On the Isometries of H ∞ E (B)

Abstract

Let E be a complex Banach space on which all the multipliers are trivial. Let HE (B) denote the Banach space of E-valued bounded holomorphic functions on the open unit ball B of Cn . In this paper we prove that every linear isometry T of HE (B) onto itself is of the form (TF)(z) = TF(fo(z)) for all F E HE (B), z E B, where T is a linear isometry of E onto itself and fo is a biholomorphic map of B.