Research On Non-commutative Hardy-Lorentz Type Space

This thesis adapts the method of operator theory, space theory and harmonic analysis, via the results obtained by other authors, to study Szego type factorization for noncom-mutative Hardy-Lorentz spaces, some maximal inequalities on noncommutative Lorentz spaces and some inequalities for r-measurable operators in noncommutative Lp-spaces. This thesis is divided into four chapters and is stated as follows:In chapter one, we give the research status of the thesis and present some notations together with definitions of spaces.In chapter two we establish Szego factorization and inner-outer factorization for non-commutative Hardy-Lorentz spaces.In chapter three, two sections are included. Section one involves the (p, q)-(p, q)-type inequality of the Hardy-Littlewood maximal function MT on noncommutative Lorentz spaces; and section two concerns the weak (p, q)-(p,q)-type inequality of the generalized Hardy-Littlewood maximal function Mp,qT on noncommutative Lorentz spaces.In chapter four, three sections are included. In section one, we generalize the Young and Heinz inequalities of matrix to r-measurable operators in noncommutative Lp-spaces; in section two, we give the singular inequalities for the arithmetic, geometric and Heinz mean of r-measurable operators and the reverse Young and Heinz inequalities for τ-measurable operators; in the last section, via the joint convexity and concavity of trace function, on one hand, we obtain generalizations of the convexity of certain functions involving noncommutative Lp-norms; on the other hand, we generalize the Carlen-Lieb theorem concerning concavity of certain trace functions to noncommutative Lp-cases.