A Caratheodory theorem for the bidisk via Hilbert space methods

Abstract

If $\ph$ is an analytic function bounded by 1 on the bidisk $\D^2$ and $τ\in\tb$ is a point at which $\ph$ has an angular gradient $\nabla\ph(τ)$ then $\nabla\ph(\la) \to \nabla\ph(τ)$ as $\la\toτ$ nontangentially in $\D^2$. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For $\ph$ as above, if $τ\in\tb$ is such that the $\liminf$ of $(1-|\ph(\la)|)/(1-\|\la\|)$ as $\la\toτ$ is finite then the directional derivative $D_{-\de}\ph(τ)$ exists for all appropriate directions $\de\in\C^2$. Moreover, one can associate with $\ph$ and $τ$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$.