Square Functions for Ritt Operators on Noncommutative $L^p$-Spaces

Abstract

For any Ritt operator T acting on a noncommutative L p -space, we define the notion of completely bounded functional calculus H ∞ (Bγ ) where Bγ is a Stolz domain. Moreover, we introduce the 'column square functions'and the 'row square functions'for any α > 0 and any x ∈ L p (M ). Then, we provide an example of Ritt operator which admits a completely bounded H ∞ (Bγ ) functional calculus for some γ ∈ 0, π 2 such that the square functions · p,T,c,α and · p,T,r,α are not equivalent. Moreover, assuming 1 < p < 2 and α > 0, we prove that if Ran(I − T ) is dense and T admits a completely bounded H ∞ (Bγ ) functional calculus for some γ ∈ 0, π 2 then there exists a positive constant C such that for any. Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative L p -spaces. IntroductionLet M be a semifinite von Neumann algebra equipped with a normal semifinite faithful trace. For anyConsider the following 'square function' (1.1)

Isometric dilations and 𝐻^{∞} calculus for bounded analytic semigroups and Ritt operators

“…We describe their holomorphic functional calculus and present some of their main features. There is now a vast literature on this topic in which details and complements can be found, see in particular [ALM14], [Arh13], [Blu01b], [Blu01a], [LM14], [Lyu99], [NZ99], [Nev93], [Vit05] and the references therein.…”

Dilation of Ritt operators on L p -spaces

“…We refer the reader to [3] for examples of operators with a noncommutative strict dilation, and to the paper [4] for more about square functions associated to Ritt operators on noncommutative L p -spaces.…”

Maximal ergodic inequalities for some positive operators on noncommutative Lp$L_p$‐spaces

“…Therefore, by Theorem 1.3, we have $$$$\begin{equation*} {\left\Vert {\left(\frac{1}{ n+1}\sum _{k=0}^n(U \otimes I_{L_p (\mathcal {M} ) } )^k x \right)}_{n\geqslant 0}\right\Vert} _p\leqslant C_p\Vert x\Vert _p,\quad \forall \, x\in L_p(L_\infty (\Omega ^{\prime })\overline{\otimes } \mathcal {M}). \end{equation*}$$$$Note that $$T \otimes I_{L_p (\mathcal {M} ) }$$, $$J \otimes I_{L_p (\mathcal {M} ) }$$ and $$Q \otimes I_{L_p (\mathcal {M} )}$$ are again positive contractions (see, e.g., [9, Theorem 2.17 and Proposition 2.21] and [42]). Thus, the proof is complete.$$\Box$$…”