  Let $\mathcal D$ be a Schauder decomposition on some Banach space $X$. We
prove that if $\mathcal D$ is not $R$-Schauder, then there exists a Ritt
operator $T\in B(X)$ which is a multiplier with respect to $\mathcal D$, such
that the set $\{T^n\, :\, n\geq 0\}$ is not $R$-bounded. Likewise we prove that
there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a
multiplier with respect to $\mathcal D$, such that the set $\{e^{-tA}\, : \,
t\geq 0\}$ is not $R$-bounded.
