Square functions associated with Ritt$_E$ operators

For a subset $E = \{\xi_1, ..., \xi_N\}$ 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 $\alpha$ and for any $x \in X$ $$\Vert{x}\Vert_{T,\alpha} = \lim\limits_{n\rightarrow \infty}\Bigl\Vert{\sum\limits_{k=1}^n k^{\alpha - 1/2} \varepsilon_k \otimes T^{k-1}\prod\limits_{j=1}^N(I-\overline{\xi_j}T)^\alpha(x)}\Bigr\Vert_{{\rm Rad}(X)}$$ 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 $\Vert{x}\Vert_{T,\alpha} \lesssim \Vert{x}\Vert$ hold for both $T$ and $T^*$. We finally prove the equivalence between these square functions.