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Geometric constants of Banach spaces and the matrix norms (The research of geometric structures in quantum information based on Operator Theory and related topics)

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(1)101. 数理解析研究所講究録 第2033巻 2017年 101-105. Geometric constants of Banach spaces and the matrix norms. Kichi‐Suke. Saito,. Naoto Komuro and. Ryotaro. Tanaka. introduction. 1. In the. of Banach space geometry, geometric constants which. theory. quantify. various. geometric features of Banach spaces often play fundamental roles. This paper is mainly concerned with, in particular, one of the best‐known geometric constants of Banach spaces, that is, James constant. Let X be. Banach space, and let S_{X} denote the unit sphere of X The von Neumann‐ a Banach space was first considered by Jordan and von Neumann [4]. a. .. Jordan constant of based. on. constant. their characterization of inner. C_{NJ}(X). of X is. product. spaces.. Namely,. the. von. Neumann‐Jordan. given by. C_{NJ}(X)=\displaystyle \sup\{\frac{\Vert x+y\Vert^{2}+\Vert x-y\Vert^{2} {2(| x| ^{2}+| y\Vert^{2})}:(x, y)\neq(0,0)\}. The. following. are. basic. properties of. von. Neumann‐Jordan constant:. (i) 1\leq C_{NJ}(X)\leq 2. (ii) C_{NJ}(X)=1 (iii) C_{NJ}(X). if and. only. if X is. <2 if and. only. if X is. such that. \displaystyle \min\{\Vert x+y. Moreover, C_{NJ}(X) if. we. has the. a. Hilbert space.. uniformly. \Vert x-y. <2(1- $\delta$). whenever x,. representation using the. norm. is, there exists. of certain 2\times 2 matrix.. ,. which. implies. This formula has the. following applications.. (iv) C_{NJ}(X^{*})=C_{NJ}(X) X is. -11 ) \left(\begin{ar y}{l x\ y \end{ar y}\right)\Vert^{2}\Vert\left(\begin{ar y}{l x\ y \end{ar y}\right)\Vert^{-2}. that. C_{NJ}(X)=2^{-1}\Vert( 11. uniformly. -11 ) :Y\rightar ow Y\Vert^{2}. .. non‐square if and. only if X^{*}. is. uniformly. a. $\delta$>0. y\in S_{X} ([12]).. put Y=X\otimes_{2}X then. \displaystyle \frac{\Vert x+y|^{2}+|x-y\Vert^{2} {\Vert x|^{2}+|y|^{2} =\Vert( 1. (v). non‐square, that. non‐square.. Indeed,.

(2) 102. Thus it. be worth. can. using the. norm. On the other. [2]. and Lau. considering another formulation of geometric constant,. in. particular,. of matrices.. hand, the James. as a measure. J(X). constant. of X. was. of the squareness of the unit ball.. introduced in 1990. Namely,. we. define. by. Gao. J(X) by. J(X)=\displaystyle \sup\{\min\{\Vert x+y \Vert x-y : x, y\in S_{X}\}. It is known that. J(X). has the. following properties:. (i) \sqrt{2}\leq J(X)\leq 2 ([2]). (ii). If H is. a. Hilbert space, then. J(H)=\sqrt{2}.. (iii). If \dim X \geq 3 , then J(X)=\sqrt{2} implies that X is a Hilbert space ([7]); and hence J(X)=\sqrt{2} if and only if X is a Hilbert space provided that \dim X\geq 3.. (iv). There. are. ([2, 7, 8,. (v) J(X)<2 (vi). if and. There exists. Unlike. von. J(X)=\sqrt{2}. a. only. uniformly. if X is. non‐square.. two‐dimensional normed space X with. J(X^{*})\neq J(X) ([5]).. Neumann‐Jordan constant, in general, James constant does not have repre‐ norm of matrices. However, for a certain class of norms on \mathbb{R}^{2} , we. using the. sentations can. various non‐Hilbert two‐dimensional normed spaces X with. 9. consider such. a. representation.. In this paper, we present a new representation of James constant for $\pi$/2 ‐rotation invariant norms on \mathbb{R}^{2} by using the norm of a 2 \mathrm{x}2 matrix. We also give some applications. of that. representation.. A. 2. representation. invariant Let. \Vert\cdot\Vert. be. of James constant for. $\pi$/2 ‐rotation. norms. a norm on. \mathb {R}^{2} and let $\theta$\in(0,2 $\pi$) Then \Vert\cdot\Vert .. ,. is said to be $\theta$ ‐rotation invariant. if the $\theta$ ‐rotation matrix. R($\thea$)=\left(begin{ar y}{l \mathrm{c}\mathrm{o}\mathrm{s}$\thea$&-\mathrm{s}\mathrm{i}\ athrm{n}$\thea$\ mathrm{s}\mathrm{i}\ athrm{n}$\thea$&\mathrm{c}\mathrm{o}\mathrm{s}$\thea$ \end{ar y}\right) is. an. isometry. Euclidean. on. norm. (\mathbb{R}^{2}, \Vert. All. .. norms on. Recall that the James constant the notion of isosceles to. \mathbb{R}^{2}. is $\theta$ ‐rotation invariant for each. J(X). of. are. $\theta$\in(0,2 $\pi$). ,. $\pi$ ‐rotation. invariant, and the. .. Banach space X has the simple form using An element x\in X is said to be isosceles orthogonal a. orthogonality. y\in X denoted by x1_{I}y if \Vert x+y\Vert=\Vert x-y\Vert ,. clearly. .. In Gao and Lau. [2],. it. was. shown that. J(X)=\displaystyle \sup\{\Vert x+y\Vert : x, y\in S_{X}, x\perp_{I}y\}. This formula. of the. plays very important role, especially, following result.. in the two‐dimensional. case. because.

(3) 103. (Gao. Lemma 2.1. and Lau. [2];. Alonso. normed space. Suppose that x\in S_{X} y\in S_{X} such that x\perp_{I}y.. .. [1];. Ji et al.. Then there exists. [3]). a. Let X be. unique (up. following theorem says that the class of $\pi$/2 ‐rotation study of James constant.. The. two‐dimensional. a. to the. invariant. sign). element. norms on. \mathbb{R}^{2} is. suitable for the. (Komuro,. Theorem 2.2. be. \mathbb{R}^{2}. a norm on. (i) \Vert\cdot\Vert. is. Saito and Mitani. Then the. .. $\pi$/2 ‐rotation. following. x\perp_{I}R( $\pi$/2)x for. following. The. \mathbb{R}^{2}. our. Saito and Tanaka. [8]).. Let. \Vert\cdot\Vert. equivalent.. whenever. each. \Vert x\Vert=\Vert y\Vert=1.. x.. main result.. (Komuro,. Theorem 2.3 norm on. is. [6]; Komuro,. invariant.. (ii) x\perp_{I}y if and only if \langle x, y\rangle=0 In which cases,. are. Saito and Tanaka. [10]).. Let. \Vert. \Vert. be. $\pi$/2 ‐rotation. a. invariant. Then. .. J((\mathbb{R}^{2}, \Vert. =\sqrt{2}\Vert R( $\pi$/4):(\mathbb{R}^{2}, \Vert. \rightarrow(\mathbb{R}^{2}, \Vert. Proof. By. the. preceding theorem,. we. have. J((\displaystyle \mathbb{R}^{2}, \Vert. =\sup\{\Vert x+y\Vert:x, y\in \mathbb{R}^{2}, \Vert x\Vert=\Vert y\Vert=1, x\perp_{I}y\} =\displaystyle \sup\{\Vert x+R( $\pi$/2)x\Vert : x\in \mathbb{R}^{2}, \Vert x\Vert=1\} =\displaystyle \sup\{\Vert(I+R( $\pi$/2))x\Vert:x\in \mathbb{R}^{2}, \Vert x\Vert=1\} =\Vert I+R( $\pi$/2):(\mathbb{R}^{2}, \Vert. \rightarrow(\mathbb{R}^{2}, \Vert. The conclusion follows from This. simple result. In this section, with the. Then. we. following. Proposition. 