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Square functions for Ritt operators on noncommutative $L^p$-spaces

For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_γ)$ where $B_γ$ is a Stolz domain. Moreover, we introduce the `column square functions&#39; $\norm{x}_{T,c,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1}|T^{k-1}(I-T)^α(x)|^2\Big)^{1/2}}_{L^p(M)}$ and the `row square functions&#39; $\norm{x}_{T,r,α}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2α-1} |\Big(T^{k-1}(I-T)^α(x)\Big)^*|^2\Big)^{1/2}}_{L^p(M)}$ for any $α>0$ and any $x\in L^p(M)$. Then, we provide an example of Ritt operator which admits a completely bounded $H^\infty(B_γ)$ functional calculus for some $γ\in \big]0,\fracπ{2}\big[$ such that the square functions $\norm{\cdot}_{T,c,α}$ and $\norm{\cdot}_{T,r,α}$ 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 $H^\infty(B_γ)$ functional calculus for some $γ\in \big]0,\fracπ{2}\big[$ then there exists a positive constant $C$ such that for any $x \in L^p(M)$, there exists $x_1, x_2 \in L^p(M)$ satisfying $x=x_1+x_2$ and $\norm{x_1}_{T,c,α}+\norm{x_2}_{T,r,α}\leq C \norm{x}_{L^p(M)}$. Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative $L^p$-spaces.