A Schwarz lemma for the pentablock

Abstract

In this paper we prove a Schwarz lemma for the pentablock. The set \[ \mathcal{P}=\{(a_{21}, \text{tr} \ A, \det A) : A=[a_{ij}]_{i,j=1}^2 \in \mathbb{B}^{2\times 2}\} \] where ${\mathbb{B}}^{2\times 2}$ denotes the open unit ball in the space of $2\times 2$ complex matrices, is called the pentablock. The pentablock is a bounded nonconvex domain in ${\mathbb{C}}^3$ which arises naturally in connection with a certain problem of $\mu$-synthesis. We develop a concrete structure theory for the rational maps from the unit disc $\mathbb{D}$ to the closed pentablock $\overline{\mathcal{P}}$ that map the unit circle $\mathbb{T}$ to the distinguished boundary $b\overline{\mathcal{P}}$ of $\overline{\mathcal{P}}$. Such maps are called rational $\overline{\mathcal{P}}$-inner functions. We give relations between penta-inner functions and inner functions from $\mathbb{D}$ to the symmetrized bidisc. We describe the construction of rational penta-inner functions $x=(a,s,p):\mathbb{D}\to \overline{\mathcal{P}}$ of prescribed degree from the zeroes of $a,s$ and $s^2-4p$. The proof of this theorem is constructive: it gives an algorithm for the construction of a family of such functions $x$ subject to the computation of Fejér-Riesz factorizations of certain non-negative trigonometric functions on the circle. We use properties and the construction of rational $\overline{\mathcal{P}}$-inner functions to prove a Schwarz lemma for the pentablock.

35 pages. This version includes minor revisions. It has been accepted for publication by the Journal of Geometric Analysis