We prove that, for certain domains $\mathbb{D}$, continuous product of domains $D_{\omega }$, the Carathéodory pseudodistance satisfies the following product property $C_{\mathbb{D}}\left(f,g\right)={sup}_{\omega }{C}_{D_{\omega }}\left(f\left(\omega \right),g\left(\omega \right)\right)$

We give several characterizations of the symmetrized n-disc Gₙ 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 Gₙ, we obtain necessary conditions of interpolation for the general spectral Nevanlinna-Pick problem. They also allow us to give a...

Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone $C_A$ to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.