The isometries of $H^{\infty }(K)$

Abstract:

Let K be a finite-dimensional Hilbert space. In this article a characterization is given of the linear isometries of the Banach space ${H^\infty }(K)$ onto itself. It is shown that T is such an isometry iff T is of the form $(TF)(z) = \mathcal {T}F(t(z))$, for $F \in {H^\infty }(K)$ and z belonging to the unit disc, where t is a conformal map of the disc onto itself and $\mathcal {T}$ is an isometry of K onto K.