Functional calculus and dilation for C0-groups of polynomial growth

Let U ( t )= e itB be a C 0 -group on a Banach space X . Let further satisfy ∑ n ∈ℤ ϕ (⋅− n )≡1. For α ≥0, we put which is a Banach algebra. It is shown that ∥ U ( t )∥≤ C (1+| t |) α for all t ∈ℝ if and only if the generator B has a bounded functional calculus, under some minimal hypotheses, which exclude simple counterexamples. A third equivalent condition is that U ( t ) admits a dilation to a shift group on some space of functions ℝ→ X . In the case U ( t )= A it with some sectorial operator A , we use this calculus to show optimal bounds for fractions of the semigroup generated by A , resolvent functions and variants of it. Finally, the calculus is compared with Besov functional calculi as considered in Cowling et al. (J. Aust. Math. Soc., Ser. A, 60(1):51–89, 1996 ) and Kriegler (Spectral multipliers, R -bounded homomorphisms, and analytic diffusion semigroups. PhD-thesis).