Atlas home || Conferences | Abstracts | about Atlas

18th International Conference on Operator Theory
June 27 - July 1, 2000
University of the West
Timisoara, Romania

Organizers
Dumitru Gaspar, Traian Ceausu, Aurelian Craciunescu, Aurelian Gheondea, Radu-Nicolae Gologan, Ciprian Pop, Dan Popovici, Nicolae Suciu, Alexandru Terescenco, Dan Timotin, Flavius Turcu

View Abstracts
Conference Homepage

The spectral Nevanlinna-Pick problem
by
N. J. Young
Newcastle University
Coauthors: Jim Agler (University of California at San Diego)

The H\infty approach to the robust stabilisation of uncertain systems leads to problems which are generalisations of well known problems in function theory. One problem of this type is called \mu-synthesis. It is a broadly formulated interpolation problem. A much-studied special case is the spectral Nevanlinna-Pick problem:

Given points z1, ... , zn in the open unit disc D and k ×k matrices W1, ..., Wn, determine whether there exists an analytic function F : --> Mn such that F(zi) = Wi for 1 <= i <= n and r(F(z)) <= 1 for all z in D.

Here r(.) denotes the spectral radius. I will discuss this problem, particularly for the case of 2 ×2 matrices. Operator-theoretic methods lead to a necessary condition of Pick type for the existence of F; in the case of two interpolation points, this condition is also sufficient. On the way we obtain a novel Schwarz Lemma for the symmetrised bidisc
\Gamma =
def
 
 { (z1+z2, z1 z2) : |z1| <= 1, |z2| <= 1 }.

[]
J. Agler and N. J. Young, `A commutant lifting theorem for a domain in C2 and spectral interpolation', J. Functional Analysis 161 (1999) 452-477.

[]
J. Agler and N. J. Young, `Operators having the symmetrized bidisc as a spectral set', Proc. Edinburgh Math. Soc. 43 (2000)

[]
J. Agler and N. J. Young, `The two-point spectral Nevanlinna-Pick problem', preprint, Newcastle University 1999, http://www.mas.ncl.ac.uk/[ \tilde]nnjy/abstracts/snp.html, to appear in Integral Equations and Operator Theory.

[]
J. Agler and N. J. Young, `A Schwarz Lemma for the symmetrised bidisc', preprint, Newcastle University 1999, http://www.mas.ncl.ac.uk/[ \tilde]nnjy/abstracts/schwarz.html, to appear in Bull London Math. Soc.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-32.