On linear isometries and \(\varepsilon\)-isometries between Banach spaces (Q892356)

Let \(X\) and \(Y\) be two real Banach spaces, \(\varepsilon>0\), and \(f: X\to Y\) a standard \(\varepsilon\)-isometry. In [\textit{L. Cheng} et al., J. Funct. Anal. 269, No. 1, 199--214 (2015; Zbl 1326.46008)] it was shown that, if \(\overline{\text{conv}}[f(X)\cup -f(X)]=Y\), then there is a surjective linear operator \(T: Y\to X\) with \(\|T\|=1\) such that \(\|Tf(x)-x\|\leq 2\varepsilon\) for all \(x\in X\). Assume that \(\overline{\text{conv}}[f(X)\cup -f(X)]=Y\) and let \(T\) be the surjective linear operator from above. The main result of the paper under review reads as follows: {\parindent=6mm \begin{itemize} \item[(1)] If there is a linear isometry \(S: X \to Y\) such that \(TS=\mathrm{Id}_X\), then \(T^\ast S^\ast: Y^\ast\to T^\ast(X^\ast)\) is a \(\omega^\ast\)-to-\(\omega^\ast\) continuous linear projection with \(\|T^\ast S^\ast\|=1\); \item [(2)] if there is a \(\omega^\ast\)-to-\(\omega^\ast\) continuous linear projection \(P: Y^\ast\to T^\ast(X^\ast)\) with \(\|P\|=1\), then there is a unique linear isometry \(S(P): X\to Y\) such that \(TS(P)=\mathrm{Id}_X\) and \(P=T^\ast S(P)^\ast\). \end{itemize}} As an application, the authors are able to answer affirmatively to a question posed in [\textit{I. A. Vestfrid}, J. Funct. Anal. 269, No. 7, 2165--2170 (2015; Zbl 1331.46010)]. Moreover, their result allows them to unify several known theorems concerning the stability of \(\varepsilon\)-isometries.