calculus and dilatations
Andreas M. Fröhlich; Lutz Weis
Bulletin de la Société Mathématique de France (2006)
- Volume: 134, Issue: 4, page 487-508
- ISSN: 0037-9484
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topFröhlich, Andreas M., and Weis, Lutz. "$H^\infty $ calculus and dilatations." Bulletin de la Société Mathématique de France 134.4 (2006): 487-508. <http://eudml.org/doc/272330>.
@article{Fröhlich2006,
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 $L^2([0,1],X)$.},
author = {Fröhlich, Andreas M., Weis, Lutz},
journal = {Bulletin de la Société Mathématique de France},
keywords = {$H^\infty $ functional calculus; dilation theorems; spectral operators; square functions; $C_0$ groups; umd spaces},
language = {eng},
number = {4},
pages = {487-508},
publisher = {Société mathématique de France},
title = {$H^\infty $ calculus and dilatations},
url = {http://eudml.org/doc/272330},
volume = {134},
year = {2006},
}
TY - JOUR
AU - Fröhlich, Andreas M.
AU - Weis, Lutz
TI - $H^\infty $ calculus and dilatations
JO - Bulletin de la Société Mathématique de France
PY - 2006
PB - Société mathématique de France
VL - 134
IS - 4
SP - 487
EP - 508
AB - 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 $L^2([0,1],X)$.
LA - eng
KW - $H^\infty $ functional calculus; dilation theorems; spectral operators; square functions; $C_0$ groups; umd spaces
UR - http://eudml.org/doc/272330
ER -
References
top- [1] P. Auscher, S. Hofmann, A. McIntosh & P. Tchamitchian – « The Kato square root problem for higher order elliptic operators and systems on », J. Evol. Equ.1 (2001), p. 361–385. Zbl1019.35029MR1877264
- [2] P. Auscher & P. Tchamitchian – Square root problem for divergence operators and related topics, Astérisque, vol. 249, Soc. Math. France, 1998. Zbl0909.35001MR1651262
- [3] D. L. Burkholder – « Martingale transforms and the geometry of Banach spaces », Springer Lecture Notes in Math., vol. 860, 1981, p. 35–50. Zbl0471.60012MR647954
- [4] M. Cowling, I. Doust, A. McIntosch & A. Yagi – « Banach space operators with a bounded functional calculus », J. Aust. Math. Soc. (Ser. A) 60 (1996), p. 51–89. Zbl0853.47010MR1364554
- [5] R. Denk, M. Hieber & J. Prüss – -boundedness, Fourier multipliers and problems of elliptic and parabolic type, Memoirs Amer. Math. Soc., vol. 788, 2003. Zbl1274.35002
- [6] N. Dunford & J. Schwartz – Linear Operators III — Spectral Operators, John Wiley & Sons Inc., 1972. Zbl0283.47002MR1009164
- [7] K.-J. Engel & R. Nagel – One-Parameter Semigroups for Linear Evolution Equations, Springer, 1999. Zbl0952.47036MR1721989
- [8] A. M. Fröhlich – « -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 -spaces via the transference principle », Adv. Differ. Equ.3 (1998), p. 847–872. Zbl0956.47008MR1659281
- [10] N. J. Kalton – « A remark on sectorial operators with an -calculus, Trends in Banach spaces and operator theory », Contemp. Math.321 (2003), p. 91–99. Zbl1058.47011MR1978810
- [11] N. J. Kalton, P. C. Kunstmann & L. Weis – « Perturbation and interpolation theorems for the calculus with applications to differential operators », submitted. Zbl1111.47020
- [12] N. J. Kalton & L. Weis – « Euclidean structures and their applications to spectral theory », in preparation.
- [13] —, « The -functional calculus and square function estimates », in preparation.
- [14] —, « The -calculus and sums of closed operators », Math. Ann.321 (2001), p. 319–345. Zbl0992.47005MR1866491
- [15] P. Kunstmann & L. Weis – « Maximal regularity for parabolic equations, Fourier multiplier theorems, and functional calculus », in preparation. Zbl1097.47041
- [16] G. Lancien & C. Le Merdy – « A generalized functional calculus for operators on subspaces of and application to maximal regularity », Ill. J. Math.42 (1998), p. 470–480. Zbl0906.47015MR1631256
- [17] C. Le Merdy – « -functional calculus and applications to maximal regularity », Publ. Math. UFR Sci. Tech. Besançon16 (1998), p. 41–77. Zbl0949.47012MR1768324
- [18] —, « The similarity problem for bounded analytic semigroups on Hilbert space », Semigroup Forum56 (1998), p. 205–224. Zbl0998.47028MR1490293
- [19] A. McIntosh – « Operators which have an functional calculus », Proc. Cent. Math. Anal. Aust. Natl. Univ.14 (1986), p. 210–231. Zbl0634.47016MR912940
- [20] G. Pisier – The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, Cambridge University Press, 1989. Zbl0698.46008MR1036275
- [21] J. van Neerven – The Adjoint of a Semigroup of Linear Operators, Springer Lecture Notes in Math., vol. 1529, 1993. Zbl0780.47026MR1222650
- [22] L. Weis – « The holomorphic functional calculus for sectorial operators », submitted.
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