A transference principle for general groups and functional calculus on UMD spaces

Abstract

Let -iA be the generator of a C0-group (U(S))S∈ℝ on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate f(A) in terms of the Lp(ℝ; X)-Fourier multiplier norm of (· ±iω). If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev-de Laubenfels. If X is a UMD space, one obtains a bounded H∞1-calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded H∞-calculus on sectors. © The Author(s) 2009.