Linearization of isometric embedding on Banach spaces (Q2787141)

Let \(X\) and \(Y\) be Banach spaces, and let \(f: X\rightarrow Y\) be a standard isometry, i.e., \(\|f(x)-f(y)\|=\|x-y\|\) for all \(x\) and \(y\) in \(X\) and \(f(0)=0\). A theorem due to Figiel formulates the existence and uniqueness of a norm \(1\) bounded linear operator \(T: \overline{\text{span}}(f(X)) \rightarrow X\) such that \(T\circ f=\operatorname{Id}_X\). This operator is called the Figiel operator. Work done by \textit{G. Godefroy} and \textit{N. J. Kalton} [Stud. Math. 159, No. 1, 121--141 (2003; Zbl 1059.46058)] shows that for a separable Banach space \(X\), the Figiel operator \(T\) has a linear isometric right inverse. However, there are pairs of Banach spaces \((X,Y)\) such that \(X\) can be isometrically embedded into \(Y\) without being linearly isomorphic to a subspace of \(Y\).NEWLINENEWLINEInspired by these results, the authors answer the following two problems for a standard isometry \(f:X\rightarrow Y\). {\parindent=6mm \begin{itemize} \item[1.] Find necessary and sufficient conditions for the Figiel operator \(T\) of \(f\) to admit a linear isometric right inverse \(S:X \rightarrow \overline{\text{span}}(f(X))\).\item [2.] Identify classes of non-separable Banach spaces \(Y\) with the property that \(T\) admits a linear isometric right inverse \(S\). NEWLINENEWLINE\end{itemize}} The main results in the paper are the following theorems:NEWLINENEWLINE {Theorem 2.1.} Let \(X\) and \(Y\) be Banach spaces. Let \(f: X\rightarrow Y\) be a standard isometry and \(T: \overline{\text{span}}(f(X)) \rightarrow X\) be the Figiel operator of \(f\).{\parindent=6mm \begin{itemize} \item[(I)] If there exists a linear isometry \(S:X \rightarrow \overline{\text{span}}(f(X))\) such that \(T\circ S=\operatorname{Id}_X,\) then \(T^*\circ S^*: \overline{\text{span}}(f(X))^* \rightarrow T^*(X^*)\) is a \(w^*\)-to-\(w^*\) continuous projections with \(\|T^*\circ S^*\|=1\).\item [(II)] If there exists a \(w^*\)-to-\(w^*\) continuous projection \(P:\overline{\text{span}}(f(X))^* \rightarrow T^*(X^*)\) with \(\|P\|=1,\) then there exists a unique linear isometric right inverse \(S:X \rightarrow \overline{\text{span}}(f(X))\) of \(T\) such that \(T\circ S=\operatorname{Id}_X\) and \(P=T^*\circ S^*\).NEWLINENEWLINE\end{itemize}} A Banach space is said to be (resp., weakly) nearly strictly convex if, given \(x^* \in X^*\) with \(\|x^*\|=1\), the set \(\{ x \in X: \|x\|=1, \, x^*(x)=1\}\) is (resp., weakly) compact.NEWLINENEWLINE {Theorem 2.6.} Let \(X\) and \(Y\) be Banach spaces. Let \(f: X\rightarrow Y\) be a standard isometry and \(T: \overline{\text{span}}(f(X)) \rightarrow X\) be the Figiel operator of \(f\). If \(\overline{\text{span}}(f(X))\) is a weakly nearly strictly convex space, then there exists a linear isometry \(S:X \rightarrow \overline{\text{span}}(f(X))\) such that \(T\circ S=\operatorname{Id}_X\).