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

## TL;DR

This paper establishes a bounded $H^ty$-calculus for the generator of certain semigroups on BMO spaces, extending to noncommutative settings under specific curvature conditions.

## Contribution

It proves the bounded $H^ty$-calculus for semigroup generators on BMO spaces under Bakry-Emry's  criterion, including noncommutative cases.

## Key findings

- Bounded $H^ty$-calculus established for semigroup generators.
- Introduces a quasi monotone property for subordinated semigroups.
- Results applicable in noncommutative analysis.

## 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 $T_{t,\alpha}=e^{-tL^\alpha},0<\alpha<1$, that is proved in the first half of the article.