ePrints

Rational dilation problems associated with constrained algebras

Lookup NU author(s): Dr Michael DritschelORCiD, Batzorig Undrakh

Downloads

• Accepted version [.pdf]

Licence

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).

Abstract

© 2018 Elsevier Inc. A set Ω is a spectral set for an operator T if the spectrum of T is contained in Ω, and von Neumann's inequality holds for T with respect to the algebra R(Ω) of rational functions with poles off of Ω‾. It is a complete spectral set if for all r∈N, the same is true for Mr(C)⊗R(Ω). The rational dilation problem asks, if Ω is a spectral set for T, is it a complete spectral set for T? There are natural multivariable versions of this. There are a few cases where rational dilation is known to hold (eg, over the disk and bidisk), and some where it is known to fail, for example over the Neil parabola, a distinguished variety in the bidisk. The Neil parabola is naturally associated to a constrained subalgebra of the disk algebra C+z2A(D). Here it is shown that such a result is generic for a large class of varieties associated to constrained algebras. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler–Pick interpolation theorem is given and it is proved that there exist Kaijser–Varopoulos style examples of non-contractive unital representations where the generators are contractions.

Publication metadata

Author(s): Dritschel MA, Undrakh B

Publication type: Article

Publication status: Published

Journal: Journal of Mathematical Analysis and Applications

Year: 2018

Volume: 467

Issue: 1

Pages: 95-131

Print publication date: 01/11/2018

Online publication date: 26/06/2018

Acceptance date: 02/04/2018

Date deposited: 06/08/2018

ISSN (print): 0022-247X

ISSN (electronic): 1096-0813

Publisher: Academic Press Inc.

URL: https://doi.org/10.1016/j.jmaa.2018.06.057

DOI: 10.1016/j.jmaa.2018.06.057

Altmetrics

Share