Intersections of several disks of the Riemann sphere as K-spectral sets

Catalin Badea; Bernhard Beckermann; Michel Crouzeix

paper

Intersections of several disks of the Riemann sphere as K-spectral sets

arXiv:0807.3136

Abstract

We prove that if $n$ closed disks $D_{1}, D_{2}, ..., D_{n}$ , of the Riemann sphere are spectral sets for a bounded linear operator $A$ on a Hilbert space, then their intersection $D_{1} \cap D_{2} ... \cap D_{n}$ is a complete $K$ -spectral set for $A$ , with $K \leq n + n (n - 1) / 3$ . When $n = 2$ and the intersection $X_{1} \cap X_{2}$ is an annulus, this result gives a positive answer to a question of A.L. Shields (1974).

arXiv comment4 figures, a remark suggested by Vern Paulsen was added