$H^{\infty }$ calculus and dilatations

Fröhlich, Andreas M.; Weis, Lutz

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 $H^{\infty }$ d’un opérateur sectoriel soit borné. Nous montrons par exemple que, si $A$ engendre un semigroupe $C_0$ analytique borné $\left(T_t\right)$ sur un espace UMD, alors le calcul $H^{\infty }$ de $A$ est borné si et seulement si $\left(T_t\right)$ 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 $L_2([0,1],X)$.

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 $\left(T_t\right)$ on a UMD space, then the $H^{\infty }$ calculus of $A$ is bounded if and only if $\left(T_t\right)$ 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 $L^2([0,1],X)$.

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

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

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

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

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

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

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

[8] A. M. Fröhlich - « $H^{\infty }$-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 | Zbl

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

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

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

[13] -, « The $H^{\infty }$-functional calculus and square function estimates », in preparation.

[14] -, « The $H^{\infty }$-calculus and sums of closed operators », Math. Ann. 321 (2001), p. 319-345. | MR | Zbl

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

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

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

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

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

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

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

[22] L. Weis - « The $H^{\infty }$ holomorphic functional calculus for sectorial operators », submitted.