In this paper, we are interested in studying Bishop–Phelps–Bollobás type properties related to the denseness of the operators which attain their numerical radius. We prove that every Banach space with a micro-transitive norm and the second numerical index strictly positive satisfies the Bishop–Phelps–Bollobás point property, and we see that the one-dimensional space is the only one with both the numerical index 1 and the Bishop–Phelps–Bollobás point property. We also consider two weaker properties L${}_{p,p}$-nu and L${}_{o,o}$-nu, the local versions of Bishop–Phelps–Bollobás point and operator properties respectively, where the η which appears in their definition does not depend just on $\epsilon >0$ but also on a state $(x,x^{\ast })$ or on a numerical radius one operator T. We address the relation between the L${}_{p,p}$-nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and $c_0$ are examples of Banach spaces satisfying the L${}_{p,p}$-nu, and we exhibit an example of a Banach space with a strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the L${}_{o,o}$-nu and that, if X has a strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.

在本文中，我们有兴趣研究 Bishop–Phelps–Bollobás 类型属性，这些属性与达到其数值半径的算子的密度有关。我们证明每个具有微传递范数和第二个数值索引严格为正的 Banach 空间都满足 Bishop-Phelps-Bollobás 点属性，并且我们看到一维空间是唯一具有数值索引 1 和Bishop–Phelps–Bollobás 点属性。我们还考虑了两个较弱的属性L${}_{p,p}$-nu 和L${}_{o,o}$-nu，分别是 Bishop–Phelps–Bollobás 点和运算符属性的本地版本，其中出现在它们定义中的η不仅仅取决于$\epsilon >0$但也对一个国家$(X,X^{＊})$或在数值半径上一个运算符T。我们解决L之间的关系${}_{p,p}$-nu 和空间X范数的强子可微性。我们证明有限维空间和$C_0$是满足L的 Banach 空间的例子${}_{p,p}$-nu，我们展示了一个 Banach 空间的例子，它有一个强可微范数失败。我们通过证明有限维空间满足L${}_{o,o}$-nu 并且，如果X具有严格的正数值索引并具有近似属性，则该属性等效于有限维数。