Functional calculus is an important tool to study operator classes and their use in evolution equations. The ubiquitous problem of establishing well-posedness of equations can be transformed into the question under which conditions an operator of the form f(A) is bounded (where A is a given partial differential operator and f is a scalar function). A so-called transference principle reduces the question - by means of a factorization of the operator f(A) - to a Fourier multiplier theorem on a vector-valued function space. Recently we established a new and surprising factorization, by a very flexible proof technique. The aim of the proposed project is to explore these new ideas, which will allow us to go far beyond the classical case. (Technically speaking, from generators of bounded to generators of unbounded operator groups and even semigroups.) More precisely, the project consists in applying the new factorization with varying choices of function spaces and hypotheses on A and in generating new factorizations, in particular in the semigroup case, by applying the new method. This will require to find new suitable function spaces, the most challenging part of the project. If successful, this project will greatly deepen our understanding of the operators relevant for evolution equations. The step from groups to semigroups is a milestone, and we hope to find an original ?harmonic analysis for operator semigroups? not relying on groups any more.