New Properties of the Multivariable H∞ Functional Calculus of Sectorial Operators

This paper is devoted to the multivariable H-infinity functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if (A(1), ... , A(d)) is such a family, if A(k) is R-sectorial of R-type omega(k) is an element of (0, pi), k = 1, ... , d, and if (A(1), ... , A(d)) admits a bounded H-infinity (Sigma(theta 1) x ... x Sigma(theta d)) joint functional calculus for some theta(k) is an element of (omega(k), pi), then it admits a bounded H-infinity (Sigma(theta 1) x ... x Sigma(theta d)) joint functional calculus for all theta(k) is an element of (omega(k), pi), k = 1,..., d. Second we introduce square functions adapted to the multivariable case and extend to this setting some of the well-known one-variable results relating the boundedness of H-infinity functional calculus to square function estimates. Third, on K-convex reflexive spaces, we establish sharp dilation properties for d-tuples (A(1), ... , A(d)) which admit a bounded H-infinity (Sigma(theta 1) x ... x Sigma(theta d)) joint functional calculus for some theta(k) < pi/2.