Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains

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. 0.1. Definitions. LetX be a compact, path connected subset ofC, with interior R, and analytic boundary B composed of n+1 disjoint curves, B0, . . . , Bn, where n ≥ 2. By analytic boundary, we mean that for each boundary curve Bi there is some biholomorphic map φi on a neighbourhood Ui of X which maps Bi to the unit circle T. By convention B0 is the outer boundary. We write Π = B0 × · · · × Bn. We say a Riemann surface Y is hyperelliptic if there is a meromorphic function with two poles on Y (see [FK92]). We say R is symmetric if there exists some anticonformal involution ̟ on Rwith 2n + 2 fixed points on B. We say a domain in C ∪ {∞} (that is, the Riemann sphere S2) is a real slit domain if its complement is a finite union of closed intervals in R ∪ {∞}. We define R(X) ⊆ C(X) as the space of all rational functions that are continuous onX. The definitions of contractivity and complete contractivity are the usual definitions, and can be found in [Pau02]. 0.2. Introduction. A key problem that this paper deals with is the rational dilation conjecture, which is as follows. Conjecture 0.1. If X ⊆ C is a compact domain, T ∈ B(H) is a Hilbert space operator with σ(T) ⊆ X and ∥∥ f (T) ∥∥ ≤ 1 for all f ∈ R(X), then there is some normal operator N ∈ B(K), K ⊇ H, such that σ(N) ⊆ B (= ∂X), and f (T) = PHN|H. A classical result of Sz.-Nagy shows that the rational dilation conjecture holds if X is the unit disc. A generalisation by Berger, Foias and Lebow Date: Submitted on 2nd September 2008. 2000 Mathematics Subject Classification. Primary 47A20; Secondary 47A25, 47A48, 46E22, 30C20, 30H05, 30F10, 30E05.