Tests for complete K-spectral sets

Let Φ be a family of functions analytic in some neighborhood of a complex domain Ω, and let T be a Hilbert space operator whose spectrum is contained in Ω‾. Our typical result shows that under some extra conditions, if the closed unit disc is complete K′-spectral for φ(T) for every φ∈Φ, then Ω‾ is complete K-spectral for T for some constant K. In particular, we prove that under a geometric transversality condition, the intersection of finitely many K′-spectral sets for T is again K-spectral for some K≥K′. These theorems generalize and complement results by Mascioni, Stessin, Stampfli, Badea-Beckermann-Crouzeix and others. We also extend to non-convex domains a result by Putinar and Sandberg on the existence of a skew dilation of T to a normal operator with spectrum in ∂Ω. As a key tool, we use the results from our previous paper [11] on traces of analytic uniform algebras.