A Carathéodory theorem for the bidisk via Hilbert space methods

If φ is an analytic function bounded by 1 on the bidisk D ² and τ ε ∂(D ²) is a point at which φ has an angular gradient ∇φ(τ) then ∇φ(λ) → ∇φ(τ) as λ → τ nontangentially in D ². This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if τ ε ∂(D ²) is such that the lim inf of (1-{pipe}φ(λ){pipe})/(1-{double pipe}λ{double pipe}) as λ → τ is finite then the directional derivative D _-δφ(τ) exists for all appropriate directions δ 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.