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 phi is an element of C-c(infinity)(R) satisfy Sigma(n is an element of Z)phi(center dot - n) equivalent to 1. For alpha >= 0, we put E-infinity(alpha) = {f is an element of C-b(R) : parallel to f parallel to(E infinity alpha) = Sigma(n is an element of Z)(1 + vertical bar n vertical bar)(alpha) parallel to f * [phi(center dot - n)](boolean OR)parallel to(L infinity(R)) < infinity}, which is a Banach algebra. It is shown that parallel to U(t)parallel to <= C(1 + vertical bar t vertical bar)(alpha) for all t is an element of R if and only if the generator B has a bounded E-infinity(alpha) 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 E-infinity(alpha) 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).