Bounded holomorphic mappings and the compact approximation property in Banach spaces (Q1430619)

A complex Banach space \(E\) is said to have the compact approximation property (CAP) if for every compact set \(K \subset E\) and every \(\varepsilon > 0,\) there is a compact operator \(T:E \to E\) such that \(| T(x) - x | < \varepsilon \) for all \(x \in K.\) As the author observes, \textit{G. Willis} [Stud. Math. 103, 99--108 (1992; Zbl 0814.46017)] has shown that CAP \(\not\Rightarrow\) AP ( approximation property). If \(U \subset E\) is open and \(F\) is another complex Banach space, let \({\mathcal H}^\infty(U;F)\) denote the Banach space of bounded holomorphic mappings \(f:U \to F.\) In [Trans. Am. Math. Soc. 324, 867--887 (1991; Zbl 0747.46038)], \textit{J. Mujica} studied \({\mathcal H}^\infty(U) = {\mathcal H}^\infty(U;C),\) constructing its predual \(G^\infty(U),\) and showing that the following properties are equivalent if \(U\) is balanced and bounded: (i) ~ \(E\) has the AP; (ii) \(G^\infty(U)\) has the AP; (iii) for every \(F,\) every \(f \in {\mathcal H}^\infty(U;F)\) can be approximated (with respect to a certain locally convex topology \(\tau_\gamma\)) by functions in \({\mathcal H}^\infty(U) \otimes F.\) The author extends Mujica's work to obtain analogous characterizations for the CAP. Theorem: Let \(U \subset E\) be an open, bounded, balanced subset of \(E.\) Then the following are equivalent: (i) \(E\) has the CAP; (ii) \(G^\infty(U)\) has the CAP; (iii) for every \(F,\) every \(f \in {\mathcal H}^\infty(U;F)\) can be approximated by polynomials \(P:E \to F\) that are weakly continuous on bounded sets in \(E,\) with respect to \(\tau_\gamma.\) As a consequence, the author shows that \(E\) has the CAP if and only if for every \(F\) and every polynomial \(P:E \to F\), \(P\) can be approximated in the compact open topology by polynomials from \(E\) to \(F\) that are weakly continuous on bounded sets.