On semi-R-boundedness and its applications

R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier's property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the H^∞-calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions.