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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Isometric dilations and $H^\infty$ calculus for bounded analytic semigroups and Ritt operators
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by Cédric Arhancet, Stephan Fackler and Christian Le Merdy PDF
Trans. Amer. Math. Soc. 369 (2017), 6899-6933 Request permission

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 $(R_t)_{t\geqslant 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t\geqslant 0$. 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
  • Email: cedric.arhancet@univ-fcomte.fr
  • Stephan Fackler
  • Affiliation: Institute of Applied Analysis, University of Ulm, Helmholtzstrasse 18, 89069 Ulm, Germany
  • MR Author ID: 1008455
  • Email: stephan.fackler@uni-ulm.de
  • Christian Le Merdy
  • Affiliation: Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté, 25030 Besançon Cedex, France
  • MR Author ID: 308170
  • Email: christian.lemerdy@univ-fcomte.fr
  • Received by editor(s): April 28, 2015
  • Received by editor(s) in revised form: September 28, 2015
  • Published electronically: March 17, 2017
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 6899-6933
  • MSC (2010): Primary 47A60; Secondary 47D06, 47A20, 22D12
  • DOI: https://doi.org/10.1090/tran/6849
  • MathSciNet review: 3683097