Some new observations on interpolation in the spectral unit ball
Abstract
Description
We present several results associated to a holomorphic-interpolation problem for the spectral unit ball Ω_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;Ω_n)$-interpolant (D here being the unit disc in the complex plane), 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\geq 2.
Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/0608177; to appear in Integral Eqns. Operator Theory
Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/0608177; to appear in Integral Eqns. Operator Theory