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 \((A_1,\ldots , A_d)\) is such a family, if \(A_k\) is R-sectorial of R-type \(\omega _k\in (0,\pi )\), \(k=1,\ldots ,d\), and if \((A_1,\ldots , A_d)\) 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 \(\theta _k\in (\omega _k,\pi )\), \(k=1,\ldots ,d\). 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 \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for some \(\theta _k<\frac{\pi }{2}\).