Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains

TL;DR

This paper demonstrates that for certain multiply connected domains with specific symmetries, there exist operators with these domains as spectral sets that cannot be dilated to normal operators, providing counterexamples to rational dilation.

Contribution

It constructs explicit counterexamples to rational dilation on symmetric multiply connected domains, expanding understanding of spectral set dilation limitations.

Findings

Counterexamples to rational dilation on symmetric multiply connected domains.

Operators with spectral set R that do not dilate to normal operators.

Domains with hyperelliptic Schottky doubles exhibit these counterexamples.

Abstract

We show that if R is a compact domain in the complex plane with two or more holes and an anticonformal involution onto itself (or equivalently a hyperelliptic Schottky double), then there is an operator T which has R as a spectral set, but does not dilate to a normal operator with spectrum on the boundary of R.