| Abstrakt: |
This paper is devoted to the multivariable H ∞ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if (A 1 , ... , A d) is such a family, if A k is R-sectorial of R-type ω k ∈ (0 , π) , k = 1 , ... , d , and if (A 1 , ... , A d) admits a bounded H ∞ (Σ θ 1 × ⋯ × Σ θ d) joint functional calculus for some θ k ∈ (ω k , π) , then it admits a bounded H ∞ (Σ θ 1 × ⋯ × Σ θ d) joint functional calculus for all θ k ∈ (ω k , π) , k = 1 , ... , 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 ∞ functional calculus to square function estimates. Third, on K-convex reflexive spaces, we establish sharp dilation properties for d-tuples (A 1 , ... , A d) which admit a bounded H ∞ (Σ θ 1 × ⋯ × Σ θ d) joint functional calculus for some θ k < π 2 . [ABSTRACT FROM AUTHOR] |
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