Composition, numerical range and Aron-Berner extension

Abstract

Given an entire mapping $f\in \mathcal{H}_b(X,X)$ of bounded type from a Banach space $X$ into $X$, we denote by $\overline{f}$ the Aron-Berner extension of $f$ to the bidual $X^{\ast\ast}$ of $X$. We show that $\overline{g\circ f} = \overline{g}\circ \overline{f}$ for all $f, g\in \mathcal{H}_b(X,X)$ if $X$ is symmetrically regular. We also give a counterexample on $l_1$ such that the equality does not hold. We prove that the closure of the numerical range of $f$ is the same as that of $\bar{f}$.