### A Caratheodory theorem for the bidisk via Hilbert space methodsA Caratheodory theorem for the bidisk via Hilbert space methods

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 Author Agler, Jim ♦ McCarthy, John E. ♦ Young, Nicholas J. Source arXiv.org Content type Text File Format PDF Date of Submission 2010-02-19 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Complex Variables ♦ Mathematics - Functional Analysis ♦ 32A40 ♦ math Abstract If $\ph$ is an analytic function bounded by 1 on the bidisk $\D^2$ and $\tau\in\tb$ is a point at which $\ph$ has an angular gradient $\nabla\ph(\tau)$ then $\nabla\ph(\la) \to \nabla\ph(\tau)$ as $\la\to\tau$ nontangentially in $\D^2$. This is an analog for the bidisk of a classical theorem of Carath\'eodory for the disk. For $\ph$ as above, if $\tau\in\tb$ is such that the $\liminf$ of $(1- \ph(\la) )/(1-\ \la\ )$ as $\la\to\tau$ is finite then the directional derivative $D_{-\de}\ph(\tau)$ exists for all appropriate directions $\de\in\C^2$. Moreover, one can associate with $\ph$ and $\tau$ an analytic function $h$ in the Pick class such that the value of the directional derivative can be expressed in terms of $h$. Educational Use Research Learning Resource Type Article