NEW COUNTEREXAMPLES ON RITT OPERATORS, SECTORIAL OPERATORS AND -BOUNDEDNESS

Source 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.