New properties of the multivariable $H^{\infty }$ functional calculus of sectorial operators

Abstract

This paper is devoted to the multivariable $H^{\infty }$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $(A_1,\dots ,A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type ${\omega }_k\in (0,\pi )$, $k=1,\dots ,d$, and if $(A_1,\dots ,A_d)$ admits a bounded $H^{\infty }({\Sigma }_{{\theta }_1}\times \cdots \times {\Sigma }_{{\theta }_d})$ joint functional calculus for some ${\theta }_k\in ({\omega }_k,\pi )$, then it admits a bounded $H^{\infty }({\Sigma }_{{\theta }_1}\times \cdots \times {\Sigma }_{{\theta }_d})$ joint functional calculus for all ${\theta }_k\in ({\omega }_k,\pi )$, $k=1,\dots ,d$. 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 $H^{\infty }$ functional calculus to square function estimates. Third, on $K$-convex reflexive spaces, we establish sharp dilation properties for $d$-tuples $(A_1,\dots ,A_d)$ which admit a bounded $H^{\infty }({\Sigma }_{{\theta }_1}\times \cdots \times {\Sigma }_{{\theta }_d})$ joint functional calculus for some ${\theta }_k<\frac{\pi }{2}$.

Acceted for publication in Integral Equations and Operator Theory