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A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

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Acta Mathematica Sinica,English Series 2012年第10期28卷 2107-2114页

作者： li xin cheng Yun Bai DONG School of Mathematical Sciences Xiamen University

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.

关键词： Norm attaining operator Hilbert space Bishop-Phelps theorem

Banach Spaces Which are Isometric to Subspaces of c_(0)(Г)

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Acta Mathematica Sinica,English Series 2021年第8期37卷 1171-1178页

作者： li xin cheng Jian Jian WANG School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

In this paper,we give a number of characterizations for a Banach space X which is isometric to a subspace of c_(0),or,c_(0)(Г),successively,in terms of extreme points of its dual unit ball Bx*,Frechet and Gateaux der... 详细信息

关键词： Isometric embedding to c_(0) Frechet derivative Gateaux derivative extreme point Banach space

On Hereditarily Indecomposable Banach Spaces

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Acta Mathematica Sinica,English Series 2006年第3期22卷 751-756页

作者： li xin cheng Huai Jie ZHONG Department of Mathematics Xiamen University The Institute of Mathematical Sciences The Chinese University of Hang Kong Department of Mathematics Fujian Normal University

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily i... 详细信息

关键词： decomposition of spaces hereditarily indecomposable Banach space renormings

Stability Characterizations of ε-isometries on Certain Banach Spaces

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Acta Mathematica Sinica,English Series 2019年第1期35卷 123-134页

作者： li xin cheng Long Fa SUN School of Mathematical Sciences Xiamen University

Suppose that X, Y are two real Banach Spaces. We know that for a standard ε-isometry f : X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of *. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ε-isometry to be stable in assuming that N is w*-closed in Y*.Making use of this result, we improve several known results including Figiel’s theorem in reflexive *** also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span[f(X)] contains a complemented linear isometric copy of X;Moreover, if X =Y, then for every e-isometry f: X → X, there exists a surjective linear isometry S:X → X such that f-S is uniformly bounded by 2ε on X.

关键词： ε-isometry stability hereditarily indecomposable space quasi-reflexive space Banach space

On Absolute Uniform Retracts, Uniform Approximation Property and Super Weakly Compact Sets of Banach Spaces

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Acta Mathematica Sinica,English Series 2021年第5期37卷 731-739页

作者： li xin cheng Qing Jin cheng Jian Jian WANG School of Mathematical Sciences Xiamen UniversityXiamen 361005P.R.China

In this paper,we show that every super weakly compact convex subset of a Banach space is an absolute uniform retract,and that it also admits the uniform compact approximation *** can be regarded as extensions of linde... 详细信息

关键词： Super weak compactness absolute uniform retract Banach spaces

On Universally Left-stability of ε-Isometry

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Acta Mathematica Sinica,English Series 2013年第11期29卷 2037-2046页

作者： ling xin BAO li xin cheng Qing Jin cheng Duan Xu DAI School of Mathematical Sciences Xiamen University

Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Ban... 详细信息

Let X, Y be two real Banach spaces and ε≥0. A map f ： X → Y is said to be a standard ε-isometry if│││f/（x） - f（y）││ - ]ix - Y││x-y││ ε for all x,y C X and with f（O） = O. We say that a pair of Banach spaces （X, Y） is stable if there exists γ〉 0 such that, for every such ε and every standard v-isometry f ： X → Y, there is a bounded linear operator T ： L（f） → f（X） → X so that ││Tf（x） - x││ ≤γε for all x E X. X（Y） is said to be universally left-stable if （X, Y） is always stable for every Y（X）. In this paper, we show that if a dual Banach space X is universally left-stable, then it is isometric to a complemented w＊-closed subspace of ∞ （1） for some set F, hence, an injective space; and that a Banach space is universally left-stable if and only if it is a cardinality injective space; and universally left-stability spaces are invariant.

关键词： ε-Isometry linear isometry stability injective space Banach space

Certain Subsets on Which Every Bounded Convex Function Is Continuous

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Acta Mathematica Sinica,English Series 2007年第6期23卷 1063-1066页

作者： li xin cheng Yan Mei TENG School of Mathematical Sciences Xiamen University Xiamen 361005 P. R. China Department of Mathematics Beijing University of Aeronautics and Astronautics Beijing 100083 P. R. China

To guarantee every real-valued convex function bounded above on a set is continuous, how ＂thick＂ should the set be？ For a symmetric set A in a Banach space E,the answer of this paper is： Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold： i） spanA has finite co-dimentions and ii） coA has nonempty relative interior. This paper also shows that a subset A C E satisfying every real-valued convex function bounded above on A is continuous on E if （and only if） every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E.

关键词： Convex function Boundedness Continuity Banach space

Statistical Convergence and Measure Convergence Generated by a Single Statistical Measure

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Acta Mathematica Sinica,English Series 2016年第6期32卷 668-682页

作者： li xin cheng li Hua liN Xian Geng ZHOU School of Mathematical Sciences Xiamen University Institute of Mathematical and Information Engineering Sanming University Department of Mathematics Ningde Normal University

The purpose of this paper is to discuss those kinds of statistical convergence, in terms of F , or ideal Z-convergence, which are equivalent to measure convergence defined by a single statistical measure. We prove a number of characterizations of a single statistical measure μ-convergence by using properties of its corresponding quotient Banach space l∞/l∞ （Iμ）. We also show that the usual sequential convergence is not equivalent to a single measure convergence.

关键词： Statistical measure ultrafilter extreme points statistical convergence

Every Weakly Compact Set Can Be Uniformly Embedded into a Reflexive Banach Space

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Acta Mathematica Sinica,English Series 2009年第7期25卷 1109-1112页

作者： li xin cheng Qing Jin cheng Zheng Hua LUO Wen ZHANG Department of Mathematics Xiamen University Xiamen 361005 P. R. China

Based on an application of the Davis-Figiel-Johnson-Pelzyski procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell- Schlumprecht and Hajek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space.

关键词： Banach space weakly compact set renorming