Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (3): 823-838.

### ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

Duanxu Dai1, Haixin Que1, Longfa Sun2,*, Bentuo Zheng3

1. 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).

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 $u,v\in X$,
$$|\; |\|g(u)+g(v)\|\pm \|g(u)-g(v)\||-|\|u+v\|\pm\|u-v\||\;|\leq\varepsilon.$$
The mapping $g$ is said to be a phase isometry provided that $\varepsilon=0$.
In this paper, we show the following universal inequality of $g$: for each $u^*\in w^*$-exp $\|u^*\|B_{X^*}$, there exist a phase function $\sigma_{u^*}: X\rightarrow \{-1,1\}$ and $\varphi$ $\in$ $Y^*$ with $\|\varphi\|= \|u^*\|\equiv \alpha$ satisfying that
$$\;\;\;\;\; |\langle u^*,u\rangle-\sigma_{u^*} (u)\langle \varphi, g(u)\rangle |\leq\frac{5}{2}\varepsilon\alpha ,\;\;{\rm for\;all\;}u\in X.$$ In particular, let $X$ be a smooth Banach space. Then we show the following:
(1) the universal inequality holds for all $u^*\in X^*$;
(2) the constant $\frac{5}{2}$ can be reduced to $\frac{3}{2}$ provided that $Y^\ast$ is strictly convex;
(3) the existence of such a $g$ implies the existence of a phase isometry $\Theta:X\rightarrow Y$ such that $\Theta(u)=\lim\limits_{n,\mathcal{U}}\frac{g(nu)}{n}$ provided that $Y^{**}$ has the $w^*$-Kadec-Klee property (for example, $Y$ is both reflexive and locally uniformly convex).

CLC Number:

• 46B04
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