Nevanlinna–Pick Problem and Uniqueness of Left Inverses in Convex Domains, Symmetrized Bidisc and Tetrablock

Łukasz Kosiński,Włodzimierz Zwonek

doi:10.1007/s12220-015-9611-9

Łukasz Kosiński, Włodzimierz Zwonek

Open Access

https://doi.org/10.1007/s12220-015-9611-9

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Abstract

In the paper we discuss the problem of uniqueness of left inverses (solutions of two-point Nevanlinna–Pick problem) in bounded convex domains, strongly linearly convex domains, the symmetrized bidisc and the tetrablock.

Highlights

The Nevanlinna-Pick problem for a domain D of Cn may be stated as follows: Given distinct points z1, . . . , zN in D and numbers λ1, . . . , λN in the unit disc D decide whether there is an analytic function F : D → D that interpolates, i.e., F = λj, j = 1, . . . , N
This problem has been considered in different domains, and many attempts have been made to extend it in different directions
In a sequence of influential papers Agler and McCarthy used the operator theory approach to carry out an analysis of the Nevanlinna-Pick problem for the bidisc

Summary

Introduction

The first author of this paper found an alternate approach to the Nevanlinna-Pick problem in the polydisc (see [16]) and N = 3, which resulted in solving the problem in this situation. This approach allowed Knese to prove the von Neumann inequality for 3 × 3 matrices (see [15]). Speaking we show that a 3-point Nevanlinna-Pick problem in the Euclidean ball may be expressed in terms of a dual problem D → Bn. We find a class of rational functions of degree at most 2 interpolating every such problem. The precise statement of the result is postponed to the section

Definitions and results

Degenerate case - proof of Theorem 3

Non-degenerate case - proof of Theorem 5

Proof of Theorem 1

Findings

Relations with the Green function with two poles