The failure of rational dilation on a triply connected domain

Abstract

For R R a bounded triply connected domain with boundary consisting of disjoint analytic curves there exists an operator T T on a complex Hilbert space H \mathcal H so that the closure of R R is a spectral set for T T , but T T does not dilate to a normal operator with spectrum in B B , the boundary of R R . There is considerable overlap with the construction of an example on such a domain recently obtained by Agler, Harland and Raphael using numerical computations and work of Agler and Harland.