$H^{\mathrm{\infty }}$ 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 $1<p<\mathrm{\infty }$ that determine when a commuting pair of ${\text{Ritt}}_{\text{E}}$ operators admits a joint bounded functional calculus.