| This thesis is devoted to the study of the representation of bounded linear operators on Banach spaces and vector-valued concentration inequalities,and the specific contents are as follows:(1)Pelczynski’s property V of spaces of compact operators;(2)The Lipschitz projection from the space of bounded linear operators onto the space of compact operators;(3)Vector-valued concentration inequalities and their relations with the Rademacher type of Banach spaces.The Pelczynski’s property V is one of the most significant concepts in Banach space theory,and "for what kind of Banach space X,the space of compact operator K(X)has the Pelczyński’s property V" is a long-standing open problem.On the other hand,due to the Lindenstrauss-Tzafriri theorem,studying linear projections on general Banach spaces will be challenging,which imposes that it will be more convenient to study nonlinear projections,especially the Lipschitz projections,in Banach spaces.The study of Lipschitz retractions of c0 and the space of continuous functions has been well developed,while the noncommutative counterpart is desired for further exploring.The concentration inequalities quantify the degree of concentration for random variables around their medians,which is one of the essential tools in Banach space theory,probability theory,convex geometry and more.However,studying concentration inequalities for vector-valued random variables has not been well developed.Hence,this thesis aims to extend concentration inequalities in the vector-valued setting and characterize the Rademacher type in terms of vector-valued concentration inequalities.Motivated by the above problems,we obtain the following results.Ⅰ.(Pelczynski’s property V for spaces of compact operators)Let X be a reflexive Banach space with an unconditional basis,then K(X)the space of compact operators on X has Pelczynski’s property V.In particular,K(Lp[0,1])(1<p<∞)the space of compact operators on Lp[0,1](1<p<∞)has Pelczynski’s property V.Ⅱ.(Lipschitz projection)Let 1≤ p,q<∞,then there exists a Lipschitz projection from B(lp,lq)onto K(lp,lq).Ⅲ.(Vector-valued Hoeffding inequality)Let B be a Banach space,1≤p≤2.Then B has Rademacher type p if and only if there exists D>0 such that for every sequence(Xj)j=1n of independent mean zero B-valued random variables,the following inequality holds#12In this thesis,Ⅰ)based on the topological structures of K(Lp[0,1]),we use the representation theory of tensor products to study Pelczynski’s property V of K(Lp[0,1]),which overcomes the limitation of classical geometric method.Ⅱ)We make full use of the relevant results of Banach spaces to construct explicitly the Lipschitz projection from B(lp)to K(lp),which avoids the dependence on operator theory(such as functional calculus)in the case of p=2,and this kind of Lipschitz projection map has a better Lipschitz constant.Ⅲ)Through the study of vector-valued concentration inequalities,we characterize many concepts in Banach space theory in a new point of view,establish a connection between measure of concentration and Banach space theory,and overcome the limitation that measure of concentration can only be used to study finite dimensional subspaces of Banach spaces. |
