The H∞-functional calculus is an extension of the Riesz-Dunford functional calculus for bounded operators to unbounded sectorial operators, and it was introduced by A. McIntosh in [165]; see also [5]. This calculus is connected with pseudodifferential operators, with Kato’s square root problem, and with the study of evolution equations and, in particular, the characterization of maximal regularity and with the fractional powers of differential operators. For an overview and more problems associated with this functional calculus for the classical case, see the book [156] and the references therein.