H calculus and dilatations
Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, pp. 487-508.

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 C 0 semigroup (T t ) on a UMD space, then the H 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).

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 C 0 analytique borné (T t ) sur un espace UMD, alors le calcul H 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).

DOI: 10.24033/bsmf.2520
Classification: 47A60, 47A20, 47D06
Keywords: $H^\infty $ functional calculus, dilation theorems, spectral operators, square functions, $C_0$ groups, umd spaces
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Fröhlich, Andreas M.; Weis, Lutz. $H^\infty $ calculus and dilatations. Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, pp. 487-508. doi : 10.24033/bsmf.2520. http://archive.numdam.org/articles/10.24033/bsmf.2520/

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