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Von Neumann‐Jordan constant of generalizedBanaś‐Fr4czek spaces l
岡山県立大学
新潟大学

情報工学部

三谷健一 (Ken‐Ichi Mitani)

Okayama Prefectural University
自然科学系 斎藤 吉助 (Kichi‐Suke Saito)
Niigata University

岡山県立大学名誉教授

高橋泰嗣 * (Yasuji Takahashi)
Okayama Prefectural University

概要

バナッハ空間の von Neumann‐Jordan 定数に関する最近の結果について報告する.特に,Banaś−Frączek 空間を含む2次元ノルム空間を導入し,Banach‐Mazur
距離の概念を用いてこの空間における von Neumann‐Jordan 定数を計算する.

Definition 1 ([2]). The von Neumann‐Jordan (NJ‐) constant of a Banach space X,
denoted by C_{NJ}(X) , is the smallest constant C for which\frac{1}{C}\leq\frac{\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2} {2(|x|^{2}+ |y\Vert^{2}) \leq C
holds for all x, y\in X not both 0.

Definition 2 ([1, 10]). For \lambda>1 , the Banaś‐Frqczek space \mathb {R}_{\lambda}^{2} is the space \mathb {R}^{2} with
the norm | . |_{\lambda} defined by|(x, y)|_{\lambda}= \max\{\lambda|x|, \Vert(x, y)\Vert_{2}\},where \Vert\Vert_{2} is the \ell_{2} ‐norm on \mathbb{R}^{2}.

Yang‐Wang [13] は幾何学的定数 \gamma_{X}(t) を導入し,これを用いて \ell_{2}-P_{1} と \ell_{\infty}-\ell_{1} におけるNJ 定数を計算した.この方法と同様にして Yang [10] はBanaś‐Fr§czek space のNJ定数を計算した.

Theorem 3 ([10]) For \lambda>1,C_{NJ}( \mathb {R}_{\lambda}^{2})=2-\frac{1}{\lambda^{2} .

1 Keywords. von Neumann‐Jordan constant, Banach‐MaLur distance*2017 年5月逝去

199
本研究では,

B anaŚ‐Fr4czek space を含む次の2次元ノルム空間を導入し,Banach‐Mazur distance を用いてこの空間の NJ 定数を計算する.
Definition 4. For a\geq b\geq 1 and 1\leq p<\infty,\Vert\mathb {R}_{a,b,p}^{2} is defined as \mathb {R}^{2} with the norm

\Vert on \mathbb{R}^{2} by\Vert(x, y)\Vert=\max\{a|x| , b|y|, \Vert(x, y)\Vert_{p}\},where\Vert . \Vert_{p} is the P_{p} ‐norm on \mathbb{R}^{2}.

Definition 5. For isomorphic Banach spaces
between

X

and

Y,

X

and

Y,

the Banach‐Mazur distancedenoted by d(X, Y) , is defined to be the infimum of \Vert T\Vert . \Vert T^{-1}\Verttaken over all bicontinuous linear operators T from X onto Y.

Lemma 6 ([4]). If X and Y are isomorphic Banach spaces, then\frac{C_{NJ}(X)}{d(X,Y)^{2} \leq C_{NJ}(Y)\leq C_{NJ}(X)d(X, Y)^{2}.
In particular, if X and Y are isometric, then C_{NJ}(X)=C_{NJ}(Y) .

Lemma 7 ([4]). Let X=(X, \Vert\cdot\Vert) be a Banach space and let X_{1}=(X, \Vert\cdot\Vert_{1}) , where\Vert\Vert_{1} is an equivalent norm on X satisfying, for \alpha, \beta>0,\alpha\Vert x\Vert\leq\Vert x\Vert_{1}\leq\beta\Vert x\Vert, x\in X.
Then\frac{\alpha^{2} {\beta^{2} C_{NJ}(X)\leq C_{NJ}(X_{1})\leq\frac{\beta^{2} {\alpha^{2} C_{NJ}(X)Definition 8. A norm.

\Vert . \Vert on \mathb {R}^{2} is said to be absolute if \Vert(|x|, |y|)\Vert=\Vert(x, y)\Vert forany x, y\in \mathbb{R}.

Lemma 7を用いて,absolute norm のNJ 定数に関する公式を得る.簡単のため

C_{NJ}((\mathbb{R}^{2}, \Vert\cdot\Vert)) を C_{NJ}(\Vert\cdot\Vert) とかく.
Theorem 9. Let \Vert\Vert, \Vert\Vert_{H} be absolute norms on \mathb {R}^{2} . Assume that the followinghold:(i) (\mathbb{R}^{2}, \Vert . \Vert_{H}) is an inner product space.

200(ii) \alpha\Vert(x, y)\Vert_{H}\leq\Vert(x, y)\Vert\leq\beta\Vert(x, y)\Vert_{H} for any (x, y)\in \mathbb{R}^{2}(\alpha, \beta are the best con‐stants).

(iii) In (ii) it satisfies either \alpha\Vert(1, 0)\Vert_{H}=\Vert(1, 0)\Vert and \alpha\Vert(0,1)\Vert_{H}=\Vert(0,1)\Vert , or\beta\Vert ( 1, 0)\Vert_{H}=\Vert(1, 0)\Vert and \beta\Vert(0,1)\Vert_{H}=\Vert(0,1)\Vert.
Then

C_{NJ}( \Vert\cdot\Vert)=\frac{\beta^{2} {\alpha^{2} .
上の公式を用いて,\mathb {R}_{a,b,p}^{2} のNJ 定数を計算する. 1/a^{p}+1/b^{p}\leq 1 のとき明らかにCNJ (\mathbb{R}_{a,b,p}^{2})=2 . また a=b=1 のとき \Vert(x, y)\Vert=\Vert(x, y)\Vert_{p} であるから a>1,1/a^{p}+1/b^{p}>1 の場合のみ考えればよい.

