New properties of the multivariable $H^\infty$ functional calculus of sectorial operators
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 $(A_1,\ldots, A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type $ω_k\in(0,π)$, $k=1,\ldots,d$, and if $(A_1,\ldots, A_d)$ admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k\in (ω_k,π)$, then it admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for all $θ_k\in (ω_k,π)$, $k=1,\ldots,d$. 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 $(A_1,\ldots, A_d)$ which admit a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k<\fracπ{2}$.