$H^\infty $ calculus and dilatations

[Calcul $H^\infty $ et dilatations]

Fröhlich, Andreas M. ; Weis, Lutz

Bulletin de la Société Mathématique de France, Tome 134 (2006), p. 487-508 / Harvested from Numdam

Résumé

Nous donnons une condition nécessaire et suffisante en termes de théorèmes de dilatation pour que le calcul $H^{\infty }$ d’un opérateur sectoriel soit borné. Nous montrons par exemple que, si $A$ engendre un semigroupe $C_0$ analytique borné $(T_t)$ sur un espace UMD, alors le calcul $H^{\infty }$ de $A$ est borné si et seulement si $(T_t)$ admet une dilatation en un groupe borné sur $L_2([0,1], X)$. Ceci généralise un résultat de C. Le Merdy sur les espaces de Hilbert. Si $X$ est un espace $L_p$, on peut choisir un autre espace $L_p$ à la place de $L_2([0,1], X)$.

Abstract

We characterise the boundedness of the $H^\infty $ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if $-A$ generates a bounded analytic $C_0$ semigroup $(T_t)$ on a UMD space, then the $H^\infty $ calculus of $A$ is bounded if and only if $(T_t)$ has a dilation to a bounded group on $L^2([0,1],X)$. This generalises a Hilbert space result of C.LeMerdy. If $X$ is an $L^p$ space we can choose another $L^p$ space in place of $L^2([0,1],X)$.

Détail

Publié le : 2006-01-01

MR 2364942 | Zbl 1168.47015

DOI : https://doi.org/10.24033/bsmf.2520
Classification:  47A60,  47A20,  47D06

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