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-infinity-functional calculus with respect to the unit disc D (that is, T is polynomially bounded) but T does not have any bounded H-infinity-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 and Portal is equivalent to the existence of a bounded H-infinity-functional calculus with respect to such a Stolz domain.