On linear isometries and ε-isometries between Banach spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:435

Article

Zhou, Yu¹  Zhang, Zihou¹  Liu, Chunyan¹ 

[1] Shanghai Univ Engn Sci, Sch Fundamental Studies, Shanghai 201620, Peoples R China

关键词: Linear isometry;    epsilon-Isometry;    Stability;    Banach space;   

DOI  :  10.1016/j.jmaa.2015.10.035

来源: Elsevier

【 摘 要 】

Let X, Y be two Banach spaces, and f : X -> Y be a standard e-isometry for some epsilon >= 0. Recently, Cheng et al. showed that if (co) over bar [f (X) U -f(X)] = Y, then there exists a surjective linear operator T: Y -> X with parallel to T parallel to = 1 such that the following sharp inequality holds: parallel to Tf(x) - x parallel to <= 2 epsilon for all x is an element of X. Making use of the above result, we prove the following results: Suppose that (co) over bar [f(X) U - f (X)] = Y. Then (1) if there is a linear isometry S : X -> Y such that TS = Id(X), then T*S* : Y* -> T*(X*) is a w*-to-w* continuous linear projection with parallel to T*S*parallel to = 1, (2) if there exists a w*-to-w* continuous linear projection P : Y* -> T* (X*) with parallel to P parallel to = 1, then there is an unique linear isometry S(P) : X -> Y such that TS(P) = Id(X) and P = T*S(P)*. Furthermore, if P-1 not equal P-2 are two w*-to-w* continuous linear projection from Y* onto T*(X*) with parallel to P-1 parallel to = parallel to P-2 parallel to = 1, then S(P-1)not equal S(P-2). We apply these results to provide an alternative proof of a recent theorem, which gives an affirmative answer of a question proposed by Vestfrid. We also unify several known theorems concerning the stability of epsilon-isometries. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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