In this paper 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 a generalized $H^\infty$-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 $H^\infty$-functional calculus and to the maximal regularity problem.

1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.