The square functional calculus

This is joint work with Markus Haase (Kiel). Many "classical" square functions in harmonic analysis can be linked to functional calculus expressions, the Littlewood-Paley theory is for example essentially about the functional calculus of the Laplacian. Generalising, this lead to the notion of "abstract" square functions, a well studied subject by now. Initial ideas from harmonic analysis are mixed with concepts from the geometry of Banach spaces, namely the use of random sums (Rademacher or Gaussian). The surprising point we add to this theory, is that square functions themselves can (and to our point of view: should) be seen as functional calculi themselves, but for vector-valued functions. I go over the principle ideas and show how this gives a relatively simple access to the whole subject.