A Carathéodory theorem for the bidisk via Hilbert space methods

Abstract

If φ is an analytic function bounded by 1 on the bidisk ${{\mathbb D}^2}$ and ${\tau\in\partial({\mathbb D}^2)}$ is a point at which φ has an angular gradient ${\nabla\varphi(\tau)}$ then ${\nabla\varphi(\lambda) \to \nabla\varphi(\tau)}$ as λ → τ nontangentially in ${{\mathbb D}^2}$. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if ${\tau\in\partial({\mathbb D}^2)}$ is such that the lim inf of ${(1-|\varphi(\lambda)|)/(1-\|\lambda\|)}$ as λ → τ is finite then the directional derivative D _−δ φ(τ) exists for all appropriate directions ${\delta\in{\mathbb C}^2}$. Moreover, one can associate with φ 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.