Carathéodory interpolation on the non-commutative polydisk

AbstractThe 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×n matrices, for all n∈N