SPECTRAL SETS AND OPERATOR RADII

Catalin Badea (1, 2) , Michel Crouzeix (3) , Hubert Klaja (4)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 Université de Lille
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 Centrale Lille

We study different operator radii of homomorphisms from an operator algebra into B(H) and show that these can be computed explicitly in terms of the usual norm. As an application, we show that if Ω is a K-spectral set for a Hilbert space operator, then it is a M-numerical radius set, where M = 1/2 (K + K ^{−1}). This is a counterpart of a recent result of Davidson, Paulsen and Woerdeman. More general results for operator radii associated with the class of operators having ρ-dilations in the sense of Sz.-Nagy and Foias are given. A version of a result of Drury concerning the joint numerical radius of non-commuting n-tuples of operators is also obtained.

Domaines: Analyse fonctionnelle [math.FA]

Catalin Badea, Michel Crouzeix, Hubert Klaja. SPECTRAL SETS AND OPERATOR RADII. Bulletin of the London Mathematical Society, 2018, 50 (6), pp.986-996. ⟨10.1112/blms.12191⟩. ⟨hal-01821886⟩