H calculus and dilatations
Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, p. 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 : https://doi.org/10.24033/bsmf.2520
Classification:  47A60,  47A20,  47D06
Keywords: H functional calculus, dilation theorems, spectral operators, square functions, C 0 groups, umd spaces
@article{BSMF_2006__134_4_487_0,
     author = {Fr\"ohlich, Andreas M. and Weis, Lutz},
     title = {$H^\infty $ calculus and dilatations},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {134},
     number = {4},
     year = {2006},
     pages = {487-508},
     doi = {10.24033/bsmf.2520},
     zbl = {1168.47015},
     mrnumber = {2364942},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2006__134_4_487_0}
}
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://www.numdam.org/item/BSMF_2006__134_4_487_0/

[1] P. Auscher, S. Hofmann, A. Mcintosh & P. Tchamitchian - « The Kato square root problem for higher order elliptic operators and systems on n », J. Evol. Equ. 1 (2001), p. 361-385. | MR 1877264 | Zbl 1019.35029

[2] P. Auscher & P. Tchamitchian - Square root problem for divergence operators and related topics, Astérisque, vol. 249, Soc. Math. France, 1998. | MR 1651262 | Zbl 0909.35001

[3] D. L. Burkholder - « Martingale transforms and the geometry of Banach spaces », Springer Lecture Notes in Math., vol. 860, 1981, p. 35-50. | MR 647954 | Zbl 0471.60012

[4] M. Cowling, I. Doust, A. Mcintosch & A. Yagi - « Banach space operators with a bounded H functional calculus », J. Aust. Math. Soc. (Ser. A) 60 (1996), p. 51-89. | MR 1364554 | Zbl 0853.47010

[5] R. Denk, M. Hieber & J. Prüss - -boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs Amer. Math. Soc., vol. 788, 2003. | Zbl 1274.35002

[6] N. Dunford & J. Schwartz - Linear Operators III - Spectral Operators, John Wiley & Sons Inc., 1972. | MR 1009164 | Zbl 0283.47002

[7] K.-J. Engel & R. Nagel - One-Parameter Semigroups for Linear Evolution Equations, Springer, 1999. | MR 1721989 | Zbl 0952.47036

[8] A. M. Fröhlich - « H -Kalkül und Dilatationen », Thèse, University of Karlsruhe, 2003, http://www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=2003/ mathematik/8.

[9] M. Hieber & J. Prüss - « Functional calculi for linear operators in vector-valued L p -spaces via the transference principle », Adv. Differ. Equ. 3 (1998), p. 847-872. | MR 1659281 | Zbl 0956.47008

[10] N. J. Kalton - « A remark on sectorial operators with an H -calculus, Trends in Banach spaces and operator theory », Contemp. Math. 321 (2003), p. 91-99. | MR 1978810 | Zbl 1058.47011

[11] N. J. Kalton, P. C. Kunstmann & L. Weis - « Perturbation and interpolation theorems for the H calculus with applications to differential operators », submitted. | Zbl 1111.47020

[12] N. J. Kalton & L. Weis - « Euclidean structures and their applications to spectral theory », in preparation.

[13] -, « The H -functional calculus and square function estimates », in preparation.

[14] -, « The H -calculus and sums of closed operators », Math. Ann. 321 (2001), p. 319-345. | MR 1866491 | Zbl 0992.47005

[15] P. Kunstmann & L. Weis - « Maximal regularity for parabolic equations, Fourier multiplier theorems, and H functional calculus », in preparation. | Zbl 1097.47041

[16] G. Lancien & C. Le Merdy - « A generalized H functional calculus for operators on subspaces of L p and application to maximal regularity », Ill. J. Math. 42 (1998), p. 470-480. | MR 1631256 | Zbl 0906.47015

[17] C. Le Merdy - « H -functional calculus and applications to maximal regularity », Publ. Math. UFR Sci. Tech. Besançon 16 (1998), p. 41-77. | MR 1768324 | Zbl 0949.47012

[18] -, « The similarity problem for bounded analytic semigroups on Hilbert space », Semigroup Forum 56 (1998), p. 205-224. | MR 1490293 | Zbl 0998.47028

[19] A. Mcintosh - « Operators which have an H functional calculus », Proc. Cent. Math. Anal. Aust. Natl. Univ. 14 (1986), p. 210-231. | MR 912940 | Zbl 0634.47016

[20] G. Pisier - The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge University Press, 1989. | MR 1036275 | Zbl 0698.46008

[21] J. Van Neerven - The Adjoint of a Semigroup of Linear Operators, Springer Lecture Notes in Math., vol. 1529, 1993. | MR 1222650 | Zbl 0780.47026

[22] L. Weis - « The H holomorphic functional calculus for sectorial operators », submitted.