Research On Non-commutative Lorentz Type Space

This thesis is devoted to the study of noncommutative Lorentz type space associated with a semifinite von Neumann algebra M, which consists of the following four parts:In the first part, we introduce some basic concepts about noncommutative integration theory and the main results about this thesis.In the second part, we consider the noncommutative weak Lp spaces, plus two the-orems as main results of our work. One concerns mainly about the dual spaces of non-commutative weak Lp spaces the other is about the compact multiplication operators on Lp,∞(M).In the third part, we consider the noncommutative weighted Lorentz spaces. After a brief introduction of noncommutative weighted Lorentz spaces, we prove some noncom-mutative analogues of the classical results about dual spaces of weighted Lorentz spaces. In this part, we also introduction the noncommutative Hardy-Lorentz spaces and give some results about Riesz factorization, Riesz projection and Herglotz transform on these spaces.In part four, we consider the noncommutative Orlicz Lorentz spaces and noncommu-tative weak Orlicz Lorentz spaces. After a brief introduction of noncommutative (weak) Orlicz Lorentz spaces, we presents some results about dual spaces Marcinkiewicz interpo-lation theorem about noncommutative (weak) Orlicz Lorentz spaces and noncommutative Orlicz Lorentz spaces. At the end of this part, we presents some results about the general associate spaces of noncommutative Calderon-Lozanovskii spaces Eφ(M).