Function theory of the pentablock

Abstract

The pentablock is the set in C 3 P = {(a21,tr A, det A) : A = [aij ] 2 i,j=1 ∈ B 2×2 }, where B 2×2 denotes the open unit ball in the space of 2×2 complex matrices. The closure of P is denoted by P. The sets P and P are polynomially convex and starlike about (0,0,0), but not convex. In this thesis we identify the singular set of P which is SP = {(0, s, p) ∈ P : s 2 = 4p} and show that SP is invariant under the automorphism group Aut P of P and is a complex geodesic in P. We provide a description of rational maps from the unit disc D to P that map the unit circle T to the distinguished boundary bP of P, where bP =

(a, s, p) ∈ C 3 : |s| ≤ 2, |p| = 1, s = sp and |a| = q 1 − 1 4 |s| 2

. These functions are called rational P-inner functions. We establish relations between P-inner functions and Γ-inner functions from D to the symmetrized bidisc Γ. We give a method of constructing rational P-inner functions starting from a rational Γ-inner function. We describe an algorithm to construct rational P-inner functions x = (a, s, p) : D → P of prescribed degree from the zeros of a, s and s 2−4p. We use a result of Agler and Young to construct an interpolating rational P-inner function x : D → P such that x(0) = (0, 0, 0) and x(λ0) = (a0, s0, p0) for suitable points λ0 ∈ D and (a0, s0, p0) ∈ P. We prove a Schwarz lemma for the pentablock.