New counterexamples on Ritt operators, sectorial operators and R-boundedness

Abstract: 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.