@book{3148983461fb46ffa3b50af0ae4c4a67,

title = "H∞ functional calculus and square functions on noncommutative Lp-spaces",

abstract = "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, McIntoch and Yagi concerning commutative Lp-spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative Lp-spaces. We mainly focus on noncommutative diﬀusion semigroups, that is, semigroups (Tt)t≥0 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 q-deformed von Neumann algebras of Bozejko-Speicher, and the noncommutative Poisson semigroup acting on the group von Neumann algebra of a free group.",

keywords = "H∞ functional calculus, noncommutative Lp-spaces, square functions, sectorial operators, diﬀusion semigroups, completely bounded maps, multipliers",

author = "Marius Junge and {Le Merdy}, Christian and Quanhua Xu",

year = "2006",

doi = "10.24033/ast.698",

language = "English (US)",

isbn = "9782856291894",

series = "Ast{\'e}risque",

publisher = "Soci{\'e}t{\'e} math{\'e}matique de France",

}