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∞‐functional calculus associated with the extension to Λ(H) of a sectorial operator on a B‐convex Banach lattice Λ, where H is a Hilbert space. We apply our results to a new construction of operators with a bounded H∞‐functional calculus and to the maximal regularity problem. 1991 Mathematics Subject Classification: 47A60, 47D06, 46C15.

We compare various functional calculus properties of Ritt operators. We show the existence of a Ritt operator T : X → X on some Banach space X with the following property: T has a bounded H ∞ functional calculus with respect to the unit disc D (that is, T is polynomially bounded) but T does not have any bounded H ∞ functional calculus with respect to a Stolz domain of D with vertex at 1. Also we show that for an R-Ritt operator, the unconditional Ritt condition of Kalton-Portal is equivalent to the existence of a bounded H ∞ functional calculus with respect to such a Stolz domain.

Let X be a (closed) subspace of L p with 1 ≤ p < ∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the formand we show that if A admits a bounded H ∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X = H 1 (R N ) is the usual Hardy space, for an appropriate choice of F . For example if N = 1, the right choice is the sum h F = ∞ 0 F (tA)h 2 dt t 1 2 L 1 + ∞ 0 H F (tA)h 2 dt t 1 2 L 1 for h ∈ H 1 (R), where H denotes the Hilbert transform.

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