On functional calculus properties of Ritt operators

TL;DR

This paper investigates the functional calculus properties of Ritt operators, demonstrating that certain bounded calculus properties do not imply others and establishing equivalences under specific conditions.

Contribution

It provides the first example of a Ritt operator with bounded $ ext{H}^\infty$ calculus on the unit disk but not on Stolz domains, and links the unconditional Ritt condition to bounded calculus on Stolz domains.

Findings

Existence of a Ritt operator with bounded $ ext{H}^\infty$ calculus on the unit disk but not on Stolz domains.

Equivalence between the unconditional Ritt condition and bounded $ ext{H}^\infty$ calculus on Stolz domains for R-Ritt operators.

Clarification of the relationship between different functional calculus properties of Ritt operators.

Abstract

We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X --> X on some Banach space X with the following property: T has a bounded $\H^\infty$ functional calculus with respect to the unit disc $\D$ (that is, T is polynomially bounded) but T does not have any bounded $\H^\infty$ functional calculus with respect to a Stolz domain of $\D$ with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded $\H^\infty$ functional calculus with respect to such a Stolz domain.