Tests for complete $K$-spectral sets

TL;DR

This paper establishes conditions under which spectral properties of operators on complex domains imply similar properties for the domain itself, generalizing previous results and extending to non-convex domains.

Contribution

It introduces new criteria for when the spectral set of an operator implies the spectral set of the domain, including intersection and dilation results, expanding the theory of spectral sets.

Findings

Finiteness of spectral set intersections under geometric conditions

Extension of spectral set results to non-convex domains

Existence of skew dilations to normal operators with boundary spectrum

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 $K'$-spectral for $\phi(T)$ for every $\phi\in \Phi$, 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 $K'$-spectral sets for $T$ is again $K$-spectral for some $K\ge K'$. 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 $\partial\Omega$. As a key tool, we use the results from our previous paper on traces of analytic uniform algebras.