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On functional calculus properties of Ritt operators

Part of: General theory of linear operators

Published online by Cambridge University Press:  26 November 2015

Florence Lancien  and
Christian Le Merdy
Florence Lancien
Affiliation:
Laboratoire de Mathématiques UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France (florence.lancien@univ-fcomte.fr; christian.lemerdy@univ-fcomte.fr)
Christian Le Merdy
Affiliation:
Laboratoire de Mathématiques UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France (florence.lancien@univ-fcomte.fr; christian.lemerdy@univ-fcomte.fr)
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Abstract

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We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T: X → X on some Banach space X with the following property: T has a bounded H∞-functional calculus with respect to the unit disc 𝔻(that is, T is polynomially bounded) but T does not have any bounded H∞-functional calculus with respect to a Stolz domain of 𝔻 with vertex at 1. Also we show that for an R-Ritt operator the unconditional Ritt condition of Kalton and Portal is equivalent to the existence of a bounded H∞-functional calculus with respect to such a Stolz domain.

Keywords

Ritt operatorssectorial operatorsfunctional calculusR-boundedness

MSC classification

Secondary: 47A60: Functional calculus
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 6 , December 2015 , pp. 1239 - 1250
Copyright
Copyright © Royal Society of Edinburgh 2015