ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

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doi:10.1007/s10473-024-0303-z

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (3): 823-838.doi: 10.1007/s10473-024-0303-z

Duanxu Dai¹, Haixin Que¹, Longfa Sun^2,*, Bentuo Zheng³

1. School of Science, Jimei University, Xiamen 361021, China;
2. Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China;
3. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

收稿日期:2022-10-26 修回日期:2023-04-15 出版日期:2024-06-25 发布日期:2024-05-21

Duanxu Dai¹, Haixin Que¹, Longfa Sun^2,*, Bentuo Zheng³

1. School of Science, Jimei University, Xiamen 361021, China;
2. Hebei Key Laboratory of Physics and Energy Technology, School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China;
3. Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA

Received:2022-10-26 Revised:2023-04-15 Online:2024-06-25 Published:2024-05-21
Contact: *Longfa Sun, E-mail:sun.longfa@163.com
About author:Duanxu Dai,E-mail:dduanxu@163.com; Haixin Que, 1131871505@qq.com
Supported by:
Dai's research was supported by the NSFC (12126329, 12171266, 12126346), the NSF of Fujian Province of China (2023J01805) and the Research Start-Up Fund of Jimei University (ZQ2021017); Sun's research was supported by the NSFC (12101234), the NSF of Hebei Province (A2022502010), the Fundamental Research Funds for the Central Universities (2023MS164) and the China Scholarship Council; Zheng's research was supported by the Simons Foundation (585081).

摘要： Let $X$ and $Y$ be two normed spaces. Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$ . Let $g: X\rightarrow Y$ be a standard $\varepsilon$ -phase isometry for some $\varepsilon\geq 0$ , i.e., $g(0)=0$ , and for all $u,v\in X$ ,

| | ‖ g (u) + g (v) ‖ \pm ‖ g (u) - g (v) ‖ | - | ‖ u + v ‖ \pm ‖ u - v ‖ | | \leq ε .

$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$
The mapping

g

$g$ is said to be a phase isometry provided that

ε = 0

$\varepsilon=0$ .
In this paper, we show the following universal inequality of

g

$g$ : for each

u^{*} \in w^{*}

$u^*\in w^*$ -exp

‖ u^{*} ‖ B_{X^{*}}

$\|u^*\|B_{X^*}$ , there exist a phase function

σ_{u^{*}} : X \to {- 1, 1}

$\sigma_{u^*}: X\rightarrow \{-1,1\}$ and

φ

$\varphi$

\in

$\in$

Y^{*}

$Y^*$ with

‖ φ ‖ = ‖ u^{*} ‖ \equiv α

$\|\varphi\|= \|u^*\|\equiv \alpha$ satisfying that

| ⟨ u^{*}, u ⟩ - σ_{u^{*}} (u) ⟨ φ, g (u) ⟩ | \leq \frac{5}{2} ε α, f o r a l l u \in X .

$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$ In particular, let

X

$X$ be a smooth Banach space. Then we show the following:
(1) the universal inequality holds for all

u^{*} \in X^{*}

$u^*\in X^*$ ;
(2) the constant

\frac{5}{2}

$\frac{5}{2}$ can be reduced to

\frac{3}{2}

$\frac{3}{2}$ provided that

Y^{*}

$Y^\ast$ is strictly convex;
(3) the existence of such a

g

$g$ implies the existence of a phase isometry

Θ : X \to Y

$\Theta:X\rightarrow Y$ such that

Θ (u) = lim_{n, U} \frac{g (n u)}{n}

$\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that

Y^{* *}

$Y^{**}$ has the

w^{*}

$w^*$ -Kadec-Klee property (for example,

Y

$Y$ is both reflexive and locally uniformly convex).

关键词: $\varepsilon$ -phase isometry, phase isometry, Banach space

Abstract: Let $X$ and $Y$ be two normed spaces. Let $\mathcal{U}$ be a non-principal ultrafilter on $\mathbb{N}$ . Let $g: X\rightarrow Y$ be a standard $\varepsilon$ -phase isometry for some $\varepsilon\geq 0$ , i.e., $g(0)=0$ , and for all $u,v\in X$ ,

| | ‖ g (u) + g (v) ‖ \pm ‖ g (u) - g (v) ‖ | - | ‖ u + v ‖ \pm ‖ u - v ‖ | | \leq ε .

$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$
The mapping

g

$g$ is said to be a phase isometry provided that

ε = 0

$\varepsilon=0$ .
In this paper, we show the following universal inequality of

g

$g$ : for each

u^{*} \in w^{*}

$u^*\in w^*$ -exp

‖ u^{*} ‖ B_{X^{*}}

$\|u^*\|B_{X^*}$ , there exist a phase function

σ_{u^{*}} : X \to {- 1, 1}

$\sigma_{u^*}: X\rightarrow \{-1,1\}$ and

φ

$\varphi$

\in

$\in$

Y^{*}

$Y^*$ with

‖ φ ‖ = ‖ u^{*} ‖ \equiv α

$\|\varphi\|= \|u^*\|\equiv \alpha$ satisfying that

| ⟨ u^{*}, u ⟩ - σ_{u^{*}} (u) ⟨ φ, g (u) ⟩ | \leq \frac{5}{2} ε α, f o r a l l u \in X .

$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$ In particular, let

X

$X$ be a smooth Banach space. Then we show the following:
(1) the universal inequality holds for all

u^{*} \in X^{*}

$u^*\in X^*$ ;
(2) the constant

\frac{5}{2}

$\frac{5}{2}$ can be reduced to

\frac{3}{2}

$\frac{3}{2}$ provided that

Y^{*}

$Y^\ast$ is strictly convex;
(3) the existence of such a

g

$g$ implies the existence of a phase isometry

Θ : X \to Y

$\Theta:X\rightarrow Y$ such that

Θ (u) = lim_{n, U} \frac{g (n u)}{n}

$\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that

Y^{* *}

$Y^{**}$ has the

w^{*}

$w^*$ -Kadec-Klee property (for example,

Y

$Y$ is both reflexive and locally uniformly convex).

Key words: $\varepsilon$ -phase isometry, phase isometry, Banach space

中图分类号:

46B04

Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng. ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES[J]. 数学物理学报(英文版), 2024, 44(3): 823-838.

Duanxu Dai, Haixin Que, Longfa Sun, Bentuo Zheng. ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES[J]. Acta mathematica scientia,Series B, 2024, 44(3): 823-838.

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