Abstract
For any Ritt operator T: L p(Ω) → L p(Ω), for any positive real number α, and for any x ∈ L p(Ω), we consider
. We show that if T is actually an R-Ritt operator, then the square functions \({\left\| {} \right\|_{T,\alpha }}\) are pairwise equivalent. Then we show that T and its adjoint T*: L p′(Ω) → L p′(Ω) both satisfy uniform estimates \({\left\| x \right\|_{T,1}} \leqslant {\left\| x \right\|_{{L^p}}}\) and \({\left\| y \right\|_{T*,1}} \leqslant {\left\| y \right\|_{{L^{p'}}}}\) for x ∈ L p(Ω) and y ∈ L p′(Ω) if and only if T is R-Ritt and admits a dilation in the following sense: there exist a measure space \(\tilde \Omega \), an isomorphism \(U:{L^p}\tilde \Omega \to {L^p}\tilde \Omega \) such that {U n: n ∈ ℤ} is bounded, as well as two bounded maps \({L^p}(\Omega )\buildrel J \over \longrightarrow {L^p}(\tilde \Omega )\buildrel Q \over \longrightarrow {L^p}(\Omega )\) such that T n = QU n J for any n ≥ 0. We also investigate functional calculus properties of Ritt operators and analogs of the above results on noncommutative L p-spaces.
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Both authors are supported by the research program ANR 2011 BS01 008 01.
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Arhancet, C., Le Merdy, C. Dilation of Ritt operators on L p-spaces. Isr. J. Math. 201, 373–414 (2014). https://doi.org/10.1007/s11856-014-1030-6
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DOI: https://doi.org/10.1007/s11856-014-1030-6