Linearization of holomorphic Lipschitz functions (Q6606938)

Given complex Banach spaces \(X\) and \(Y\), the authors use \(\mathcal{H}L_0(B_X,Y)\) to denote the space of all Lipschitz functions \(f\colon B_X\to Y\) with \(f(0)=0\) which are holomorphic on \(B_X\) and endow it with the Lipschitz norm, \(L(f):=\sup_{x,y\in B_X,x\not=y}\frac{\|f(x)-f(y)\|}{\|x-y\|}\). The authors prove that for any \(f\) in \(\mathcal{H}L_0(B_X,Y)\), \(L(f)=\|df \|\). They show that for each complex Banach space \(X\) there are a Banach space \(\mathcal{G}_0(B_X)\) and \(\delta\in \mathcal{H}L_0(B_X,\mathcal{G}_0(B_X))\) with \(L( \delta)=1\) which have the property that given any Banach space \(Y\)and any \(f\) in \(\mathcal{H}L_0(B_X,Y)\), there is a unique linear operator \(T_f\colon\mathcal{G}_0(B_X) \to Y\) such that \(T_f\circ\delta=f\). Moreover, \(\|T_f\|=L(f)\). \N\NIt is shown that \(\mathcal{G}_0(B_X)\) has the approximation property (resp. metric approximation property) if and only if \(X\) has the approximation property (resp. metric approximation property). When \(X\) is symmetrically regular and \(X^{**}\) has the metric approximation property it is shown that \(\mathcal{G}_0(B_{X^{**}})\) is a \(1\)-complemented subspace of \(\mathcal{G}_0(X)^{**}\) and that \(f\) in \(\mathcal{H}L_0 (B_X)\) with \(L(f)=1\) has a unique norm preserving extension to \(\tilde f\in\mathcal{H}L_0(B_{X^{**}})\) if and only if the canonical extension from \((B_{\mathcal{H}L_0(B_X)},w^*)\) to \((B_{\mathcal{H}L_0(B_{X^{**}})},w^*)\) is continuous at~\(f\).