A Carath,odory theorem for the bidisk via Hilbert space methods

If. is an analytic function bounded by 1 on the bidisk D-2 and tau is an element of partial derivative(D-2) is a point at which. has an angular gradient del phi(tau) then del phi(lambda) -> del phi(tau) as lambda -> tau nontangentially in D-2. This is an analog for the bidisk of a classical theorem of Caratheodory for the disk. For. as above, if tau is an element of partial derivative(D-2) is such that the lim inf of (1 - |phi(lambda)|)/(1 - parallel to lambda parallel to) as lambda -> tau is finite then the directional derivative D_(delta phi)(tau) exists for all appropriate directions delta is an element of C-2. Moreover, one can associate with. and t an analytic function h in the Pick class such that the value of the directional derivative can be expressed in terms of h.