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

Catalin Badea (1) , Bernhard Beckermann (1) , Michel Crouzeix (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 IRMAR - Institut de Recherche Mathématique de Rennes

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).

Domaines: Théorie spectrale [math.SP]; Analyse fonctionnelle [math.FA]

Catalin Badea, Bernhard Beckermann, Michel Crouzeix. K-spectral sets and intersections of disks of the Riemann sphere. 2007. ⟨hal-00905874⟩