Some New Observations on Interpolation in the Spectral Unit Ball

Abstract.

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ω _n , n ≥ 2. We begin by showing that a known necessary condition for the existence of a ${\mathcal{O}} ({\mathbb{D}};\Omega_{n})$-interpolant (${\mathbb{D}}$ here being the unit disc in ${\mathbb{C}}$), given that the matricial data are non-derogatory, is not sufficient. We provide next a new necessary condition for the solvability of the two-point interpolation problem – one which is not restricted only to non-derogatory data, and which incorporates the Jordan structure of the prescribed data. We then use some of the ideas used in deducing the latter result to prove a Schwarz-type lemma for holomorphic self-maps of Ω _n , n ≥ 2.