On the spectral Nevanlinna–Pick problem
Volume 170 / 2005
Studia Mathematica 170 (2005), 23-55
MSC: Primary 30E05; Secondary 32F45.
DOI: 10.4064/sm170-1-2
Abstract
We give several characterizations of the symmetrized $n$-disc $G_{n}$ which generalize to the case $n\geq 3$ the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna–Pick problem in ${\mathcal M}_{2}( {\mathbb C})$. Using these characterizations of the symmetrized $n$-disc, which give necessary and sufficient conditions for an element to belong to $G_{n}$, we obtain necessary conditions of interpolation for the general spectral Nevanlinna–Pick problem. They also allow us to give a method to construct analytic functions from the open unit disc of ${\mathbb C}$ into $G_{n}$ and to obtain some of the complex geodesics on $G_{n}$.
