On James and Jordan von Neumann Constants of Lorentz Sequence Spaces

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1 Journal of Mathematical Analysis Applications 58, ) doi:0.006/jmaa , available online at on On James Jordan von Neumann Constants of Lorentz Sequence Spaces Mikio Kato Department of Mathematics, Kyushu Institute of Technology, Kitakyushu , Japan Lech Maligra Department of Mathematics, LuleåUniversity of Technology, S Luleå, Sweden Submitted by Muhammad Aslam Noor Received September 8, 000 The nonsquare or James constant JX the Jordan von Neumann constant C NJ X are computed for two-dimensional Lorentz sequence spaces d w q in the case where q<.the Jordan von Neumann constant is also calculated in the case where q<. 00 Academic Press Key Words: uniformly nonsquare spaces; James constant; Jordan von Neumann constant; n-dimensional Lorentz sequence spaces; Lorentz sequence spaces. Several results on the nonsquare constant JX of James also the Jordan von Neumann constant C NJ X [which is usually called the von Neumann Jordan constant, so we use the notation C NJ X] of a Banach space X have been recently obtained by Casini [], Gao-Lau [, 3], Kato Takahashi [8], Kato, Maligra Takahashi [6, 7] see also [5] for the classical result).in particular, they calculated JX C NJ X for various spaces X showed that some properties of X, such as uniform nonsquareness, superreflexivity, type, cotype, can be described in terms of the constant C NJ X. The aim of this paper is to compute the constants JX C NJ X for two-dimensional Lorentz sequence spaces X = d w q in the case where q<.the paper is organized as follows.in Section we collect X/0 $35.00 Copyright 00 by Academic Press All rights of reproduction in any form reserved.

2 458 kato maligra properties of constants JX C NJ X, also relations between them. In Section we present results on JX C NJ X for two-dimensional Lorentz sequence spaces X = d w q in the case where q<.in Section 3 we give the precise value of C NJ d w q in the case where q<..preliminaries Let X =X be a real Banach space with dim X B X =x X x its unit ball S X =x X x = its unit sphere.the constant JX =supminx + y x y x y S X is called the James constant,orthenonsquare constant, of a Banach space X. We collect properties of the James constant JX see Casini [], Gao Lau [, 3], Kato, Maligra, Takahashi [6, 7]): i) JX =supminx + y x y x y B X. ii) JX JX = ifx is a Hilbert space the converse is not true. iii) If p dim L p µ, then JL p µ = max /p /p. iv) JX =supɛ 0 δ X ɛ ɛ/, where δ X ɛ = inf x + y/ x y S X x y ɛ is the modulus of convexity of X. v) JX < if only if the space X is uniformly nonsquare; that is, there exists a δ 0 such that for any x y S X either x + y/ δ or x y/ δ. vi) JX =JX JX JX JX/ +, there exists X such that JX JX, where X X are the dual bidual spaces of X, respectively. The Jordan von Neumann constant, of a Banach space X C NJ X, is defined by { x + y +x y } C NJ X =sup x y X not both 0 x +y We again collect its properties see Jordan von Neumann [5], Kato Takahashi [8], Kato, Maligra, Takahashi [6, 7]): vii) C NJ X ; X is a Hilbert space C NJ X =. viii) C NJ X =C NJ X.

3 james jordan von neumann constants 459 ix) If p dim L p µ, then C NJ L p µ = /r with r = minp p. x) X is uniformly nonsquare C NJ X <. xi) JX / C NJ X JX /JX + ; If X is not uniformly nonsquare, then we have equalities, there exists a twodimensional Banach space X for which JX / <C NJ X. Let w = w w w n with w w w n > 0 n = 3 For q<, the n-dimensional Lorentz sequence space, d n w q, is n with norm x wq =w x q + w x q + +w n x nq /q where x x x n is the nonincreasing rearrangement of x x x n ; that is, x x x n cf.[9]).in the case when w k = k q/p k= n q<, we have the classical n-dimensional Lorentz sequence space l n pq. Next, we compute the constants JX C NJ X for two-dimensional Lorentz sequence spaces X = d w q..lorentz SEQUENCE SPACES d w q, FOR THE CASE WHERE q Our computation of the James the Jordan von Neumann constants for two-dimensional Lorentz sequence spaces X = d w q in the case where q<, begins with the following theorem. Theorem. If q, then Jd w q = w w + w ) /q.) C NJ d w q = ) /q Jd w q w =.) w + w In the proof, we will need the following lemma. Lemma. If q<, then ) w + w /q x q x wq w /q x q.3) for all x.

