By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop-Phelps-Bollobés theorem: Suppose that X is a Banach space, T: X → C ( K ) is an Asplund operator with ∥T∥ = 1, and that x 0 ∈ S X , 0 < ε satisfy $$\left\| {T\left({{x_0}} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ ‖ T ( x 0 ) ‖ > 1 − ε 2 2 . Then there exist x ε ∈ S X and an Asplund operator S: X → C ( K ) of norm one so that $$\left\| {S\left({{x_\varepsilon}} \right)} \right\| = 1,\,\,\,\,\,\,\left\| {{x_0} - {x_\varepsilon}} \right\| < \varepsilon \,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\left\| {T - S} \right\| < \varepsilon .$$ ‖ S ( x ε ) ‖ = 1 , ‖ x 0 − x ε ‖ < ε and ‖ T − S ‖ < ε . Making use of this theorem, we further show a dual version of Bishop-Phelps-Bollobas property for a strong Radon-Nikodym operator T : ℓ 1 → Y of norm one: Suppose that y 0 * ∈ S y * , ε ≥ 0 satisfy $$\left\| {{T^*}\left({y_0^*} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ ‖ T * ( y 0 * ) ‖ > 1 − ε 2 2 . Then there exist y * µ ∈ S y * , x ε ∈ (±e n ), y ε ∈ S y , and a strong Radon-Nikodym operator S: ℓ 1 → Y of norm one so that $$\begin{array}{*{20}{c}} {(i)}&{\left\| {S({x_\varepsilon })} \right\| = 1;}&{(ii)S({x_\varepsilon }) = {y_\varepsilon };}&{(iii)\left\| {T - S} \right\| < \varepsilon ;} \end{array}$$ $$\begin{array}{*{20}{c}} {(iv)\left\| {{S^*}(y_\varepsilon ^*)} \right\| = \left\langle {y_\varepsilon ^*,{y_\varepsilon }} \right\rangle = 1;}&{(v)\left\| {y_0^* - y_\varepsilon ^*} \right\| < \varepsilon }&{and}&{(vi)\left\| {{T^*} - {S^*}} \right\| < \varepsilon ,} \end{array}$$ where ( e n ) denotes the standard unit vector basis of ℓ 1 .

中文翻译：

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Asplund 算子的 Bishop-Phelps-Bollobás 定理

通过凸函数的 Fréchet 可微性特性表征 Asplund 算子，我们展示了以下 Bishop-Phelps-Bollobés 定理：假设 X 是 Banach 空间，T: X → C ( K ) 是一个 Asplund 算子，其中 ∥T∥ = 1，并且 x 0 ∈ SX , 0 < ε 满足 $$\left\| {T\left({{x_0}} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ ‖ T ( x 0 ) ‖ > 1 − ε 2 2 。那么存在 x ε ∈ SX 和一个 Asplund 算子 S: X → C ( K ) 的范数 1 使得 $$\left\| {S\left({{x_\varepsilon}} \right)} \right\| = 1,\,\,\,\,\,\,\left\| {{x_0} - {x_\varepsilon}} \right\| < \varepsilon \,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\left\| {T - S} \right\| < \varepsilon .$$ ‖ S ( x ε ) ‖ = 1 , ‖ x 0 − x ε ‖ < ε 和‖ T − S ‖ < ε 。利用这个定理，我们进一步展示了强 Radon-Nikodym 算子 T 的 Bishop-Phelps-Bollobas 性质的双重版本： ℓ 1 → Y 范数 1：假设 y 0 * ∈ S y * , ε ≥ 0 满足 $$\left\| {{T^*}\left({y_0^*} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$$ ‖ T * ( y 0 * ) ‖ > 1 − ε 2 2 。那么存在 y * µ ∈ S y * , x ε ∈ (±en ), y ε ∈ S y 和一个强 Radon-Nikodym 算子 S: ℓ 1 → Y 范数 1 使得 $$\begin{array} {*{20}{c}} {(i)}&{\left\| {S({x_\varepsilon })} \right\| = 1;}&{(ii)S({x_\varepsilon }) = {y_\varepsilon };}&{(iii)\left\| {T - S} \right\| < \varepsilon ;} \end{array}$$ $$\begin{array}{*{20}{c}} {(iv)\left\| {{S^*}(y_\varepsilon ^*)} \right\| = \left\langle {y_\varepsilon ^*,{y_\varepsilon }} \right\rangle = 1;}&{(v)\left\| {y_0^* - y_\varepsilon ^*} \right\| < \varepsilon }&{and}& {(vi)\左\| {{T^*} - {S^*}} \right\| < \varepsilon ,} \end{array}$$ 其中 ( en ) 表示 ℓ 1 的标准单位向量基。