H 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

Abstract

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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 ) .

How to cite

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Fröhlich, 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

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