Spectral properties and norm estimates associated to the Cc(k)-functional calculus

We show that the Davies functional calculus and the AC((nu))-calculus coincide under common hypotheses. Then we apply the calculus to operators on Banach spaces, to investigate spectral invariance and norm estimates linked to abstract Cauchy equations. This extends some previous results in the area, and unifies diverse approaches. The theory is applied to operators related to multiplier theory on Lie groups.