Joint functional calculus for sectorial operators

In this Note we study the notion of joint functional calculus associated with a couple of resolvent commuting sectorial operators on a Banach space X. We present some positive results when X is for example a Banach lattice or a quotient of subspaces of a B-convex Banach lattice. Furthermore, we develop a notion of generalized H-infinity functional calculus associated with the extension to Lambda(H) of a sectorial operator on a B-convex Banach lattice Lambda, where H is a Hilbert space. We apply our results to a new construction of operators with a bounded H-infinity functional calculus and to the maximal regularity problem.