On linear isometries and ε-isometries between Banach spaces
On linear isometries and ε-isometries between Banach spaces
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DOI：
10.1016/j.jmaa.2015.10.035
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发表时间：
2016-03
影响因子：
1.3
通讯作者：
Yu Zhou;Zihou Zhang;Chunyan Liu
中科院分区：
文献类型：
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作者：
Yu Zhou;Zihou Zhang;Chunyan Liu
设X，Y是两个Banach空间，f：X→Y是某个ε0的标准ε≥等距。最近，程等人提出了一个新的观点。证明了：如果co‾[f(X)∪−f(X)]=Y，则存在一个满射线性算子T：Y→X，且‖T‖=1，使得对所有x个∈X都成立：−T f(X)‖≤xε2∈X。利用上述结果，我们证明了如下结果：设co‾[f(X)∪−f(X)]=Y，则(1)如果存在线性等距算子S：X→Y使得T S=ID X，则T⁎S⁎：Y⁎→T⁎(X⁎)是一个w⁎-to-w⁎连续线性投影，且‖T⁎S⁎‖=1；(2)如果存在w⁎-to-w⁎连续线性投影P：y⁎→T⁎(X⁎)且‖P‖=1，则存在唯一的线性等距投影S(P)：X→Y使得T S(P)=ID X且P=T⁎S(P)⁎.进一步地，如果P1≠P2是两个w⁎to w⁎从Y⁎到T⁎(X⁎)的连续线性投影，且‖P1‖=‖P2‖=1，则S(P1)≠S(P2).我们应用这些结果为最近的一个定理提供了另一种证明，该定理对VestFrid提出的一个问题给出了肯定的回答。我们还统一了关于ε-等距稳定性的几个已知定理。
Let X, Y be two Banach spaces, and f: X→ Y be a standard ε-isometry for some ε≥ 0. Recently, Cheng et al. showed that if co‾[f (X)∪− f (X)]= Y, then there exists a surjective linear operator T: Y→ X with‖ T‖= 1 such that the following sharp inequality holds:‖ T f (x)− x‖≤ 2 ε for all x∈ X. Making use of the above result, we prove the following results: Suppose that co‾[f (X)∪− f (X)]= Y. Then (1) if there is a linear isometry S: X→ Y such that T S= Id X, then T⁎ S⁎: Y⁎→ T⁎(X⁎) is a w⁎-to-w⁎ continuous linear projection with‖ T⁎ S⁎‖= 1,(2) if there exists a w⁎-to-w⁎ continuous linear projection P: Y⁎→ T⁎(X⁎) with‖ P‖= 1, then there is an unique linear isometry S (P): X→ Y such that T S (P)= Id X and P= T⁎ S (P)⁎. Furthermore, if P 1≠ P 2 are two w⁎-to-w⁎ continuous linear projection from Y⁎ onto T⁎(X⁎) with‖ P 1‖=‖ P 2‖= 1, then S (P 1)≠ 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 ε-isometries.