The H∞−calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an H∞−calculus.We then apply this to the joint functional calculus of two sectorial operators when one has an H∞calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of Lp−maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of L1−spaces and C(K)−spaces, to obtain some more detailed results in this setting. Mathematics Subject Classification (2000): 47A60, 47D06