Abstract
We prove that the negative generator L of a semigroup of positive contractions on L∞ has bounded H∞(Sη)-calculus on the associated Poisson semigroup-BMO space for any angle η>π/2, provided L satisfies Bakry-Émery's Γ2≥0 criterion. Our arguments only rely on the properties of the underlying semigroup and work well in the noncommutative setting. A key ingredient of our argument is a type of quasi monotone properties for the subordinated semigroup Tt,α=e−tLα,0<α<1, that is proved in the first part of this article.
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