$H^{\infty }$-calculus for semigroup generators on BMO

Abstract

We prove that the negative infinitesimal generator $L$ of a semigroup of positive contractions on $L^{\infty }$ has a bounded $H^{\infty }(S_{\eta }^0)$-calculus on the associated Poisson semigroup-BMO space for any angle $\eta >\pi /2$, provided the semigroup satisfies Bakry-Emry's ${\Gamma }_2$ criterion. Our arguments only rely on the properties of the underlying semigroup and works well in the noncommutative setting. A key ingredient of our argument is a quasi monotone property for the subordinated semigroup $T_{t,\alpha }=e^{-t{L}^{\alpha }},0<\alpha <1$, that is proved in the first half of the article.

32 pages

References in corpus (3)

• Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi
• A non-commutative Path Space approach to stationary free Stochastic Differential Equations
• Complete boundedness of the Heat Semigroups on the von Neumann Algebra of hyperbolic groups

Cited by in corpus (2)

• Paley's inequality for nonabelian groups
• An operator-valued $T1$ theory for symmetric CZOs