On the isometries of $H^ \infty _ E(B)$

Abstract:

Let $E$ be a complex Banach space on which all the multipliers are trivial. Let $H_E^\infty (B)$ denote the Banach space of $E$-valued bounded holomorphic functions on the open unit ball $B$ of ${{\mathbf {C}}^n}$. In this paper we prove that every linear isometry $T$ of $H_E^\infty (B)$ onto itself is of the form $(TF)(z) = \mathfrak {T}F(\varphi (z))$ for all $F \in H_E^\infty (B),\;z \in B$, where $\mathfrak {T}$ is a linear isometry of $E$ onto itself and $\varphi$ is a biholomorphic map of $B$.