Operator valued Hardy spaces

Abstract

We give a systematic study on the Hardy spaces of functions with values in the non-commutative L p -spaces associated with a semifinite von Neumann algebra M.This is motivated by the works on matrix valued Harmonic Analysis (operator weighted norm inequalities, operator Hilbert transform), and on the other hand, by the recent development on the non-commutative martingale inequalities.Our non-commutative Hardy spaces are defined by the non-commutative Lusin integral function.The main results of this paper include:(i) The analogue in our setting of the classical Fefferman duality theorem between H 1 and BMO.(ii) The atomic decomposition of our non-commutative H 1 .(iii) The equivalence between the norms of the non-commutative Hardy spaces and of the noncommutative L p -spaces (1 < p < ∞).(iv) The non-commutative Hardy-Littlewood maximal inequality.(v) A description of BMO as an intersection of two dyadic BMO.(vi) The interpolation results on these Hardy spaces.