TY - JOUR

T1 - A Carathéodory theorem for the bidisk via Hilbert space methods

AU - Agler, Jim

AU - McCarthy, John E.

AU - Young, N. J.

N1 - Funding Information:
J. Agler was partially supported by National Science Foundation Grant DMS 0801259; J. E. McCarthy was partially supported by National Science Foundation Grants DMS 0501079 and DMS 0966845; and N. J. Young was supported by EPSRC Grant EP/G000018/1.

PY - 2012/3

Y1 - 2012/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84856210812&partnerID=8YFLogxK

U2 - 10.1007/s00208-011-0650-7

DO - 10.1007/s00208-011-0650-7

M3 - Article

AN - SCOPUS:84856210812

SN - 0025-5831

VL - 352

SP - 581

EP - 624

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3

ER -