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A spectral multiplier theorem for non-self-adjoint operators


Author: El Maati Ouhabaz
Journal: Trans. Amer. Math. Soc. 361 (2009), 6567-6582
MSC (2000): Primary 42B15; Secondary 47F05
DOI: https://doi.org/10.1090/S0002-9947-09-04754-0
Published electronically: July 17, 2009
MathSciNet review: 2538605
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Abstract: We prove a spectral multiplier theorem for non-self-adjoint operators. More precisely, we consider non-self-adjoint operators $ A: D(A) \subset L^2 \to L^2$ having numerical range in a sector $ \Sigma(w)$ of angle $ w,$ and whose heat kernel satisfies a Gaussian upper bound. We prove that for every bounded holomorphic function $ f$ on $ \Sigma(w),$ $ f(A)$ acts on $ L^p$ with $ L^p-$norm estimated by the behavior of a finite number of derivatives of $ f$ on the boundary of $ \Sigma(w).$


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Additional Information

El Maati Ouhabaz
Affiliation: Institut de Mathématiques de Bordeaux, CNRS UMR 5251, Equipe d’Analyse et Géométrie, Université Bordeaux 1, 351, Cours de la Libération, 33405 Talence, France
Email: Elmaati.Ouhabaz@math.u-bordeaux1.fr

DOI: https://doi.org/10.1090/S0002-9947-09-04754-0
Received by editor(s): June 28, 2007
Received by editor(s) in revised form: January 30, 2008
Published electronically: July 17, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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