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.