On stability of nonlinear non-surjective \(\varepsilon \)-isometries of Banach spaces
Let \(X, Y\) be two Banach spaces and \(\varepsilon \geqslant 0\). A mapping \(f: X \to Y\) is said to be an \(\varepsilon\)-isometry provided that \(|\,\|f (x)-f (y)\| - \|x-y\|\,|\leq \varepsilon\) for all \(x, y\in X\). For such a function \(f\) with \(f(0)=0\), the authors show that for every \(x^{\ast}\in X^{\ast}\), there exists \(\phi \in Y^{\ast}\) with \(\|\phi \| =\|x^{\ast}\| = r\) such that \( |\langle\phi, f(x)\rangle-\langle x^\ast, x\rangle|\leqslant 4 \varepsilon r\) for all \(x\in X\). This is an improvement of a one-dimensional lemma which was presented by \textit{S.-W. Qian} [Proc. Am. Math. Soc. 123, No. 6, 1797--1803 (1995; Zbl 0827.47022)]. Making use of the above result, they prove that if \(Y\) is reflexive and if the annihilator \(E\subset Y\) of the subspace \(F\subset Y^\ast\) consisting of all functionals bounded on \(\overline{co}(f(X), -f(X))\) is \(\alpha\)-complemented in \(Y\), then there is a bounded linear operator \(T:Y \to X\) with \(\|T\|\leqslant \alpha\) such that \( \|Tf(x)-x\|\leqslant 4 \varepsilon\) for all \(x\in X\). If, in addition, \(Y\) is Gâteaux smooth, strictly convex and has the Kadec-Klee property, then they prove the sharp estimate \(\|T f(x)-x\|\leqslant 2 \varepsilon\) for all \(x \in X\). Motivated by a result of \textit{P. Šemrl} and \textit{J. Väisälä} [J. Funct. Anal. 198, No. 1, 268--278 (2003; Zbl 1033.46019)], they also discuss the existence of certain limits in general reflexive Banach spaces.
scientific article; zbMATH DE number 6140423
scientific article