Functional calculus and dilation for C0-groups of polynomial growth

Let U(t)=eitB be a C0-group on a Banach space X. Let further \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi\in C^{\infty}_{c}(\mathbb {R})$\end{document} satisfy ∑n∈ℤϕ(⋅−n)≡1. For α≥0, we put \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ E^\alpha _\infty }= \biggl\{ f \in C_b(\mathbb {R}):\: \|f\|_{ E^\alpha _\infty }= \sum_{n \in \mathbb {Z}} (1 +|n|)^\alpha\| f \ast [\phi(\cdot- n)]\check{\phantom{i}} \|_{L^\infty(\mathbb {R})} < \infty\biggr\},$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${ E^{\alpha}_{\infty}}$\end{document} 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)=Ait 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${ E^{\alpha}_{\infty}}$\end{document} 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).