Functional calculus and dilation for C-groups of polynomial growth

Kriegler, C.
June 2012
Semigroup Forum;Jun2012, Vol. 84 Issue 3, p393
Academic Journal
Let U( t)= e be a C-group on a Banach space X. Let further $\phi\in C^{\infty}_{c}(\mathbb {R})$ satisfy ∑ ϕ(⋅− n)≡1. For α≥0, we put which is a Banach algebra. It is shown that ∥ U( t)∥≤ C(1+| t|) for all t∈ℝ if and only if the generator B has a bounded ${ E^{\alpha}_{\infty}}$ 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 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 ${ E^{\alpha}_{\infty}}$ calculus is compared with Besov functional calculi as considered in Cowling et al. (J. Aust. Math. Soc., Ser. A, 60(1):51-89, ) and Kriegler (Spectral multipliers, R-bounded homomorphisms, and analytic diffusion semigroups. PhD-thesis).


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