Open Access Open Access  Restricted Access Subscription Access

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

Cédric Arhancet


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.

Full Text:




  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library