The Duality And Interpolation For Noncommutative Quasi-martingale Spaces

The theory of noncommutative mathematics is leading in the word in the math-ematical research at the moment.Since noncommutative martingales is an important part of noncommutative mathematics,many researchers have devoted themselves into it and have achieved many results.Quasi-martingales is the significant generalization of martingales.However,the study of noncommutative quasi-martingales is still blank.In this thesis,we devote to the theory of noncommutative quasi-martingales and ex-tend many important results of noncommutative martingales to the noncommutative quasi-martingale setting.The framework of this thesis consists of the following six parts:Chapter 1 is an introduction on the background,motivation and the principle results of the dissertation.Chapter 2 is the preliminary,it contains the main notion of the thesis.In chapter 3,we prove the convergence and fundamental inequalities for noncom-mutative quasi-martingales.These include Cuculescu’s weak type(1,1)inequality,Doob’s inequality and Burkholder-Gundy inequalities and so on.Also,we prove the Gundy’s decomposition for noncommutative quasi-martingales.As an application,we solve a problem about basis sequences.In chapter 4,we define three new noncommutative spaces BDp(M),SP(M)and E∞(M)in order to describe the dual spaces of Lp(M)and Hp(M),where Lp(M)and Hp(M)are respectively the Lp space and Hardy space of noncommutative quasi-martingales.In chapter 5,we prove a real interpolation theorem between the spaces qLp(M).We also prove real interpolation theorem between the spaces BMO(M)andHq(M).In the final,using the connection between real and complex interpolation,we obtain some corresponding results for complex interpolation.In chapter 6,we devote to extend the results discussed above of noncommutative Lp spaces to the noncommutative symmetric space setting.It should be pointed out that we encounter substantial difficulties in dealing with quasi-martingales in noncommutative symmetric spaces.This is often highly non-trivial and requires additional tool and techniques.For instance,the proof of Burkholder-Gundy inequalities makes use of the tensor product M(?)B(l2)and the corresponding symmetric spaces E(M(?)B(l2));the proof of the duality theorems makes use of the point product of operators;the proof of the interpolation theorems makes use of the power theorem of the K-Method.