Infinite products of holomorphic mappings

Abstract

Let $X$ be a complex Banach space, $𝒩$ a norming set for $X$, and $D\subset X$ a bounded, closed, and convex domain such that its norm closure $\overline{D}$ is compact in $\sigma \left(X,𝒩\right)$. Let $\varnothing \ne C\subset D$ lie strictly inside $D$. We study convergence properties of infinite products of those self-mappings of $C$ which can be extended to holomorphic self-mappings of $D$. Endowing the space of sequences of such mappings with an appropriate metric, we show that the subset consisting of all the sequences with divergent infinite products is $\sigma$-porous.