Finite Blaschke products and the construction of rational $\Gamma$-inner functions

Abstract

Let \[ Γ= \{(z+w, zw): |z|\leq 1, |w|\leq 1\} \subset \mathbb{C}^2. \] A $\Gamma$-inner function is defined to be a holomorphic map $h$ from the unit disc $\mathbb{D}$ to $\Gamma$ whose boundary values at almost all points of the unit circle $\mathbb{T}$ belong to the distinguished boundary $b\Gamma$ of $\Gamma$. A rational $\Gamma$-inner function $h$ induces a continuous map $h{|}_{\mathbb{T}}$ from the unit circle to $b\Gamma$. The latter set is topologically a Möbius band and so has fundamental group $\mathbb{Z}$. The {\em degree} of $h$ is defined to be the topological degree of $h{|}_{\mathbb{T}}$. In a previous paper the authors showed that if $h=(s,p)$ is a rational $\Gamma$-inner function of degree $n$ then $s^2-4p$ has exactly $n$ zeros in the closed unit disc ${\mathbb{D}}^{-}$, counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational $\Gamma$-inner functions of degree $n$ with the $n$ zeros of $s^2-4p$ and the corresponding values of $s$, prescribed.

35 pages. This is the revised version after referees'reports, Journal of Mathematical Analysis and Applications, 2016

References in corpus (5)

• Extremal holomorphic maps and the symmetrised bidisc
• From Stinespring dilation to Sz.-Nagy dilation on the symmetrized bidisc and operator models
• Lifting maps from the symmetrized polydisk in small dimensions
• Some analysable instances of mu-synthesis
• Algebraic and geometric aspects of rational $\Gamma$-inner functions