Constants and Normal Structure in Banach Spaces

Transcription

1 Constants and Normal Structure in Banach Spaces Satit Saejung Department of Mathematics, Khon Kaen University, Khon Kaen 4000, Thailand Franco-Thai Seminar in Pure and Applied Mathematics October 9 31, 009

2 This talk is based on the following papers: A. Jimenez-Melado, E. Llorens-Fuster and S. Saejung, The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces, Proc. Amer. Math. Soc. 134 (006), no., S. Saejung, On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property. J. Math. Anal. Appl. 33 (006), no., S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals. J. Math. Anal. Appl. 330 (007), no. 1,

3 S. Saejung, The characteristic of convexity of a Banach space and normal structure, J. Math. Anal. Appl., 337, (008) E. Casini, P. L. Papini, and S. Saejung, Some estimates for the weakly convergent sequence coefficient in Banach spaces, J. Math. Anal. Appl. 346 (008) J. Gao and S. Saejung, Normal structure and the generalized James and Zbaganu constants, Nonlinear Analysis Series A: Theory, Methods & Applications, 71(7-8) (009), J. Gao and S. Saejung, Some geometric measures of spheres in Banach spaces, Appl. Math Comp. 14 (009)

4 This talk is organized as follows: Some definitions and historical remarks James and von Neumann Jordan constants parameterized James and von Neumann Jordan constants generalized James and von Neumann Jordan constants Quantitative result

5 Let X be a Banach space. The research theme of this talk: properties or conditions on a Banach space X normal structure of X

6 Recall that the Banach space X has normal structure 1 if every nonempty bounded closed convex subset C of X, with diam C > 0, contains a non-diametral point, that is, there exists x 0 C such that sup{ x x 0 : x C} < diam C. 1 Brodskĭı, M. S.; Mil man, D. P. On the center of a convex set. (Russian) Doklady Akad. Nauk SSSR (N.S.) 59, (1948)

7 Kirk s fixed point theorem X has normal structure and is reflexive X has the fixed point property Recall that X has the fixed point property if for every bounded closed convex subset C of X and every nonexpansive self-mapping T : C C, that is, Tx Ty x y for all x,y C, there exists a point x 0 C such that that is, x 0 is a fixed point of T. x 0 = Tx 0, A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 7 (1965),

8 Spaces with/without normal structure Spaces with normal structure Finite dimensional spaces Uniformly convex spaces (Clarkson, 1936) Uniformly smooth spaces Spaces without normal structure C[a, b] (the space of real-valued continuous functions on [a, b]) Bynum spaces (Bynum, 1980) To test whether a given Banach space has normal structure is not an easy task.

9 Two starting points: Gao and Lau 3 proved that J(X) < 3 X has normal structure. Kato, Maligranda, and Takahashi 4 proved that C NJ (X) < 5 4 X has normal structure. Recall that J(X) = sup{min{ x + y, x y } : x = y = 1} { x + y + x y } C NJ (X) = sup x + y : x + y 0. 3 On two classes of Banach spaces with uniform normal structure. Studia Math. 99 (1991), no. 1, On James and Jordan-von Neumann constants and the normal structure coefficient of Banach spaces. Studia Math. 144 (001), no. 3,

10 Some facts on J(X) and C NJ (X): J(X) X is a Hilbert space J(X) = 1 C NJ (X) X is a Hilbert space C NJ (X) = 1 J(X) C NJ (X) J(X) J(l p ) = J(L p [a,b]) = 1/p if 1 < p C NJ (l p ) = C NJ (L p [a,b]) = /p 1 if 1 < p Recall that l p = {(x n ) R : n=1 x n p < } L p [a,b] = {f : f is a real-valued function on [a,b] and [a,b] f p dµ < }

11 Let X be a Banach space. The research theme of this talk: conditions on a Banach space X in terms of J(X) or C NJ (X) normal structure of X

12 The strongest results so far: J(X) < 3 J(X) < X has normal structure C NJ (X) < 5 4 C NJ(X) < X has normal structure 5 Dhompongsa, Keawkhao and Tasena, J. Math. Anal. Appl. 85 (003), no., S. Saejung, J. Math. Anal. Appl. 33 (006), no.,

13 The strongest results so far: J(X) < 3 J(X) < X has normal structure C NJ (X) < 5 4 C NJ(X) < X has normal structure Remark: Both sufficient conditions cannot be applied for l p or L p [a,b] where p is near 1. In fact, it is known that all l p or L p [a,b] where 1 < p < have normal structure and J(l p ) = J(L p [a,b]) = 1/p if 1 < p ; C NJ (l p ) = C NJ (L p [a,b]) = /p 1 if 1 < p. 5 Dhompongsa, Keawkhao and Tasena, J. Math. Anal. Appl. 85 (003), no., S. Saejung, J. Math. Anal. Appl. 33 (006), no.,

14 Parameterized James and von Neumann Joradan constants We study these constants: 7 for 0 t 1 J t (X) = sup{min{ x + ty, x ty } : x = y = 1} CNJ(X) t 1 = (1 + t ) sup{ x + ty + x ty : x = y = 1}. Remark: J(X) = J 1 (X) C NJ (X) = sup{cnj t (X) : 0 t 1} 7 S. Saejung, Sufficient conditions for uniform normal structure of Banach spaces and their duals. J. Math. Anal. Appl. 330 (007), no. 1,

15 Better sufficient conditions: J(X) < ( J(X) < ) J(X) J t t (X) < 1 + J t for some 0 t 1 (X) + 1 t X has normal structure Remark: If X = l p or X = L p [a,b] where 1 < p <, then J t (X) < 1 + for some 0 t 1. t J t (X)+1 t

16 Better sufficient conditions: C NJ (X) < (1 + t )C t NJ (X) < (1 + ts) (1 + s )C s NJ (X ) (1 + s) for some 0 t,s 1 X has normal structure Remark: If X = l p or X = L p [a,b] where 1 < p <, then (1 + t )C t NJ (X) < (1+ts) (1+s )C s NJ (X ) (1+s) for some 0 t,s 1.

