Nonisotropically balanced domains, Lempert function estimates, and the spectral Nevanlinna-Pick problem

2006/01/06 by Gautam Bharali, Bharali, Gautam
Mathematics · #Algebraic and Geometric Analysis #Differential Equations and Boundary Problems #Holomorphic and Operator Theory #math.CV #math.FA #msc:30C80 #msc:32F45

paper · pdf · doi:10.48550/arxiv.math/0601107

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

arxiv created 2006/03/30 · arxiv updated 2009/12/01

Abstract

We introduce the notion of a λ-nonisotropically balanced domain and show that the symmetrized polydisc in Cⁿ, n ≥ 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)∈Ω×Ω. 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)∈Ω_n×Ω_n; and b) a necessary and sufficient condition for realising a two-point Nevanlinna- Pick interpolation in the spectral unit ball.

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