\(\mathcal I\)-bounded holomorphic functions (Q2789085)

Given a Banach space \(F\) and an operator ideal \({\mathcal I}\), the authors use \({\mathcal C}_{\mathcal I}(F)\) to denote the collection of all sets \(A\subset F\) such that there is a Banach space \(Z\) and an operator \(T\in {\mathcal I}(Z,F)\) with \(A\subset T(B_Z)\). An entire holomorphc function \(f: E\to F\) is said to be locally \({\mathcal I}\)-bounded at \(x\) if there is a neighbourhood \(V_x\) of \(x\) such that \(f(V_x)\in {\mathcal C}_{\mathcal I}(F)\) and locally \({\mathcal I}\)-bounded if it is locally \({\mathcal I}\)-bounded at each point of \(E\). For \(f: E\to F\) locally \({\mathcal I}\)-bounded at \(x\), the radius of \({\mathcal I}\)-boundedness is \(\sup\{t>0:f(x+t B_E)\in {\mathcal C}_{\mathcal I}(F)\}\). The collection of all locally \({\mathcal I}\)-bounded holomorphic functions from \(E\) to \(F\) is denoted by \({\mathcal H}_{\mathcal I}(E,F)\), whereas the collection of all locally \({\mathcal I}\)-bounded \(n\)-homogeneous polynomials is denoted by \({\mathcal P}_{\mathcal I}(^nE,F)\). The authors prove that, if \(f: E\to F\) is locally \({\mathcal I}\)-bounded at \(x\) and the closed convex hull of a set in \({\mathcal C}_{\mathcal I}(F)\) remains in \({\mathcal C}_{\mathcal I}(F)\), then \({\hat d}^nf(x)/n!\) belongs to \({\mathcal P}_{\mathcal I}(^nE, F)\) for each \(n\) in \({\mathbb N}\) and NEWLINE\[NEWLINEr_{\mathcal I}(f,x)={1\over \limsup_{n\to\infty}\|{\hat d}^nf(x)/n!\|_{\mathcal I}^{1/n}}.NEWLINE\]NEWLINE As a converse, it is shown that, if \(f: E\to F\) is holomorphic, \({\hat d}^n f(x)/n!\) belongs to \({\mathcal P}_{\mathcal I}(^nE,F)\), and there is \(R>0\) so that \(\sum_{n=0}^\infty \|{\hat d}^nf(x)/n!\|_{\mathcal I}r^n < \infty\) for \(0<r<R\), then \(f\) is locally \({\mathcal I}\)-bounded at \(x\). It is also shown that \({\mathcal H}_{\mathcal I}(E,F)\) can be given a locally convex Hausdorff topology with natural properties.