A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions,we show the following Bishop–Phelps–Bollobás theorem:Suppose that X is a Banach space,T:X→C(K)is an Asplund operator with║T║=1,and that x0∈SX,0<εsatisfy║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<εand║T-S║<ε.Making use of this theorem,we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T:?1→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 strong Radon–Nikodym operator S:?1→Y of norm one so that(ⅰ)║S(xε)║=1;(ⅱ)S(xε)=yε;(ⅲ)║T-S║<ε;(ⅳ)║S*(yε*)║=<yε*,yε>=1;(ⅴ)║y0*-yε*║<εand(ⅵ)║T*-S*║<ε,where(en)denotes the standard unit vector basis of?1.