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