3.1. J((\mathbb{R}^{2}, \Vert.. We. can. make. interesting applications.. present. two. apphcations of Theorem 2.3. [2, Proposition 2.8].. The first. one. is concerned. result of Gao and Lau. (Gao and =\sqrt{2}. use. of. our. Lau. [2]).. Let. \Vert\cdot\Vert. main result for. be. a. giving. a. $\pi$/4 ‐rotation. invariant. norm on. simple proof of partial. \mathbb{R}^{2}.. converse. to. preceding proposition.. (Komuro, J((\mathbb{R}^{2}, \Vert.. Theorem 3.2 on. some. \square. .. Applications. 3. the. has. I+R( $\pi$/2)=\sqrt{2}R( $\pi$/4). \mathbb{R}^{2}. .. Then. [8]). Let \Vert\cdot\Vert be a $\pi$/2 ‐rotation invariant =\sqrt{2} if and only if \Vert\cdot\Vert is $\pi$/4 ‐rotation invariant.. Saito and Tanaka. norm.

(4) 104. The “if”’ part is the statement of. Proof.. =\sqrt{2} Then \Vert R( $\pi$/4)\Vert J((\mathbb{R}^{2}, \Vert. \Vert R(- $\pi$/2)\Vert=1 it follows that .. Proposition 3.1. For the converse, suppose that by Theorem 2.3. Moreover, since | R( $\pi$/2)\Vert. 1. =. =. ,. \Vert R( $\pi$/4)^{-1}\Vert=\Vert R(- $\pi$/4)\Vert=\Vert R(- $\pi$/2)R( $\pi$/4)\Vert\leq 1. Thus, for each. has. x , one. \Vert x\Vert=\Vert R(- $\pi$/4)R( $\pi$/4)x\Vert\leq\Vert R( $\pi$/4)x\Vert\leq\Vert x\Vert, that. is, \Vert R( $\pi$/4)x\Vert=\Vert x\Vert This. proves that. .. \Vert\cdot\Vert. is. $\pi$/4‐rotation. \square. invariant.. application is concerned with the sufficient condition that the equality J(X^{*})= Naturally, the dual space X^{*} of X can be identified with (\mathbb{R}^{2}, \Vert\cdot\Vert_{*}) under J(X) X^{*} \ni f \leftrightar ow (f(1,0), f(0,1)) \in \mathb {R}^{2} In this manner, the adjoint operator (as a Banach A^{*} of a A can be represented by the transpose A^{T} Moreover, it matrix space operator) Another. holds.. .. .. \Vert A^{*}\Vert_{X^{*}}=\Vert A\Vert_{X}.. is well‐known that. following. We need the. Lemma 3.3. Let $\theta$\in \mathbb{R} the dual. norm. Using. \Vert\cdot\Vert_{*}. Proof.. \mathbb{R}^{2}. Let. dual. norm. that. J(X). isometry. Suppose. .. (Komuro,. =. \Vert\cdot\Vert. is. we. have the. a. $\theta$ ‐rotation invariant. following. Saito and Tanaka. J((\mathbb{R}^{2}, \Vert\cdot\Vert)^{*})=J((\mathbb{R}^{2}, \Vert.. Then. norm on. \mathb {R}^{2}. .. Then. [10]).. theorem.. Let. \Vert. \Vert. be. a. $\pi$/2 ‐rotation. invariant. Then its $\pi$/2 ‐rotation invariant norm on \mathbb{R}^{2} Put X= (\mathbb{R}^{2}, \Vert invariant Lemma It follows from Theorem 2.3 ‐rotation 3.3. by $\pi$/2 and since is R( $\pi$/2) an \sqrt{2}\Vert