Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny

Norm convergence of some power series of operators in

L^{p}

with applications in ergodic theory

Christophe Cuny

Studia Mathematica, Tome 196 (2010), p. 1-29 / Harvested from The Polish Digital Mathematics Library

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Résumé

Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie.

Publié le : 2010-01-01

Zbl 1207.47033

EUDML-ID : urn:eudml:doc:285679

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     title = {Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory},
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     year = {2010},
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Christophe Cuny. Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory. Studia Mathematica, Tome 196 (2010) pp. 1-29. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm200-1-1/