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. [ABSTRACT FROM AUTHOR]