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Classification of \(\varepsilon\)-isometries by stability. (English) Zbl 1457.46013

Summary: Assume that \(X\), \(Y\) are two Banach spaces, and \(f:X\to Y\) is a standard \(\varepsilon\)-isometry. In this paper, we first study properties of the two specific subspaces \(L\) and \(N^\bot\) of \(L(f)^{\ast\ast}\equiv\overline{\operatorname{span}}[f(X)]^{\ast\ast}\), which are deduced from the weak stability formula and play an essential rule in study of stability of \(f\). Then we show that \(f\) is \(w^\ast\)-\(\gamma\)-stable if and only if there is a \(w^\ast\)-to-\(w^\ast\) projection \(P:L(f)^{\ast\ast}\to N^\bot\) so that \(Pf:X\to N^\bot\) is a \(\gamma\)-approximate linear isometry, i.e., there exists a surjective linear isometry \(U:X^{\ast\ast}\to N^\bot\) so that \[ \|Pf(x)-Ux\|\le\gamma\varepsilon,\quad\text{for all }\; x\in X. \] In particular, if \(\varepsilon=0\), then \(Pf\) is simply a linear isometry. Finally, we show a new characterization for \(f\) to be (resp. \(w^\ast\)) stable in terms of linear bounded selections respect to the weak stability formula of \(f\).

MSC:

46B04 Isometric theory of Banach spaces
46B20 Geometry and structure of normed linear spaces
47A58 Linear operator approximation theory
46A20 Duality theory for topological vector spaces
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