This means that semigroup theory may be applied directly to the study of the equation I'!.f = h on M. In  Yau proves that, for h ~ 0, there are nconstant, nnegative solutions f in [j' for 1 00. From this, Yau gets the geometric fact that complete ncom pact Riemannian manifolds with nnegative Ricci curvature must have infinite volume, a result which was anunced earlier by Calabi . 6. Concluding Remarks In several of the above results, positivity of the semigroup plays an important role. This was also true, although only implicitly, for the early work of Hille and Yosida on the Fokker-Planck equation, i.e., Equation (4) with c = O. But it was Phillips , and Lumer and Phillips  who first called attention to the importance of dissipative and dispersive properties of the generator in the context of linear operators in a Banach space. The generation theorems in the Batty-Robinson paper appear to be the most definitive ones, so far, for this class of operators. The fundamental role played by the infinitesimal operator, also for the understanding of order properties, in the commutative as well as the ncommutative setting, are highlighted in a number of examples and applications in the different papers, and it is hoped that this publication will be of interest to researchers in a broad spectrum of the mathematical sub-divisions.