2015/10/28 by Dritschel, Michael A., Estévez, Daniel, Yakubovich, Dmitry · 1 citation
#30H50 #46J15 #47A12 #47A20 (Secondary) #47A25 (Primary) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1510.08350
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-Beckerman-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 on traces of analytic uniform algebras.