A Joint Functional Calculus for Sectorial Operators with Commuting Resolvents

Abstract

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.

The $H^{\infty}-$ calculus and sums of closed operators

“…Any subspace of a Banach lattice with nontrivial cotype has property (α) while any Banach lattice has property (A). It is also observed in [28] that L 1 /H 1 has (α). The Schatten ideals C p when 1 ≤ p ≤ ∞ fail to have (A).…”

“…We say that X has property (α) (see [36] and [28]) if there is a constant C so that for any (x jk ) n j,k=1 ⊂ X and any (α jk ) n j,k=1 ⊂ C we have…”

“…(2) For Hilbert spaces and certain operators on L 2 (X) such an operatorvalued functional calculus is constructed in [28] Theorem 5.2 and [30]. These are cases when the U-boundedness condition is satisfied automatically.…”

“…This estimate allows a definition of f (A) as a bounded operator for functions f which are bounded and analytic on the sector Σ σ = {ζ : | arg ζ| < σ} and which obey a condition of the type |f (ζ)| ≤ C(|ζ|/(1 + |ζ| 2 ) ǫ ) for some ǫ > 0. This is described in detail in [35] and [28] and we give a somewhat different approach in Section 2 below. If we then have an estimate…”

“…This then is a special case of the problem of constructing a joint functional calculus of A, B. In the case of Hilbert spaces and some related situations it was shown in [20], [28] and [30] that one can construct an operatorvalued functional calculus associated to an operator with H ∞ −calculus and this permits a solution; however, it was also shown that such an approach cannot work in general Banach spaces and additional conditions are therefore needed.…”