Stability Characterizations of ε-isometries on Certain Banach Spaces  

作　　者：Li Xin CHENG Long Fa SUN 

机构地区：[1]School of Mathematical Sciences,Xiamen University

出　　处：《Acta Mathematica Sinica,English Series》2019年第1期123-134,共12页数学学报（英文版）

基　　金：supported in part by the Natural Science Foundation of China(Grant Nos.11731010,11471270&11471271)

摘　　要：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 spaces.We 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.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 spaces.We 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 

分 类 号：O177.2[理学—数学]