Description

We investigate sectorial operators and semigroups acting on non-commutative L^p-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 L^p-spaces. This requires natural variants of Rademacher sectoriality and the use of the matricial structure of noncommutative L^p-spaces. We mainly focus on non-commutative diffusion semigroups, that is, semigroups (T_t)(...) of normal selfadjoint operators on a semifinite von Neumann algebra (M, t) such that T_t : L^p(M) (...) L^p(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.