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 \(\theta _k<\frac{\pi }{2}\).

本文致力于多变量\(H^\infty \) 泛函演算与 Banach 空间上扇形算子的有限交换族相关。首先，我们证明了如果\（（A_1，\ ldots，A_D）\）是这样的家庭，如果\（A_k \）是- [R的-sectorial ř型\（\欧米加_K \在（0，\ PI）\ ) , \(k=1,\ldots ,d\)，并且如果\((A_1,\ldots , A_d)\)承认有界\(H^\infty (\Sigma _{\theta _1}\times \) cdots \times \Sigma _{\theta _d})\)联合泛函演算对于某些\(\theta _k\in (\omega _k,\pi )\)，那么它承认一个有界\(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\)所有\(\theta _k\in (\omega _k,\ pi )\) , \(k=1,\ldots ,d\)。其次，我们介绍了适用于多变量情况的平方函数，并将一些众所周知的单变量结果扩展到这种设置，这些结果将\(H^\infty \)函数演算的有界性与平方函数估计相关联。第三，在K凸自反空间上，我们为d元组\((A_1,\ldots , A_d)\)建立了尖锐的膨胀特性，它允许有界\(H^\infty (\Sigma _{\theta _1}\ times \cdots \times \Sigma _{\theta _d})\)一些联合函数演算\(\theta _k<\frac{\pi }{2}\)。