The numerical range is a $(1+\sqrt{2})$ spectral set

Michel Crouzeix (1) , César Palencia (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LPI - Laboratorio de Procesado de Imagen [Valladolid]

It is shown that the numerical range of a linear operator operator in a Hilbert space is a (complete) $(1{+}\sqrt2)$-spectral set. The proof relies, among other things, in the behavior of the Cauchy transform of the conjugates of holomorphic functions.

Mots clés: Numerical range; Spectral set

Domaines: Analyse fonctionnelle [math.FA]

Michel Crouzeix, César Palencia. The numerical range is a $(1+\sqrt{2})$ spectral set. SIAM Journal on Matrix Analysis and Applications, 2017, 38 (2), pp.649-655. ⟨10.1137/17M1116672⟩. ⟨hal-01455243⟩