$H^\infty$ Functional Calculus for a Commuting Pair of $\text{Ritt}_{\text{E}}$ Operators

In this article, we develop a framework for the joint functional calculus of commuting pair of $\text{Ritt}_{\text{E}}$ operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for pair of $\text{Ritt}_{\text{E}}$ operators to that of their associated sectorial counterparts. In addition, we prove a joint dilation theorem for commuting tuples of $\text{Ritt}_{\text{E}}$ operators on a broad class of Banach spaces. As a key application, we obtain an equivalent set of criteria on $L^p$-spaces for $1<p< \infty$ that determine when a commuting pair of $\text{Ritt}_{\text{E}}$ operators admits a joint bounded functional calculus.