Ritt property and functional calculus of subordinated operators in the group case

Given $T$ a power bounded operator on a Banach space $X$ one can consider the discrete subordinated operator $S=\sum_{k=0}^\infty c_k T^k$ where $c_k\geq 0$ and $\sum_{k=0}^\infty c_k=1.$ N. Dungey gave conditions on $(c_k)$ for $S$ to be a Ritt operator i.e. $S$ powerbounded and $\sup_n n IIS^n-S^{n+1}II<\infty.$ With C. Le Merdy we concentrate on the following subordination situation : for $\nu$ a probability measure on a locally compact abelian group and $\pi: G\to B(X)$ a bounded representation we define the operator $S=\int_G \pi(t) d\vu(t).$ We show that under certain conditions on $\vu$ as well as on the geometry of the space $X$ the subordinated operator $S$ is a Ritt operator or admits a bounded $H^\infty$ functional calculus. This is joint work with Christian Le Merdy.