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

## TL;DR

This paper constructs counterexamples of Ritt and sectorial operators that are multipliers but lack R-boundedness, highlighting limitations in the theory of operator boundedness on Banach spaces.

## Contribution

It demonstrates that non-$R$-Schauder decompositions can produce operators with non-$R$-bounded powers or semigroups, revealing new limitations in operator theory.

## Key findings

- Existence of Ritt operators that are multipliers but not $R$-bounded
- Existence of sectorial operators with non-$R$-bounded semigroups
- Counterexamples challenge assumptions about $R$-boundedness in Banach spaces

## 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.