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 $\{T^n:n\ge 0\}$ 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 $\{e^{-tA}:t\ge 0\}$ is not $R$-bounded.