In Section 2.1 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation. In Section 2.2 we introduce some notation for certain spaces of holomorphic functions on sectors. A functional calculus for sectorial operators is constructed in Section 2.3 along the lines of the abstract framework of Chapter 1. Fundamental properties like the composition rule are proved. In Section 2.5 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible. In this way a panorama of functional calculi is developed. In Section 2.6 some mixed topics are discussed, e.g., adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results. Section 2.7 contains a spectral mapping theorem.

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A general approach to transference principles for discrete and continuous operator (semi)groups is described. This allows to recover the classical transference results of Calderón, Coifman and Weiss and of Berkson, Gillespie and Muhly and the more recent one of the author. The method is applied to derive a new transference principle for (discrete and continuous) operator semigroups that need not be groups. As an application, functional calculus estimates for bounded operators with at most polynomially growing powers are derived, culminating in a new proof of classical results by Peller from 1982. The method allows a generalization of his results away from Hilbert spaces to L p -spaces and -involving the concept of γ-boundedness -to general Banach spaces. Analogous results for strongly-continuous one-parameter (semi)groups are presented as well. Finally, an application is given to singular integrals for one-parameter semigroups.In the classical transference principle(s) the objects under investigation are derived operators of the formwhere G is a locally compact group and T = (T (s)) s∈G : G → L(X) is a bounded strongly continuous representation of G on a Banach space X. The integral (1.1) has to be understood in the strong sense, i.e.,Since such operators occur in a variety of situations, the applications of transference principles are manifold, and the literature on this topic is vast. We therefore restrict ourselves to mentioning only a few 'landmarks' which we regard as most important for the understanding of the present paper.

Let −i A be the generator of a C 0 -group (U (s)) s∈R on a Banach space X and ω > θ(U ), the group type of U . We prove a transference principle that allows to estimateIf X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev-de Laubenfels. If X is a UMD space, one obtains a bounded H ∞ 1 -calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded H ∞ -calculus on sectors.