Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem
Abstract
Description
We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in C^n, n \geq 2, is an example of such a domain. Given a λ-nonisotropically balanced domain Ω, we derive effective estimates from above and from below for the Lempert function at (0,z)\inΩ\timesΩ. We use these estimates to derive certain conditions for realising a two-point Nevanlinna-Pick interpolation in the symmetrized polydisc. Applying the ideas used in the derivation of our Lempert function estimates to the so-called spectral unit ball Ω_n, we deduce: a) a formula for the Lempert function at (0,W)\inΩ_n\timesΩ_n; and b) a necessary and sufficient condition for realising a two-point Nevanlinna- Pick interpolation in the spectral unit ball.
12 pages. Added clarifications to Thm.1.8; added the important remark (Rmrk.1.3): the "only if" part of Thm.1.8 is a special case of a result by Globevnik
12 pages. Added clarifications to Thm.1.8; added the important remark (Rmrk.1.3): the "only if" part of Thm.1.8 is a special case of a result by Globevnik