Isometries of some smooth normed spaces of analytic functions

Abstract

Let H(D) denote the linear space of analytic functions on the open unit disc D, and let N be a norm on H(D). Let H N and S N denote the spaces of functions f in H(D) such that respectively. Let H N have the norm N. and give S N norm . We show that when is smooth, the linear isometries of H N can be described in terms of the linear isometries of H N. Applications are given to the weighted Hardy space, the weighted Bergman space, and the Bloch space.