17 Some improvement in terms of these constants In 006, Jimenez-Melado, Llorens-Fuster and Saejung 8 proved the following: Recall that 9 µ(x) = inf J(X) < X has normal structure µ(x) C NJ (X) < X has normal structure µ(x) r > 0 : lim sup n x + x n r lim sup n x x n for all x X and all weakly null sequences {x n } in X.. 8 Proc. Amer. Math. Soc. 134 (006), no., B. Sims, A class of spaces with weak normal structure, Bull. Austral. Math. Soc. 50 (1994),

18 J(X) < X has normal structure µ(x) C NJ (X) < X has normal structure µ(x) Note: Both results are sharp in the sense that there is a Banach space X such that X fails to have normal structure and J(X) = µ(x) and C NJ (X) = µ(x).

19 We can prove the following results 10 : J(X) < µ(x) J t (X) < 1 + t µ(x) X has normal structure for some 0 t 1 C NJ (X) < µ(x) ( ) 1 + t CNJ(X) t µ(x) < 1 + t for some 0 t 1 X has normal structure 10 J. Gao and S. Saejung, Appl. Math Comp. 14 (009)

20 Generalized James and von Neumann Joradan constants Let B X = {x x : x 1}. Based on the Hexagonal Lemma of Gao and Lau, 11 the following constants are introduced 113 : { J(a,X) = sup min{ x + y, x z } : x,y,z B X, } y z a x { x + y + x z C NJ (a,x) = sup x + y : x,y,z X, + z } x + y + z 0, y z a x. 11 On two classes of Banach spaces with uniform normal structure. Studia Math. 99 (1991), no. 1, Dhompongsa, Kaewkhao, and Tasena, J. Math. Anal. Appl. 85 (003), no., Dhompongsa, Piraisangjun, and Saejung, Bull. Austral. Math. Soc. 67 (003), no., 5 40.

21 { J(a,X) = sup min{ x + y, x z } : x,y,z B X, } y z a x { x + y + x z C NJ (a,x) = sup x + y : x,y,z X, + z } x + y + z 0, y z a x. Remark: J(0,X) = J(X) C NJ (0,X) = C NJ (X)

22 Improved sufficient conditions: 15 The following is an improvement of 14 : J(a,X) < 3 + a for some 0 a 1 J(a,X) < 1 a + (1 a) + 4(1 + a) X has normal structure for some 0 a 1 Remark: For all 0 a < 1, 3 + a < 1 a + (1 a) + 4(1 + a). 14 Dhompongsa, Kaewkhao, and Tasena, J. Math. Anal. Appl. 85 (003), no., J. Gao and S. Saejung, Nonlinear Analysis Series A: Theory, Methods & Applications, 71(7-8) (009),

23 Remark: We also obtain an improvement for the generalized NJ-constant.

24 An answer of Llorens-Fuster s question 16 Llorens-Fuster proved that where C Z (X) < 16 X has normal structure, 13 { } x + y x y C Z (X) = sup x + y : x + y 0. He asked this question: Is 16/13 sharp in this situation? Remark: C Z (X) C NJ (X) and there is a Banach space X such that C Z (X) < C NJ (X). 16 E. Llorens-Fuster, Zbăganu constant and normal structure, Fixed Point Theory 9 (008)

25 The main tool that Llorens-Fuster used in his paper is the modified Hexagonal Lemma 17. A careful application of this lemma gives the following result: C Z (X) < C Z (X) < X has normal structure Remark: It seems to be unknown whether 1+ 3 is sharp in this situation. 17 S. Saejung, J. Math. Anal. Appl. 330 (007), no. 1,

26 Quantitative results: Bynum 18 defined the weakly convergent sequence coefficient of X by { lim k sup{ x n x m : n,m k} } WCS(X) = inf inf { lim sup n x n y : y co({x n }) } where the infimum is taken over all weakly convergent sequences {x n } which are not norm convergent. It is clear that for reflexive spaces X WCS(X) > 1 X has normal structure. 18 Normal structure coefficients for for Banach spaces. Pacific J. Math. 86 (1980),

27 Suppose that a Banach space X fails the Schur property, that is, X contains a weakly convergent sequence which is not norm convergent. Then 19 1 WCS(X). J(X)+ 1 5 ( WCS(X) 1 J(X) WCS(X) 1 C NJ (X) µ(x) ). ( µ(x) ). In particular, X has normal structure if J(X) < 1+ 5 or J(X) < µ(x), or C NJ(X) < µ(x). 19 E. Casini, P. L. Papini, and S. Saejung, J. Math. Anal. Appl. 346 (008)