A Schwarz Lemma for the Pentablock

Abstract

In this paper, we prove a Schwarz lemma for the pentablock. The pentablock mathcal {P} is defined by P={(a21,trA,detA):A=[aij]i,j=12∈B2×2}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathcal {P}=\\{(a_{21}, {\ ext {tr}}A, \\det A) : A=[a_{ij}]_{i,j=1}^2 \\in \\mathbb {B}^{2\ imes 2}\\} \\end{aligned}$$\\end{document}where mathbb {B}^{2times 2} denotes the open unit ball in the space of 2times 2 complex matrices. The pentablock is a bounded non-convex 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 boverline{mathcal {P}} of overline{mathcal {P}}. Such maps are called rational {overline{mathcal {P}}}-inner functions. We give relations between {overline{mathcal {P}}}-inner functions and inner functions from mathbb {D} to the symmetrized bidisc. We describe the construction of rational {overline{mathcal {P}}}-inner functions x = (a, s, p) : mathbb {D} rightarrow 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.