#### Document Type

Article

#### Publication Date

2012

#### Abstract

If *φ* is an analytic function bounded by 1 on the bidisk D 2 and τ∈∂(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∈∂(D 2 ) is such that the lim inf of (1−|φ(λ)|)/(1−∥λ∥) as λ → *τ* is finite then the directional derivative *D*_{−δ}*φ*(*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*.

#### Recommended Citation

Agler, Jim; McCarthy, John E.; and Young, N J., "A Carathéodory theorem for the bidisk via Hilbert space methods" (2012). *Mathematics Faculty Publications*. 8.

https://openscholarship.wustl.edu/math_facpubs/8

#### Embargo Period

1-3-2013

## Comments

Author version. Original publication in Mathematische Annalen Volume 352, Issue 3 , pp 581-624 is available at link.springer.com. DOI: 10.1007/s00208-011-0650-7.