Small into isomorphisms on uniformly smooth spaces. (Q1426999)

It is known that, in some cases, a small into-isomorphism \(T\) between Banach spaces is close to an isometry (see \textit{D. E. Alspach} [Ill. J. Math. 27, 300--314 (1983; Zbl 0495.46011)], for \(T:L^p\to L^p\), \(1\leq p<+\infty\), or \textit{Y. Benyamini} [Proc. Am. Math. Soc. 83, 479--485 (1981; Zbl 0474.46012)] for \(T:{\mathcal C}(K)\to {\mathcal C}(S)\), \(K\) compact metric, \(S\) Hausdorff compact). However, it is not always true (see \textit{Y. Benyamini} [Trans. Am. Math. Soc. 277, 825--833 (1983; Zbl 0515.46023)], and \textit{G. Godefroy, N. Kalton} and \textit{D. Li} [Indiana Univ. Math. J. 49, No. 1, 245--286 (2000; Zbl 0973.46008)]), even for operators \(T:\ell_2^n \to \ell_\infty\) (\(2\leq n<\infty\)): this is pointed out in \textit{R. Wang} [Acta Math. Sci. 9, No. 1, 27--32 (1989; Zbl 0716.46025)] and recalled in the present paper. However, the author shows that if \(X\) is an infinite-dimensional uniformly smooth Banach space and \(\mu\) a \(\sigma\)-finite measure, then every operator \(T: X\to L^\infty(\mu)\) such that \((1-\varepsilon)\| x\| \leq \| Tx\|\leq \| x\|\), for all \(x\in X\), with \(0<\varepsilon<1/2\) and \(\delta (2-2\varepsilon)>13/14\) (\(\delta\) being the modulus of uniform convexity of \(X^\ast\)), then there is an isometry \(U: X\to L^\infty(\mu)\) such that \(\| T-U\|\leq 16[1-\delta(2-2\varepsilon)]+\varepsilon/2\). The infinite dimension of \(X\) is needed to have well-separated infinite sequences of norm-one elements in \(X\) (see Lemma 2.1, which comes from Wang's paper).