The H-infinity-calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an H-infinity-calculus. We then apply this to the joint functional ca lculus of two sectorial operators when one has an H-infinity calculus. Usin g this we prove theorem of Dore-Venni type on sums of sectorial operators a nd apply our results to the problem of L-p-maximal regularity. Our main ass umption is the R-boundedness of certain sets of operators. and therefore me thods from the geometry of Banach spaces are essential here. In the final s ection we exploit the special Banach space structure of L-1-space, and C(K) -spaces, to obtain some more detailed results in this setting.