Isometric dilations and 𝐻^{∞} calculus for bounded analytic semigroups and Ritt operators

Arhancet, Cédric; Fackler, Stephan; Le Merdy, Christian

doi:10.1090/tran/6849

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
DOI: https://doi.org/10.1090/tran/6849
Published electronically: March 17, 2017
MathSciNet review: 3683097
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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 $(R_t)_{t\geqslant 0}$ 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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