A Bishop-Phelps-Bollobás Theorem for Asplund Operators

Abstract

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 x₀ ∈ S_X, 0 < ε satisfy $\left\| {T\left({{x_0}} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$. Then there exist x_ε ∈ S_X and an Asplund operator S: X → C(K) of norm one so that

$$\left\| {S\left({{x_\varepsilon}} \right)} \right\| = 1,\,\,\,\,\,\,\left\| {{x_0} - {x_\varepsilon}} \right\| < \varepsilon \,\,\,\,\,{\rm{and}}\,\,\,\,\,\,\,\left\| {T - S} \right\| < \varepsilon .$$

Making use of this theorem, we further show a dual version of Bishop-Phelps-Bollobas property for a strong Radon-Nikodym operator T: ℓ₁ → Y of norm one: Suppose that y₀* ∈ S_y*, ε ≥ 0 satisfy $\left\| {{T^*}\left({y_0^*} \right)} \right\| > 1 - {{{\varepsilon^2}} \over 2}$. Then there exist y*_µ ∈ S_y*, x_ε∈ (±e_n), y_ε∈S_y, and a strong Radon-Nikodym operator S: ℓ₁→ 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 ℓ₁.