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

Authors:	Markus Haase

Institution:	(1) Delft Institute of Applied Mathematics, Delft University of Technology, PO Box 5031, 2600 GA Delft, The Netherlands

Abstract:	Let –iA be the generator of a C ₀-group ${(U(s))_{s\in \mathbb {R}}}$ on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate in terms of the ${{\rm{L}}^{p}(\mathbb {R};X)}$ -Fourier multiplier norm of ${f(\cdot \pm i \omega)}$ . 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 ${{\rm{H}}^\infty_1}$ -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 ${{\rm{H}}^\infty}$ -calculus on sectors.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 47A60 47D06 44A40 42A45
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