Tests for complete $K$-spectral sets

Abstract

Let $\Phi$ be a family of functions analytic in some neighborhood of a complex domain $\Omega$, and let $T$ be a Hilbert space operator whose spectrum is contained in $\overline{\Omega }$. Our typical result shows that under some extra conditions, if the closed unit disc is complete $K^{\prime }$-spectral for $\varphi (T)$ for every $\varphi \in \Phi$, then $\overline{\Omega }$ 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 $K^{\prime }$-spectral sets for $T$ is again $K$-spectral for some $K\ge K^{\prime }$. 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 $\partial \Omega$. As a key tool, we use the results from our previous paper on traces of analytic uniform algebras.

33 pages, 1 figure