A Carathéodory theorem for the bidisk via Hilbert space methods

If φ is an analytic function bounded by 1 on the bidisk D2 and τ ∈ ∂(D2) is a point at which φ has an angular gradient ∇φ(τ) then ∇φ(λ) → ∇φ(τ) as λ → τ nontangentially in D2. This is an analog for the bidisk of a classical theorem of Carathéodory for the disk. For φ as above, if τ ∈ ∂(D2) is such that the lim inf of (1− |φ(λ)|)/(1− ‖λ‖) as λ → τ is finite then the directional derivative D−δφ(τ) exists for all appropriate directions δ ∈ C2. 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.