1 research outputs found
A Schwarz lemma for the pentablock
In this paper we prove a Schwarz lemma for the pentablock. The set where denotes the open
unit ball in the space of complex matrices, is called the
pentablock. The pentablock is a bounded nonconvex domain in which
arises naturally in connection with a certain problem of -synthesis. We
develop a concrete structure theory for the rational maps from the unit disc
to the closed pentablock that map the unit
circle to the distinguished boundary
of . Such maps are called rational
-inner functions. We give relations between
penta-inner functions and inner functions from to the symmetrized
bidisc. We describe the construction of rational penta-inner functions of prescribed degree from
the zeroes of and . The proof of this theorem is constructive:
it gives an algorithm for the construction of a family of such functions
subject to the computation of Fej\'er-Riesz factorizations of certain
non-negative trigonometric functions on the circle. We use properties and the
construction of rational -inner functions to prove a
Schwarz lemma for the pentablock.Comment: 35 pages. This version includes minor revisions. It has been accepted
for publication by the Journal of Geometric Analysi