The square functional calculus

I will discuss the interplay between (abstract) square functions that I shall explain and bounded functional calculi. Part of these results are explicitly thought for any abstract functional calculus, parts more specific to the $H^\infty$-calculus developed by Alan McIntosh. I will try to break down the abstract results to concrete situations of (semi)groups on usual Banach spaces, like $L_p(\Omega)$. Using the main result, I will give as applications some (really) three line proofs, for example on the optimal angle of the $H^\infty$-calculus , or the Hörmander-calculus of Kriegler-Weis.