Function theory of the pentablock

Abstract

The set P={(a21, tr A, det A) : A=[aij]_i,j=1 2∈B^2×2} where B^2×2 denotes the open unit ball in the space of 2×2 complex matrices, is called the pentablock. The pentablock is a bounded nonconvex domain in C^3 which arises naturally in connection with a certain problem of μ-synthesis. In this thesis we identify the singular set S_P of P and show that S_P is invariant under the automorphism group Aut P of P and is a complex geodesic in P. We develop a concrete structure theory for the rational maps from the unit disc D to the closed pentablock P¯ that map the unit circle T to the distinguished boundary bP¯ of P¯. Such maps are called rational P¯-inner functions. We give relations between penta-inner functions and inner functions from D to the symmetrized bidisc. We describe the construction of rational penta-inner functions x=(a,s,p):D→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 P¯-inner functions to prove a Schwarz lemma for the pentablock.