Square functions associated with RittE operators

In a finite subset $E=\{{\xi }_1,\dots ,{\xi }_N\}$ of the torus $T=\{z\in ℂ:\left|z\right|=1\}$, the notion of Ritt${}_E$ operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in Bouabdillah and Le Merdy (2024).

In this paper, we define a quadratic functional calculus for a Ritt${}_E$ operator on $E_r$, by a decomposition of type Franks–McIntosh. We show that with some hypothesis on the cotype of $X$, this notion is equivalent to the existence of a bounded functional calculus on $E_r$.

We define for a Ritt${}_E$ operator on a Banach space $X$ and for any positive real number $\alpha$ and for any $x\in X$ ${\Vert x\Vert }_{T,\alpha }=\underset{n\to \infty }{lim}\Vert \sum _{k=1}^n{k}^{\alpha -1/2}{\varepsilon }_k\otimes T^{k-1}\prod _{j=1}^N{(I-\overline{{\xi }_j}T)}^{\alpha }{\left(x\right)\Vert }_{\mathrm{Rad}\left(X\right)}.$ We show that, under the condition of finite cotype of $X$, a Ritt${}_E$ operator admits a quadratic functional calculus if and only if the estimates ${\Vert x\Vert }_{T,\alpha }\lesssim \Vert x\Vert$ hold for both $T$ and $T^{\ast }$. We finally prove the equivalence between these square functions.