H∞ calculus and dilatations
[Calcul H∞ et dilatations]
Bulletin de la Société Mathématique de France, Tome 134 (2006) no. 4, pp. 487-508.

Nous donnons une condition nécessaire et suffisante en termes de théorèmes de dilatation pour que le calcul H∞ d’un opérateur sectoriel soit borné. Nous montrons par exemple que, si A engendre un semigroupe C0 analytique borné (Tt) sur un espace UMD, alors le calcul H∞ de A est borné si et seulement si (Tt) admet une dilatation en un groupe borné sur L2([0,1],X). Ceci généralise un résultat de C. Le Merdy sur les espaces de Hilbert. Si X est un espace Lp, on peut choisir un autre espace Lp à la place de L2([0,1],X).

We characterise the boundedness of the H∞ calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if -A generates a bounded analytic C0 semigroup (Tt) on a UMD space, then the H∞ calculus of A is bounded if and only if (Tt) has a dilation to a bounded group on L2([0,1],X). This generalises a Hilbert space result of C.LeMerdy. If X is an Lp space we can choose another Lp space in place of L2([0,1],X).

DOI : 10.24033/bsmf.2520
Classification : 47A60, 47A20, 47D06
Mots-clés : H∞ functional calculus, dilation theorems, spectral operators, square functions, C0 groups, umd spaces 
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Fröhlich, Andreas M.; Weis, Lutz. $H^\infty $ calculus and dilatations. Bulletin de la Société Mathématique de France, Tome 134 (2006) no. 4, pp. 487-508. doi : 10.24033/bsmf.2520. https://www.numdam.org/articles/10.24033/bsmf.2520/

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