R( $\pi$/4)\Vert_{X} J(X^{*}) \sqrt{2}\Vert R( $\pi$/4)\Vert_{X^{*}} However,. \Vert. \Vert \Vert\cdot\Vert_{*}. on. that. is also $\theta$ ‐rotation invariant.. this and Theorem 2.3,. Theorem 3.4 norm on. .. results.. be. a. .. .. is also. =. X^{*} ,. one. .. has that. \Vert R( $\pi$/4)\Vert_{X}=\Vert R( $\pi$/4)^{*}\Vert_{X^{*=}}\Vert R(- $\pi$/4)\Vert_{X^{*}} =\Vert R( $\pi$/2)R(- $\pi$/4)\Vert_{X^{*}}=\Vert R( $\pi$/4)\Vert_{X^{*}}. This proves that. J(X^{*})=J(X). ,. as. \square. desired.. References [1]. J.. Alonso, Uniqueness properties of isosceles orthogonality Quebec, 18 (1994), 25‐38.. [2]. J. Gao and K.‐S. Lau, On the geometry Math. Soc. Ser. A, 48 (1990), 101‐112.. [3]. D.. Ji, J.. in normed linear spaces, J. Aust.. Wu, On the uniqueness of isosceles orthogonality Math., 59 (2011), 157‐162.. P. Jordan and J.. Math.. of spheres. Li and S.. spaces, Results. [4]. in normed linear spaces,. Ann. Sci. Math.. (2),. 36. von. (1935),. Neumann, On 719‐723.. inner. products. in. linear,. in normed linear. metric spaces, Ann. of.

(5) 105. [5]. M.. Kato, L. Maligranda and Y. Takahashi, On James and Jordan‐von Neumann coefficient of Banach spaces, Studia Math., 144. constants and the normal structure. (2001), [6]. N.. 275‐295.. Mitani, Convex property of James and von of absolute norms on \mathbb{R}^{2} Proceedings of the 8th Interna‐ Nonlinear Analysis and Convex Analysis, 301‐308, Yokohama. Komuro, K.‐S. Saito. and K.‐I.. Neumann‐Jordan constant tional Conference. on. ,. Publ., Yokohama, 2015.. [7]. N.. Komuro, K.‐S. Saito and R. Tanaka, On the class of Banach. constant. [8]. N.. \sqrt{2}. ,. 289. (2016),. spaces with James. 1005‐1020.. Komuro, K.‐S. Saito and R. Tanaka, On the class of Banach \sqrt{2} : Part II, Mediterr. J. Math., 13 (2016), 4039−4061.. spaces with James. Komuro, K.‐S. Saito and R. Tanaka, On the class of Banach \sqrt{2} IlI, to appear in Math. Inequal. Appl.. spaces with James. constant. [9]. N.. constant. Komuro, K.‐S. Saito and R. Tanaka, A sufficient condition that J(X^{*}) for a Banach spaces X, to appear in Tokyo J. Math.. [10]. N.. [11]. K.‐I. Mitani and K.‐S.. Saito, The James. linear Convex. (2003),. [12]. =. J(X). holds. Anal.,. Y. Takahashi and M.. 4. Kato,. von. constant. of absolute. norms on. \mathbb{R}^{2} J. Non‐ ,. 399‐410. Neumann‐Jordan constant and. Banach spaces, Nihonkai Math. J., 9. (1998),. 155‐169.. Kichi‐Suke Saito. Department of Mathematical Sciences, Technology, Niigata University, Niigata 950‐2181, Japan E‐‐mail: saito@math.sc.niigata‐u.ac.jp Institute of Science and. Naoto Komuro. Department of Mathematics, Hokkaido University of Education, Asahikawa Campus, Asahikawa 070‐8621, Japan E‐‐mail: komuro@asa.hokkyodai.ac.jp Ryotaro Tanaka Faculty of Mathematics, Kyushu University, Fukuoka 819‐0395, Japan Lmail: \mathrm{r}‐tanaka@math. kyushu‐u. ac.jp. uniformly. non‐square.

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