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

TL;DR

This paper develops a framework for the joint $H^$ functional calculus of commuting pairs of $ ext{Ritt}_{ ext{E}}$ operators on Banach spaces, including transfer principles and dilation theorems.

Contribution

It introduces a transfer principle linking bounded holomorphic calculus of $ ext{Ritt}_{ ext{E}}$ pairs to sectorial operators and establishes a joint dilation theorem.

Findings

Established a transfer principle for $ ext{Ritt}_{ ext{E}}$ operator pairs.

Proved a joint dilation theorem for commuting $ ext{Ritt}_{ ext{E}}$ operators.

Provided criteria on $L^p$-spaces for joint bounded functional calculus.

Abstract

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.