A Carathéodory theorem for the bidisk via Hilbert space methods

Jim Agler, John E. McCarthy, N. J. Young

Research output: Contribution to journalArticlepeer-review

20 Scopus citations


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 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.

Original languageEnglish
Pages (from-to)581-624
Number of pages44
JournalMathematische Annalen
Issue number3
StatePublished - Mar 2012


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