# Spectral multipliers, R-bounded homomorphisms and analytic diffusion semigroups

Abstract : The thesis is concerned with the smooth functional calculus for operators with spectrum in the positive reals, more specifically spectral multiplier theorems.\\ We start with abstract and optimal functional calculi, that is, homomorphisms $u : C(K) \to B(X).$ If X is a Hilbert space, then the natural operator valued extension $C(K;[u]') \to B(X)$ is again bounded. Using $R$-boundedness, a strengthening of uniform boundedness of operators, we extend this result to general Banach spaces $X$ and apply it to the $H$ infinity calculus and to unconditional bases in $L^p$ spaces.\\ We develop calculi which are associated with sectorial operators. The classical examples are the spectral theorems of Mihlin and H\"{o}rmander giving classes of smooth functions which are Fourier multipliers on $L^p$. These theorems have already been extended to a large class of Laplace type operators. We add a unifying theme using operator theory: we compare the Mihlin and H\"{o}rmander calculus with the boundedness of classical operator families associated with the sectorial operator.\\ For the family of imaginary powers, we give a characterization of their polynomial norm growth in terms of a functional calculus which refines the Mihlin calculus.\\ We study diffusion semigroups acting on a scale of Banach spaces. If this scale is the classical $L^p$ spaces, it is known that the semigroup has an analytic extension on a sector in the complex plane. We give a generalization of this result to non-commutative $L^p$-spaces using the theory of operator spaces.
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Mathématiques [math]. Université de Franche-Comté, 2009. Français
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https://tel.archives-ouvertes.fr/tel-00461310
Contributor : Christoph Kriegler <>
Submitted on : Thursday, March 4, 2010 - 12:56:29 PM
Last modification on : Thursday, February 15, 2018 - 8:48:02 AM
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• HAL Id : tel-00461310, version 1

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Christoph Kriegler. Spectral multipliers, R-bounded homomorphisms and analytic diffusion semigroups. Mathématiques [math]. Université de Franche-Comté, 2009. Français. 〈tel-00461310〉

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