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Isometric dilations and $ H^\infty$ calculus for bounded analytic semigroups and Ritt operators

Authors: Cédric Arhancet, Stephan Fackler and Christian Le Merdy
Journal: Trans. Amer. Math. Soc. 369 (2017), 6899-6933
MSC (2010): Primary 47A60; Secondary 47D06, 47A20, 22D12
Published electronically: March 17, 2017
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Abstract: We show that any bounded analytic semigroup on $ L^p$ (with $ 1<p<\infty $) whose negative generator admits a bounded $ H^{\infty }(\Sigma _\theta )$ functional calculus for some $ \theta \in (0,\frac {\pi }{2})$ can be dilated into a bounded analytic semigroup $ (R_t)_{t\geq 0}$ on a bigger $ L^p$-space in such a way that $ R_t$ is a positive contraction for any $ t\geq 0$. We also establish a discrete analogue for Ritt operators and consider the case when $ L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.

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Additional Information

Cédric Arhancet
Affiliation: Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France

Stephan Fackler
Affiliation: Institute of Applied Analysis, University of Ulm, Helmholtzstrasse 18, 89069 Ulm, Germany

Christian Le Merdy
Affiliation: Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France

Keywords: Dilation, Ritt operator, sectorial operator, group representation, functional calculus, semigroup, amenable group
Received by editor(s): April 28, 2015
Received by editor(s) in revised form: September 28, 2015
Published electronically: March 17, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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