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Carathéodory interpolation on the non-commutative polydisk

Dmitry S. Kalyuzhny\uı-Verbovetzki\uı
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Abstract

The Carathéodory problem in the $N$-variable non-commutative Herglotz--Agler class and the Carathéodory--Fejér problem in the $N$-variable non-commutative Schur--Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory--Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set $Λ$) takes operator values with positive semidefinite real part (resp., contractive operator values) on $N$-tuples of $Λ$-jointly nilpotent contractive $n\times n$ matrices, for all $n\in\mathbb{N}$.

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Dmitry S. Kalyuzhny\uı-Verbovetzki\uı. 2005-03-17. Carathéodory interpolation on the non-commutative polydisk. https://arxiv.org/abs/math/0412161

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