Dilation of Ritt operators on L ^p-spaces

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

$${\left\| x \right\|_{T,\alpha }} = {\left\| {{{\left( {\sum\limits_{k = 1}^\infty {{k^{2\alpha - 1}}} {{\left| {{T^{k - 1}}{{(I - T)}^\alpha }x} \right|}^2}} \right)}^{\frac{1}{2}}}} \right\|_{{L^p}}}$$

. 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.