  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.
