COMPOSITION, NUMERICAL RANGE AND ARON-BERNER EXTENSION

Abstract

Given an entire mapping f is an element of H-b(X, X) of bounded type from a Banach space X into X, we denote by (f) over bar the Aron-Berner extension of f to the bidual X** of X. We show that <(g o f)over bar > = (g) over baro (f) over bar for all f, g is an element of 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 f.