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 $\mathit{\text{completely}}$ bounded functional calculus $H^{\mathrm{\infty }}(B_{\gamma })$ where $B_{\gamma }$ is a Stolz domain. Moreover, we introduce the column square functions' $\norm{x}_{T,c,\alpha}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2\alpha-1}|T^{k-1}(I-T)^{\alpha}(x)|^2\Big)^{1/2}}_{L^p(M)}$ and the row square functions’ $\norm{x}{T,r,\alpha}=\Bnorm{\Big(\sum{k=1}^{+\infty}k^{2\alpha-1} |\Big(T^{k-1}(I-T)^{\alpha}(x)\Big)^*|^2\Big)^{1/2}}{L^p(M)}$forany$\alpha>0$andany$x\in L^p(M)$. Then, we provide an example of Ritt operator which admits a completely bounded $H^{\mathrm{\infty }}(B_{\gamma })$ functional calculus for some $\gamma \in ]0,\frac{\pi }{2}[$ such that the square functions $\text{norm}{\cdot }_{T,c,\alpha }$ and $\text{norm}{\cdot }_{T,r,\alpha }$ are not equivalent. Moreover, assuming $1<p<2$ and $\alpha >0$, we prove that if $\text{Ran}(I-T)$ is dense and $T$ admits a completely bounded $H^{\mathrm{\infty }}(B_{\gamma })$ functional calculus for some $\gamma \in ]0,\frac{\pi }{2}[$ 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$\norm{x_1}{T,c,\alpha}+\norm{x_2}{T,r,\alpha}\leq C \norm{x}{L^p(M)}$. Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative $L^p$-spaces.