Newcastle University eTheses >
Newcastle University >
Faculty of Science, Agriculture and Engineering >
School of Mathematics and Statistics >
Please use this identifier to cite or link to this item:
http://hdl.handle.net/10443/1264

Title:  Operator and functional theory of the symmetrized polydisc 
Authors:  Ogle, David John 
Issue Date:  1999 
Publisher:  Newcastle University 
Abstract:  We establish necessary conditions, in the form of the positivity of Pickmatrices, for the
existence of a solution to the spectral NevanlinnaPick problem:
Let k and n be natural numbers. Choose n distinct points zj in the open unit disc, D, and
n matrices Wj in Mk(C), the space of complex k × k matrices. Does there exist an analytic
function : D ! Mk(C) such that
(zj ) = Wj
for j = 1, ...., n and
( (z)) D
for all z 2 D?
We approach this problem from an operator theoretic perspective. We restate the problem
as an interpolation problem on the symmetrized polydisc k,
k = {(c1(z), . . . , ck(z))  z 2 D} Ck
where cj(z) is the jth elementary symmetric polynomial in the components of z. We establish
necessary conditions for a ktuple of commuting operators to have k as a complete spectral set.
We then derive necessary conditions for the existence of a solution of the spectral Nevanlinna
Pick problem.
The final chapter of this thesis gives an application of our results to complex geometry. We
establish an upper bound for the Caratheodory distance on int k. 
Description:  Phd Thesis 
URI:  http://hdl.handle.net/10443/1264 
Appears in Collections:  School of Mathematics and Statistics

Items in eTheses are protected by copyright, with all rights reserved, unless otherwise indicated.
