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

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.