On functional calculus properties of Ritt operators

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 $\text{D}$ (that is, T is polynomially bounded) but T does not have any bounded \H^\infty functional calculus with respect to a Stolz domain of $\text{D}$ 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.