Definition 10. For a\geq b\geq 1, \Vert . \Vert_{H} is defined by \Vert(x, y)\Vert_{H}=\Vert (ax, by) \Vert_{2}.
p\geq 2 とする.Definition 4と Definition 10で定義した \Vert . \Vert, \Vert . \Vert_{H} はabsolute であり,さらにTheorem 9の(i) , (ii) , (iii) の条件をみたす.Theorem 9を適用することで次が得られる.

Theorem 11 ([5]) Let a>1, a\geq b\geq 1 and p\geq 2 with(i) If b\leq a(a^{p}-1)^{\frac{p-2}{2p}} , then\frac{1}{a^{p} +\frac{1}{b^{p} >1.

C_{NJ}( \mathbb{R}_{a,b,p}^{2})=1+b^{2}(1-\frac{1}{a^{p} )^{\frac{2}{p}(ii) If b>a(a^{p}-1)^{\frac{p-2}{2p}} , then

C_{NJ}( \mathb {R}_{a,b,p}^{2})=b^{2}(1+(\frac{a}{b})^{\frac{2p}{p-2} )^{1- \frac{2}{p}In particular, C_{NJ}(\mathbb{R}_{a,b,p}^{2})=d(\mathbb{R}_{a,b,p}^{2}, H)^{2} , where

H is a two‐dimensional inner productspace.

この結果は Theorem 3を含む.また,p<2 の場合も同様の方法で計算することができる.
Theorem 12 Let

a>1, a\geq b\geq 1 and p<2 with

\frac{1}{a^{p} +\frac{1}{b^{p} >1 and a^{\frac{2p}{p-2} +b^{\frac{2p}{p-2} \leq 1.

Then

C_{NJ}( \mathbb{R}_{a,b,p}^{2})=1+b^{2}(1-\frac{1}{a^{p} )^{\frac{2}{p}In particular, C_{NJ}(\mathbb{R}_{a,b,p}^{2})=d(\mathbb{R}_{a,b}^{2},{}_{p}H)^{2} , where H is a two‐dimensional inner productspace.

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201
Corollary 13 Let a>1, a\geq b\geq 1 with

\frac{1}{a}+\frac{1}{b}>1 and \frac{1}{a^{2} +\frac{1}{b^{2} \leq 1 . Then

C_{NJ}( \mathbb{R}_{a,b,1}^{2})=1+b^{2}(1-\frac{1}{a})^{2}
In particular,

C_{NJ}(\mathbb{R}_{a,a,1}^{2})=a^{2}-2a+2.

参考文献[1] J. Banaś, K. Fraczek, Deformation of Banach spaces, Comment. Math. Univ.
Carolin. 34 (1993), no. 1, 47‐53.
[2] J. A. Clarkson, The von Neumann‐Jordan constant for the Lebesgue space, Ann.
of Math. 38 (1937), 114‐115.
[3] S. Dhompongsa, P. Piraisangjun, S. Saejung, Generalized Jordan‐von Neumannconstants and uniform normal structure, Bull. Austral. Math. Soc. 67 (2003),
225‐240.

[4] M. Kato, L. Maligranda, Y. Takahashi, On James and Jordan‐von Neumann con‐stants and the normal structure coefficient of Banach spaces, Studia Math. 144(2001), 275‐295.

[5] K.‐I. Mitani, K.‐S. Saito, Y. Takahashi, On the von Neumann‐Jordan constant ofgeneralized Banaś‐Frqczek spaces, Linear Nonlinear Anal. 2 (2016), 311‐316.

[6] K.‐S. Saito, M. Kato, Y. Takahashi, Von Neumann‐Jordan constant of absolutenormalized norms on \mathb {C}^{2} , J. Math. Anal. Appl. 244 (2000), 515‐532.
[7] Y. Takahashi, Some geometric constants of Banach spaces‐A unified approach,Banach and function spaces II, 191‐220, Yokohama Publ., Yokohama, 2008.

[8] Y. Takahashi, M. Kato, Von Neumann‐Jordan constant and uniformly non‐square
Banach spaces, Nihonkai Math. J. 9 (1998), 155‐169.
[9] Y. Takahashi, M. Kato, A simple inequality for the von Neumann‐Jordan andJames constants of a Banach space, J. Math. Anal. Appl. 359 (2009), 602‐609.
[10] C. Yang, Jordan‐von Neumann constant for Banaś‐Frqczek space, Banach J. Math.
Anal. 8 (2014), 185‐192.

202[11] C. Yang, An inequality between the James type constant and the modulus ofsmoothness, J. Math. Anal. Appl. 398 (2013), 622‐629.

[12] C. Yang, H. Li, On the James type constant of \ell_{p}-\ell_{1} , J. Inequal. Appl. 2015:
Article ID 79 (2015).
[13] C. Yang, F. Wang, On a new geometric constant related to the von Neumann‐Jordan constant, J. Math. Anal. Appl. 324 (2006), 555‐565.
[14] C. Yang, F. Wang, The von Neumann‐Jordan constant for a class of Day‐JamesSpaces, Mediterr. J. Math. 13 (2016), 1127‐1133.