On the spectral Nevanlinna–Pick problem

Constantin Costara¹

¹ Département de mathématiques et de statistique Université Laval Québec (QC) G1K 7P4, Canada

Studia Mathematica, Tome 170 (2005) no. 1, pp. 23-55

Résumé

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 $G_{n}$.

Détail

DOI : 10.4064/sm170-1-2

Keywords: several characterizations symmetrized n disc which generalize geq characterizations symmetrized bidisc order solve two point spectral nevanlinna pick problem mathcal mathbb using these characterizations symmetrized n disc which necessary sufficient conditions element belong obtain necessary conditions interpolation general spectral nevanlinna pick problem allow method construct analytic functions unit disc mathbb obtain complex geodesics