x V 0 x = V 0 x 0 1+V 0 x V 0 x 0 1 V 0 x < V 0 x 0 1+V 0 x x 0 1 x V 0 x 0 1+V 0 x 0 2 = V 0 x 0 1 V 0 x 0 2 x 0 1 x 0 2 = x x From 2.2 and 2.3 we derive which contradicts 2.1. Thus we obtain 4 < x x x 0 1 x =2 x x =4, V 0 x 1 V 0 x 2 = x 1 x 2, x 1,x 2 S 1 E. 2.4

3 No. 4 1-LIPSCHITZ MAPPING 481 Secondly, from 2.4 we have V 0 x 1 V 0 x 2,V 0 x 1 V 0 x 2 = x 1 x 2,x 1 x 2, x 1,x 2 S 1 E, i.e. Then 2 V 0 x 1,V 0 x 2 V 0 x 2,V 0 x 1 = 2 x 1,x 2 x 2,x 1. V 0 x 1,V 0 x 2 + V 0 x 2,V 0 x 1 = x 1,x 2 +x 2,x 1, x 1,x 2 S 1 E. 2.5 By 2.5, for each λ R we have Hence, we obtain V 0 x 1,V 0 x 1 λv 0 x 1,V 0 x 2 λv 0 x 2,V 0 x 1 + λ 2 V 0 x 2,V 0 x 2 =1+λ 2 λv 0 x 1,V 0 x 2 λv 0 x 2,V 0 x 1 =1+λ 2 λx 1,x 2 λx 2,x 1 =x 1,x 2 λx 1,x 2 λx 2,x 1 +λ 2 x 2,x 2. V 0 x 1 λv 0 x 2 = x 1 λx 2, Next we shall give the first theorem. Theorem 2.1. λ R, x 1,x 2 S 1 E. Let E and F be inner-product spacs, and V 0 a mapping from the unit sphere S 1 E intos 1 F. If V 0 [S 1 E] V 0 [S 1 E] and V 0 x 1 V 0 x 2 x 1 x 2, x 1,x 2 S 1 E, then V 0 can be extended to be a real homogeneous isometric mapping of E into F. Let V z = { z V0 z z, if z θ, θ, if z = θ. Then, by Lemma 2.1 for every 0 λ R, wehave λz V λz = λz V 0 = λ z λ z λz λ V 0 = λv z, θ z E, z i.e. V is real homogeneous. By Lemma 2.2, if z 1 θ, z 2 θ, weget V z 1 V z 2 = z 1 V 0 = z 1 V 0 = z 1 Thus, V is isometric on E. ref. [6]. z 1 z 1 z 1 z 1 z 2 V 0 z2 z 1 V 0 z1 z z2 z 2 1 z 1 z 2 z 2 z 2 z 2 z 2 = z 1 z 2, z 1,z 2 E. 2.6 For the linearity of the above V we need to introduce the following lemma Lemma in Lemma 2.3. Let V be an isometry of a real Banach space E into a real Banach space F such that V θ =θ. Letx 0 be a smooth point of the sphere S x0 and g F be a continuous linear functional of norm one such that, for all real λ, g[v λx 0 ] = λ x 0.

4 482 SCIENCE IN CHINA Series A Vol. 45 Then g[v x] = f x0 x, x E, where f x0 E is the support functional at x 0 in the sphere S x0. Theorem 2.2. Let E and F be Hilbert spaces, and V 0 a mapping from the unit sphere S 1 E intos 1 F. If V 0 [S 1 E] V 0 [S 1 E] and V 0 x 1 V 0 x 2 x 1 x 2, x 1,x 2 S 1 E. Then V 0 can be extended to be a real linear isometric mapping from E into F. Let V z be the same as in ref. [6] in the proof of Theorem 2.1. Then we know that V z is a real homogeneous isometric mapping from E to F. Next we shall prove the additivity of V z one. First, we notice that every point x S 1 E is a smooth point of S 1 E since E is a Hilbert space. By the Riesz representation theorem, for each V x F, we can assume g = V x F, and we have g[v λx] = V λx,vx = λv x,vx = λ V x 2 = λ = λ V x, λ R. Hence, by Lemma 2.3 we obtain that is, for every x S 1 E, g[v z] = f x z, z E; V z,vx = f x z, z E, 2.7 where f x E is the support functional at x in the unit sphere S 1 E. and By 2.7, it is easy to verify that for z 1, z 2 E we have V z 1 + z 2,Vx = V z 1 +Vz 2,Vx, x S 1 E, V z 1 + z 2,αVx 1 +βv x 2 = V z 1 +Vz 2,αVx 1 +βv x 2, x 1,x 2 S 1 E, α, β R V z 1 + z 2,y=V z 1 +V z 2,y, y Span{V x x S 1 E}. 2.8 Let F 0 = Span{V x x S 1 E}. Then, F 0 is also a Hilbert space and Span{V z z E} F 0,soF0 = F 0. By 2.8 and the Hahn-Banach theorem we obtain V z 1 + z 2 =Vz 1 +Vz 2, z 1,z 2 E. Thus, V is a real linear isometric mapping from E into F. Corollary 1. Let E and F be Hilbert spaces, and V 0 a 1-Lipschitz mapping from E onto F. Then V 0 can be extended to a real linear isometric mapping from E onto F. When we notice that if F is strictly convex and V 0 is isometric, then V 0 [S 1 E] V 0 [S 1 E]. Thus we have Corollary 2. Let E and F be Hilbert spaces, and V 0 an isometric mapping from E into F. Then, V 0 can be extended to a real linear isometric mapping from E into F.

5 No. 4 1-LIPSCHITZ MAPPING 483 Remark. It is clear that the above Theorem 2.1 implies Theorem 2.2 by Baker s theorem [7]. However, for Hilbert spaces, the above independent proof seems to be better. Acknowledgements Grant No This work was supported by the National Natural Science Foundation of China References 1. Tingley, D., Isometries of the unit sphere, Geometriae Dedicata, 1987, 22: Ma Yumei, Isometries of the unit sphere, Acta Math. Sci., 1992, 12: Ding Guanggui, On the extension of isometries between unit spheres of E and CΩ, Acta Math. Sinica New Series, 2001, 17: Rassias, T. M., Properties of isometries and approximate isometries, Recent Progress in Inequalities, MIA, , Drichlet: Kluwer Academic Publishers, Rassias, Th. M., Šemrl, P., On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mapping, Proc. Amer. Math. Soc., 1993, 118: Rolewicz, S., Metric Linear Spaces, Dordrecht-Warszawa: Reidel and PWN, Baker, J. A., Isometries in normed spaces, Amer. Math. Monthly, 1971, 8: Rudin, W., Functional Analysis, New York, Toronto: McGraw-Hill Inc.,1973.