article

Title: Remarks on the operator-valued interpolation for multivariable bounded analytic functions

Authors: Calin G. Ambrozie

Issue: Volume 53 (2004), Issue 6, 1551-1578

Abstract:

$We consider a domain $D \subset \mathbb{C}^n$ defined by a uniform norm inequality $\sup_{\lambda}\|\Delta_{\lambda}(z)\| < 1$ involving a set $\Delta=(\Delta_{\lambda})_{\lambda}$ of matrix-valued analytic functions $\Delta_{\lambda}$. The associated Schur interpolation class is then $$\mathcal{S} = \{F=F(z) :\sup_{\|\Delta(Z)\| < 1}\|F(Z)\| \leq 1\},$$ where $Z$ runs the commuting $n$-tuples of matrices. We characterize by positive-definiteness conditions the existence of the solutions $F \in \mathcal{S}S$ of an operator-valued Nevanlinna-Pick type problem over $D$. Also, we describe the elements of $mathcal{S}$ as fractional transforms $F = a_{22}+a_{21}(I - \Delta a_{11})^{-1}\Delta a_{12}$, with $[a_{ij}]_{i,j=1}^2$ unitary. The results are based on a representation technique due to J. Agler.$