$H^\infty$--functional calculus for generators of semigroups that admit lower bounds

TL;DR

This paper proves bounded $H^$ functional calculus for generators of semigroups on UMD Banach spaces with a lower bound, using dilation and transference techniques.

Contribution

It introduces a dilation-based method to establish $H^$ calculus for semigroup generators with lower bounds on UMD spaces, extending previous Hilbert space results.

Findings

Bounded $H^$ calculus established for generators with lower bounds.

Dilation and transference techniques are effectively combined.

Quantitative exponential lower bounds for semigroups are derived.

Abstract

We study $C_0$-semigroups on UMD Banach spaces under the assumption that a single semigroup operator admits a lower bound. We establish boundedness of $H^\infty$ functional calculi for the negative generator of such semigroups. Our approach is based on a dilation argument: combining a recent construction due to Madani with transference results for groups on UMD spaces, we embed the semigroup into a $C_0$-group on a larger space and transfer functional calculus estimates back to the original generator. As a byproduct, we obtain quantitative exponential lower bounds for the semigroup. We also show that equivalences due to Batty and Geyer, valid in Hilbert spaces, fail in the general Banach space setting.