The H∞-functional calculus

Abstract

In this final chapter we present a substantial application of the tools developed in the earlier chapters by constructing and characterising the H ∞ -functional calculus of sectorial and bi-sectorial operators in a Banach space. We begin with a general theory of the sectorial operators and a construction of the H ∞ -calculus, complemented by several examples. In two sections, we connect the H ∞ -calculus to R-boundedness on the one hand, and to the generalised square functions on the other hand. In an independent section, we show the necessity of certain assumptions on the underlying Banach space to some of the deeper results about the H ∞ -calculus. In two final sections, we discuss the H ∞ -calculus of bi-sectorial operators, and establish the H ∞ -calculus of generators of appropriate groups and semigroups on UMD spaces.