ON A UNIVERSAL INEQUALITY FOR APPROXIMATE PHASE ISOMETRIES  

作　　者：戴端旭 阙海新 孙龙发 郑本拓 Duanxu DAI;Haixin QUE;Longfa SUN;Bentuo ZHENG(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)

机构地区：[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

出　　处：《Acta Mathematica Scientia》2024年第3期823-838,共16页数学物理学报（B辑英文版）

基　　金：supported by the NSFC(12126329,12171266,12126346);the NSF of Fujian Province of China(2023J01805);the Research Start-Up Fund of Jimei University(ZQ2021017);supported by the NSFC(12101234);the NSF of Hebei Province(A2022502010);the Fundamental Research Funds for the Central Universities(2023MS164);the China Scholarship Council;supported by the Simons Foundation(585081)。

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 someε≥ 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^(*)‖B_(x^(*)),there exist a phase function σ_(u^(*)):X→{-1,1} and φ ∈ Y^(*) with ‖φ‖=‖u^(*)‖≡α satisfying that|(u^(*),u)-σ_(u^(*))(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■ provided that Y^(**) has the w^(*)-Kadec-Klee property(for example,Y is both reflexive and locally uniformly convex).

关 键 词：ε-phase isometry phase isometry Banach space 

分 类 号：O178[理学—数学]