A Bishop-Phelps-Bollobás Theorem For Asplund Operators

The study of the Bishop-Phelps-Bollobas property for operators has become an active area,and a large number of papers have appeared in the literature.In this dissertation,we are devoted to characterizing Asplund operators and strong RadonNikodym operators in terms of Frechet differentiability of convex functions,and to research of sharp Bishop-Phelps-Bollobás theorem of C(K)-valued Asplund operators and the dual version of Bishop-Phelps-Bollobas property of strong Radon-Nikodym operators defined on l1.We introduce the following three notions:Generic Frechet differentiability operators T:X→ Y,i.e.p(x)=‖Tx‖ are densely Frechet differentiability in X,which are generalization of Asplund operators;strong Radon-Nikodym operator and w*-Asplund operators.We show that an operator T ∈ L(X,Y)is an Asplund operator if and only if for every convex function f defined on Y,the composition f o T is Frechet differentiable at each point of a dense Gδ-subset of X;and an operator T ∈ L(X,Y)is an Asplund operator if and only if T*is a strong RandonNikodym operator;T ∈ L(X,Y)is an strong Radon-Nikodym operator if and only if T*is a w*-Asplund operator.Applying these properties,incorporating of a careful discussion on the Br?ndsted-Rockafellar theorem,we show the following theorem:Suppose that X is a Banach space,T:X→C(K)is a Asplund operator with‖T‖=1,and that x0 ∈ Sx,ε>0 satisfy ‖T(x0)‖>1-ε2/2.Then there exist xε ∈ SX and an Asplund operator S:X→C(K)of norm one so that‖S(xε)‖=1,‖x0-xε‖<ε and ‖T-S‖<ε.Making use of this theorem,we further show a dual version of Bishop-PhelpsBollobas property for a strong Radon-Nikodym operator T:l1→ Y of norm one:Suppose that y0*∈SY*,ε≥ 0 satisfy ‖T*(y0*)‖>1-ε2/2.Then there exist yε*∈SY*,xε(±en),yε∈SY,and a Radon-Nikodym operator S:l1→ Y of norm one so thatⅰ)‖S(xε)‖=1;ⅱ)S(xε)=yε;ⅲ)‖T-S‖<ε;ⅳ)‖S*(yε*)‖=(yε*,yε)=1;ⅴ)‖y0*-yε*‖<ε 以及 ⅵ)‖T*-S*‖<ε,where(en)denotes the standard unit vector basis of l1.