ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES

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外文摘要：Let X and Y be two normed spaces.Let U be a non-principal ultrafilter on N.Let g:X→Y be a standard ε-phase isometry for some ε ≥ 0,i.e.,g(0)=0,and for all u,v ∈ X,||||g(u)+g(v)||±||g(u)-g(v)|||-|||u+v||±||u-v||||≤ε.The mapping g is said to be a phase isometry provided that ε=0.In this paper,we show the following universal inequality of g:for each u*∈ w*-exp||u*||Bx*,there exist a phase function σu*:X → {-1,1} and φ ∈ Y*with||φ||=||u*||≡ α satisfying that|<u*,u>-σu*(u)<φ,g(u)>|≤5/2εα,for all u∈X.In particular,let X be a smooth Banach space.Then we show the following:(1)the universal inequality holds for all u*∈ X*;(2)the constant 5/2 can be reduced to 3/2 provided that Y*is strictly convex;(3)the existence of such a g implies the existence of a phase isometryΘ:X → Y such that Θ(u)=limn,u g(nu)/n provided that Y**has the w*-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).

外文关键词：

ε-phase isometryphase isometryBanach space

作者：

戴端旭、阙海新、孙龙发、郑本拓

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作者单位：

School of Science,Jimei University,Xiamen 361021,China

Hebei Key Laboratory of Physics and Energy Technology,School of Mathematics and Physics,North China Electric Power University,Baoding 071003,China

Department of Mathematical Sciences,University of Memphis,Memphis,TN 38152,USA

基金：

NSFCNSFCNSFCNSF of Fujian Province of ChinaResearch Start-Up Fund of Jimei UniversityNSFCNSF of Hebei ProvinceFundamental Research Funds for the Central UniversitiesChina Scholarship CouncilSimons Foundation

项目编号：

1212632912171266121263462023J01805ZQ202101712101234A20225020102023MS164585081

出版年：

2024

DOI：

10.1007/s10473-024-0303-z

数学物理学报(英文版)

中科院武汉物理与数学研究所

数学物理学报(英文版)

CSTPCD

影响因子：0.256

ISSN：0252-9602

年,卷(期)：2024.44(3)