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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.
MSC (2010): Primary 47A60; Secondary 47D06, 47A20, 22D12
DOI: https://doi.org/10.1090/tran/6849
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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Cédric Arhancet
Affiliation: Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email: cedric.arhancet@univ-fcomte.fr

Stephan Fackler
Affiliation: Institute of Applied Analysis, University of Ulm, Helmholtzstrasse 18, 89069 Ulm, Germany
Email: stephan.fackler@uni-ulm.de

Christian Le Merdy
Affiliation: Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
Email: christian.lemerdy@univ-fcomte.fr

DOI: https://doi.org/10.1090/tran/6849
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