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

Suman Mondal, Subhajit Palai, Samya Kumar Ray

TL;DR

This work develops a comprehensive joint $H^\infty$-functional calculus for commuting tuples of $\text{Ritt}_{\text{E}}$ operators on Banach spaces. It introduces a transfer principle linking multivariate calculus for $\text{Ritt}_{\text{E}}$ tuples to sectorial tuples, and proves a joint dilation theorem on broad Banach spaces, connecting calculus, dilation, and boundedness notions. As a key application, the authors derive equivalent $L^p$-space criteria ($1<p<\infty$) for when a commuting tuple admits a joint bounded calculus, and show the equivalence with $p$-polynomial boundedness and loose dilation. By unifying discrete $Ritt_{\text{E}}$ operator theory with sectorial techniques, the results have implications for time-discretized evolution equations and maximal regularity in Banach spaces.

Abstract

In this article, we develop a framework for the joint functional calculus of commuting tuples of $\text{Ritt}_{\text{E}}$ operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for tuples 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 tuple of $\text{Ritt}_{\text{E}}$ operators admits a joint bounded functional calculus.

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

TL;DR

This work develops a comprehensive joint -functional calculus for commuting tuples of operators on Banach spaces. It introduces a transfer principle linking multivariate calculus for tuples to sectorial tuples, and proves a joint dilation theorem on broad Banach spaces, connecting calculus, dilation, and boundedness notions. As a key application, the authors derive equivalent -space criteria () for when a commuting tuple admits a joint bounded calculus, and show the equivalence with -polynomial boundedness and loose dilation. By unifying discrete operator theory with sectorial techniques, the results have implications for time-discretized evolution equations and maximal regularity in Banach spaces.

Abstract

In this article, we develop a framework for the joint functional calculus of commuting tuples of operators on Banach spaces. We establish a transfer principle that relates the bounded holomorphic functional calculus for tuples of operators to that of their associated sectorial counterparts. In addition, we prove a joint dilation theorem for commuting tuples of operators on a broad class of Banach spaces. As a key application, we obtain an equivalent set of criteria on -spaces for that determine when a commuting tuple of operators admits a joint bounded functional calculus.
Paper Structure (5 sections, 22 theorems, 138 equations, 1 figure)

This paper contains 5 sections, 22 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Definition
  3. Joint functional calculus of RE Operators.
  4. Transfer principle: Proof of Theorem \ref{['transferprin']}
  5. Joint dilation for commuting tuple of $\text{Ritt}_E$ operators

Key Result

Theorem 1.1

(Transfer Principle): Let $E=\{\xi_1,\dots,\xi_N\}$ be a finite subset of $\mathbb{T}$. Let $(T_1,T_2)$ be a commuting tuple of $\text{Ritt}_{\text{E}}$ operators on a Banach space $X$. For any $i=1,...,N$, denote $A^1_i=I_{X}-\overline{\xi}_i T_1$ and $A^2_i=I_{X}-\overline{\xi}_i T_2$. Then the fo

Figures (1)

  • Figure 1: polygon $\Delta^0_{i}$ and $\Delta_{i}$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Sector
  • Definition 2.2: Sectorial Operator
  • Definition 2.3
  • Definition 2.4: Polygonal type operator and Polygonal functional calculus
  • Definition 2.5
  • Remark 1
  • Lemma 2.6: MR4819960, Lemma 2.8
  • ...and 39 more