Facial behaviour of analytic functions on the bidisk

Abstract

We prove that if $\varphi$ is an analytic function bounded by 1 on the bidisk and $\tau$ is a point in a face of the bidisk at which $\varphi$ satisfies Caratheodory's condition then both $\varphi$ and the angular gradient $\nabla \varphi$ exist and are constant on the face. Moreover, the class of all $\varphi$ with prescribed $\varphi (\tau )$ and $\nabla \varphi (\tau )$ can be parametrized in terms of a function in the two-variable Pick class. As an application we solve an interpolation problem with nodes that lie on faces of the bidisk.

18 pages. We have replaced an erroneous proof of Theorem 5.4(1) by a valid proof