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

TL;DR

This paper develops a theory of square functions for Ritt operators on noncommutative L^p-spaces, introduces a bounded functional calculus, and explores the relationships between different square functions, providing examples and conditions for their equivalence.

Contribution

It introduces new notions of column and row square functions for Ritt operators on noncommutative L^p-spaces and analyzes their properties and relationships, including examples and conditions for bounded functional calculus.

Findings

Existence of Ritt operators with non-equivalent square functions despite bounded functional calculus.

Under certain conditions, decomposition of elements with controlled square function norms.

Application of results to selfadjoint Markov maps on noncommutative L^p-spaces.

Abstract

For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_\gamma)$ 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' $\norm{x}_{T,r,\alpha}=\Bnorm{\Big(\sum_{k=1}^{+\infty}k^{2\alpha-1} |\Big(T^{k-1}(I-T)^{\alpha}(x)\Big)^*|^2\Big)^{1/2}}_{L^p(M)}$ 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 $\gamma \in \big]0,\frac{\pi}{2}\big[$ such that the square functions $\norm{\cdot}_{T,c,\alpha}$ and $\norm{\cdot}_{T,r,\alpha}$ are not equivalent. Moreover, assuming $1<p<2$ and $\alpha>0$, we prove that if $\Ran (I-T)$ is dense and $T$ admits a completely bounded $H^\infty(B_\gamma)$ functional calculus for some $\gamma \in \big]0,\frac{\pi}{2}\big[$ 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 $\norm{x_1}_{T,c,\alpha}+\norm{x_2}_{T,r,\alpha}\leq C \norm{x}_{L^p(M)}$. Finally, we observe that this result applies to a suitable class of selfadjoint Markov maps on noncommutative $L^p$-spaces.