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