K-spectral sets and intersections of disks of the Riemann sphere

TL;DR

This paper establishes a new spectral set intersection property for bounded linear operators on Hilbert spaces, specifically when the sets are disks on the Riemann sphere, extending classical spectral set theory.

Contribution

It proves that the intersection of two spectral disks is a complete spectral set with a specific constant, answering a question posed by Shields in 1974.

Findings

Intersection of spectral disks is a complete spectral set with constant 2+2/√3.

Results apply to intersections forming annuli, confirming Shields' conjecture.

Provides new bounds for spectral sets on the Riemann sphere.

Abstract

We prove that if two closed disks X_1 and X_2 of the Riemann sphere are spectral sets for a bounded linear operator A on a Hilbert space, then the intersection X_1\cap X_2 is a complete (2+2/\sqrt{3})-spectral set for A. When the intersection of X_1 and X_2 is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).