Dual-stable counterexamples on \(\ell^1\) or \(L^1\) without an \(H^\infty\)-calculus requirement

Construct, on \(\ell^1\) or \(L^1\), an example analogous to the dual-stable counterexample in which both an operator and its adjoint are \(R\)-sectorial and neither operator has maximal \(L^p\)-regularity, without requiring the operators to have bounded \(H^\infty\)-calculi.

Background

The paper constructs a sectorial diagonal operator on a space of the form X(ℓ2)X(\ell^2), where XX is a non-UMD Banach function space with finite cotype, such that the operator and its adjoint are both RR-sectorial and have bounded H∞H^\infty-calculi of angle zero, while neither has deterministic maximal LpL^p-regularity. This provides a counterexample stable under taking adjoints.

The authors explicitly ask whether an analogous construction can be carried out on the classical endpoint spaces ℓ1\ell^1 or L1L^1 if the bounded H∞H^\infty-calculus requirement is dropped. They note that standard methods for transferring the preceding construction to these spaces appear to destroy RR-sectoriality, identifying preservation of RR-sectoriality as the main obstruction.

References

To the best of our knowledge, it remains open whether a similar example to Corollary \ref{cor:dual-stable-counterexample}, without the $H\infty$-calculus requirement, can be constructed on $\ell1$ or $L1$. Standard ways of transferring the preceding construction to these spaces appear to destroy $R$-sectoriality.

— Necessary conditions for deterministic and stochastic maximal regularity  (2608.20266 - Lorist et al., 20 Aug 2026) in Section 2, subsection “An extension of the deterministic construction,” immediately after Corollary \ref{cor:dual-stable-counterexample}