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The complex geometry of a domain related to μ-synthesis

Lookup NU author(s): Professor Jim Agler, Dr Zinaida Lykova, Professor Nicholas Young



This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).


We establish the basic complex geometry and function theory of the pentablock P,which is the bounded domainP = {(a 21 , tr A, det A) : A = [a ij ] 2 i,j=1 ∈ B}where B denotes the open unit ball in the space of 2 × 2 complex matrices. Weprove several characterisations of the domain. We show that P arises naturallyin connection with a certain robust stabilisation problem in control theory, theproblem of μ-synthesis. We describe the distinguished boundary of P and exhibita 4-parameter group of automorphisms of P. We demonstrate connections betweenthe function theories of P and B. We show that P is polynomially convex andstarlike, and we show that the real pentablock P ∩ R 3 is a convex set bounded byfive faces, three of them flat and two curved.

Publication metadata

Author(s): Agler J, Lykova Z, Young NJ

Publication type: Article

Publication status: Published

Journal: Journal of Mathematical Analysis and Applications

Year: 2015

Volume: 422

Issue: 1

Pages: 508-543

Print publication date: 01/02/2015

Online publication date: 01/10/2014

ISSN (print): 0022-247X

ISSN (electronic): 1096-0813

Publisher: Elsevier


DOI: 10.1016/j.jmaa.2014.08.051


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