4 460 kato maligra Proof. The first inequality means that w + w x q + x q w x q + w x q for any x =x x or, equivalently, w x q + w x q w x q + w x q which is true by the fact of the Hardy Littlewood type; that is, we have u k v k k= u k v k k= for any u u v v.the second inequality in.3) follows immediately from the assumption w w. Proof of Theorem Using Lemma, we have x + y wq +x y wq w/q x + y q +x y q w /q C NJ lq x q +y q C NJ lq = minqq which for q, gives, C NJ d w q /q /q { w w + w w + w + w w w + w w w + w ) /q x wq ) /q } y wq ) /q x wq +y wq ) /q ) /q w = w + w On the other h, let α>0 be a constant such that α α wq = ; that is, α = /w + w /q.for x 0 =α α y 0 =α α, we have C NJ d w q x 0 + y 0 wq +x 0 y 0 wq x 0 wq +y 0 wq Also, = αw/q +αw /q 4 = α w /q = w w + w ) /q ) /q Jd w q αw /q w = w + w

5 james jordan von neumann constants 46 by the first estimate in xi) we obtain JX the proof is complete. ) /q C NJ X = 4α w /q = αw /q w = w + w Remark. The foregoing proofs show also that for q<, we have the following estimates for the n-dimensional Lorentz spaces d n w q. Let W n = w + w + +w n.then Wn n ) /q x q x wq w /q x q for any x n, so Jd n w q max /q /q nw /W n /q C NJ d n w q /r nw /W n /q, where r = minq q. Corollary. If p q, then Jl pq = + q/p /q.4) C NJ l pq =.5) + q/p /q In the case where q =, the foregoing equalities were found in [7] cf. also [], where.5) is calculated for the case q = ). 3.LORENTZ SEQUENCE SPACES d w q, FOR THE CASE WHERE q< In the case where q<, we calculate precisely only the Jordan von Neumann constant. Theorem. If q<, then C NJ d wq= [ { w / q +w / q ] /q max In the proof we will need the following lemma. w +w /q w /q } 3.)

6 46kato maligra Lemma. a) If q<, then w + w /q x x wq w /q x 3.) b) If q<, then where a = min ax x wq bx 3.3) { w + w /q } w /q b = [ w / q + w / q ] /q / Proof. To show all of the foregoing estimates, we must calculate the supremum, A = supu + v / w u q + w v q /q = u v>0 Taking u = λv, we obtain A = supλ + / v w λ q + w /q v = λ = supλ + / /w λ q + w /q λ The function f λ =λ + / /w λ q + w /q has the derivative f λ= [ λλ + / w λ q +w /q w λ q w λ q +w /q λ + /] w λ q +w /q = [ λw λ q ] w w λ q +w /q λ + / w λ q +w /q Now in the case of q<, we have that f λ is decreasing on that x wq inf = x 0 x A = f = w + w /q x wq sup = x 0 x inff λ λ = f = w/q Let q.assume that w >w, because otherwise the estimates in 3.3) are clear. Then f λ is decreasing on λ 0 increasing on λ 0, where λ 0 =w /w / q >.Therefore, x wq inf = x 0 x A = maxf f { w + w = min /q } w /q = a

7 james jordan von neumann constants 463 x wq sup = x 0 x inff λ λ = f λ 0 or =w w /w q/ q + w /q /w /w / q + / = b Proof of Theorem Using Lemma b), we have x + y wq +x y wq b[ x + y +x ] y = b x +y b x wq +y wq /a C NJ d w q b a On the other h, if x 0 = w / q 0 ) y 0 = 0w / q ), then C NJ d w q x 0 + y 0 wq +x 0 y 0 wq x 0 wq +y 0 wq ) /q w w q/ q + w w q/ q = w w q/ q ) /q + w w q/ q ) /q / q w + w / q ) /q = / q w + w / q ) /q w = [ w / q + w / q ] /q = b w /q w /q Also, if we take two points x = w / q + w / q ) y = / q w W / q ), then x + y = w / q w / q ) x y = w / q w / q ), so C N J d w q ] /q 4 [ w w q/ q + w w q/ q ) / q + w w + w /q[ w / q + w / q ) /q = w / q + w / q = b w + w /q w + w w / q The last two estimates from below show that we have equality in 3.), the proof is complete. /q ) ]

8 464 kato maligra Remark.a) Estimates 3.) in Lemma a) also give the result in Theorem. b) Property xi) equality.) give the estimate Jd w q CNJ d w q = b/a, but we do not know whether the equality holds here.note also that ) /q ) /q w Jd w q /q w w + w w + w ) /q ) /q w C w + w NJd w q 4/q w w + w Corollary. If w = w = q/p with p q q<, then C NJ l p q = + q p/p q /q In particular, C NJ l p = + 4/p. Corollary 3. If q =, then { w C NJ d w = max + w w + w w + } w w Problem. Compute JX JX for X = d w q when q<. Note that the dual norm of d w q is not known for q> that for q = we have that the dual space to the two-dimensional Lorentz space d w is a two-dimensional Marcinkiewicz space m w given by the norm { x x mw = max x + } x w w + w Problem. Compute JX JX C NJ X for the n-dimensional Lorentz sequence spaces X = d n w q when n 3, for the infinitedimensional Lorentz sequence spaces X = dw q see [9] for the definition). ACKNOWLEDGMENTS This research was done when the second author was visiting the Department of Mathematics, Kyushu Institute of Technology KIT), Kitakyushu-Japan, in January March 000.He was supported by the Japan Society for the Promotion of Science JSPS) grant S-9989.He is deeply indebted to the Department of Mathematics at KIT for their kind hospitality during his stay to JSPS for their financial support.the research of the first author was also supported by JSPS.

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