New properties of the multivariable H^∞ functional calculus of sectorial operators

2020/07/09 by Arrigoni, Olivier, Merdy, Christian Le
#47A20 #47A60 #47D03 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2007.04580

Abstract

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 (A₁,…, A_d) is such a family, if A_k is R-sectorial of R-type ω_k∈(0,π), k=1,…,d, and if (A₁,…, A_d) admits a bounded H^∞(Σ_θ1× ⋯×Σ_θd) joint functional calculus for some θ_k∈ (ω_k,π), then it admits a bounded H^∞(Σ_θ1× ⋯×Σ_θd) 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 (A₁,…, A_d) which admit a bounded H^∞(Σ_θ1× ⋯×Σ_θd) joint functional calculus for some θ_k

Related

• H^\∞-functional calculus for commuting families of Ritt operators\n and sectorial operators
• The H^∞-Functional Calculus and Square Function Estimates
• The H^∞-calculus and sums of closed operators
• H^∞ Functional Calculus for a Commuting tuple of Ritt_E Operators
• Quadratic estimates for the H^∞-functional calculus of bisectorial Clifford operators