Square functions associated with Ritt${}_E$ operators

Abstract

For a subset $E=\{\hat{I}{¾}_1,...,\hat{I}{¾}_N\}$ of the unit circle $\mathbb{T}$, the notion of Ritt${}_E$ operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in arXiv:2203.05373. 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 $\hat{I}\pm$ and for any $x\in X$ $$\Vert x{\Vert }_{T,\hat{I}\pm }=\underset{n\to \infty }{\lim }\Vert \sum _{k=1}^n{k}^{\hat{I}\pm -1/2}{\epsilon }_k\otimes T^{k-1}\prod _{j=1}^N(I-\overline{\hat{I}{¾}_j}T{)}^{\hat{I}}\pm (x){\Vert }_{\mathrm{Rad}(X)}$$ 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 $\Vert x{\Vert }_{T,\hat{I}\pm }\lesssim \Vert x\Vert$ hold for both $T$ and $T^{\ast }$. We finally prove the equivalence between these square functions.