Some New Observations on Interpolation in the Spectral Unit Ball

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \Omega n, n ≥ 2. We begin by showing that a known necessary condition for the existence of a O(D;\Omega n)-interpolant (D here being the unit disc in 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 \Omega n, n ≥ 2