Functional calculus and dilation for C 0-groups of polynomial growth

Abstract

Let U(t) = e itB be a C 0 -group on a Banach space X.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 E α ∞ 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 α ∞ calculus is compared with Besov functional calculi as considered in Cowling et al. (J. Aust. Math. Soc., Ser. A, 60(1): 1996) and Kriegler (Spectral multipliers, R-bounded homomorphisms, and analytic diffusion semigroups. PhD-thesis).

A Besov algebra calculus for generators of operator semigroups and related norm-estimates

“…Thus the generality of Theorem 1.1 was out of reach. Related functional calculi for generators of C 0 -groups of polynomial growth (thus having their spectrum on iR) and for sectorial operators of zero angle were studied in [21], [52] and [53], again by means of Fourier analysis.…”

Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

“…exists in operator norm. Denoting this limit by µ(B) as in the assertion, the estimate finally shows (10). For the following we observe that, for ψ ∈ Φ(τ X ), we have…”

Bounds on the Phillips calculus of abstract first order differential operators

“…In this case the functions g need to have holomorphic extensions to a strip of the complex plane. Kriegler in 2012 (Theorem 4.9, [20]) proved a similar result for a group that grows at most polynomially on L p . In the case of both Coifman-Weiss and Kriegler's results the functions can be compactly supported.…”