Some new observations on interpolation in the spectral unit ball

Details Some new observations on interpolation in the spectral unit ball Some new observations on interpolation in the spectral unit ball

2007-04-16

arxiv.org/abs/0704.1966v2

Integral Equations Operator Theory 59 (2007) no. 3, 329-343

Mathematics

Complex Variables

Added a definition (Def.1.1); 2 of the 4 results herein are minor refinements of those in the author's preprint math.CV/060817

Scientific paper

10.1007/s00020-007-1534-9

We present several results associated to a holomorphic-interpolation problem for the spectral unit ball \Omega_n, n\geq 2. We begin by showing that a known necessary condition for the existence of a $\mathcal{O}(D;\Omega_n)$-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 \Omega_n, n\geq 2.

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