H functional calculus and square functions on noncommutative Lp-spaces

Marius Junge, Christian Le Merdy, Quanhua Xu

Research output: Chapter in Book/Report/Conference proceedingChapter


We investigate sectorial operators and semigroups acting on noncommutative Lp-spaces. We introduce new square functions in this context and study their connection with H functional calculus, extending some famous work by Cowling, Doust, Mclntoch and Yagi concerning commutative Lp-spaces. This requires natural variants of Rademacher sectoriality and the use of the matriciel structure of noncommutative Lp-spaces. We mainly focus on noncommutative diffusion semigroups, that is, semigroups (Tt)t≥o of normal selfadjoint operators on a semifinite von Neumann algebra (M,τ) such that Tt: Lp(M) → Lp(M) is a contraction for any p ≥ 1 and any t ≥ 0. We discuss several examples of such semigroups for which we establish bounded H functional calculus and square function estimates. This includes semigroups generated by certain Hamiltonians or Schur multipliers, q-Ornstein-Uhlenbeck semigroups acting on the g-deformed von Neumann algebras of Bozejko-Speicher, and the noncommutative Poisson semigroup acting on the group von Neumann algebra of a free group.

Original languageEnglish (US)
Title of host publicationH Functional Calculus and Square Functions on Noncommutative L- Spaces
EditorsMarius Junge, Christian Le Merdy, Quanhua Xu
Number of pages138
StatePublished - 2006

Publication series

ISSN (Print)0303-1179


  • Completely bounded maps
  • Diffusion semigroups
  • H functional calculus
  • Multipliers
  • Noncommutative L -spaces
  • Sectorial operators
  • Square functions

ASJC Scopus subject areas

  • General Mathematics


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