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 $$(A_1,\ldots , A_d)$$ such family, $$A_k$$ R-sectorial R-type $$\omega _k\in (0,\pi )$$ $$k=1,\ldots ,d$$ and admits bounded (\Sigma _{\theta _1}\times \cdots \times \Sigma _d})$$ joint for some $$\theta (\omega _k,\pi then it all . Second introduce square functions adapted case extend this setting well-known one-variable results relating boundedness function estimates. Third, K-convex reflexive spaces, establish sharp dilation properties d-tuples which admit _k<\frac{\pi }{2}$$