Abstract. We give several characterizations of the symmetrized n-disc G n which generalize to the case n ≥ 3 the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna-Pick problem in M 2 (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 C into G n and to obtain some of the complex geodesics on G n .

Let G ⊆ C 2 be the open symmetrized bidisc, namely G = {(λ 1 + λ 2 , λ 1 λ 2 ) : |λ 1 | < 1, |λ 2 | < 1}. In this paper, a proof is given that G is not biholomorphic to any convex domain in C 2 . By combining this result with earlier work of Agler and Young, the author shows that G is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set.

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A method is presented to construct interpolation functions into the 2 × 2 open spectral unit ball. For the spectral Nevanlinna-Pick problem, these functions are in some sense extremal, and the set of all these interpolation functions is enough to solve any interpolation problem, with solvable finite interpolation data. This fact is used to compute the complex geodesics for the symmetrized bidisc and for the spectral unit ball, and to solve completely the two-point interpolation problem for the two target sets.

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