The Failure of Rational Dilation on a Triply Connected Domain

Let R denote a domain in C with boundary B. Let X denote the closure of R. An operator T on a complex Hilbert space H has X as a spectral set if σ(T ) ⊂ X and ‖f(T )‖ ≤ ‖f‖R = sup{|f(z)| : z ∈ R} for every rational function f with poles off X. The expression f(T ) may be interpreted either in terms of the Riesz functional calculus, or simply by writing the rational function f as pq−1 for polynomials p and q, in which case f(T ) = p(T )q(T )−1. The operator T has a normal B-dilation if there exists a Hilbert space K containing H and a normal operator N on K so that