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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 $G_{n}$.