2020/10/01 by Sheldon Dantas, Sun Kwang Kim, Dantas, Sheldon +5
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2010.00280
openalex publication_date 2020/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we are interested in studying two properties related to the denseness of the operators which attain their numerical radius: the Bishop-Phelps-Bollobás point and operator properties for numerical radius (BPBpp-nu and BPBop-nu, respectively). We prove that every Banach space with micro-transitive norm and second numerical index strictly positive satisfy the BPBpp-nu and that, if the numerical index of X is 1, only one-dimensional spaces enjoy it. On the other hand, we show that the BPBop-nu is a very restrictive property: under some general assumptions, it holds only for one-dimensional spaces. We also consider two weaker properties, the local versions of BPBpp-nu and BPBop-nu, where the η which appears in their definition does not depend just on ε> 0 but also on a state (x, x^*) or on a numerical radius one operator T. We address the relation between the local BPBpp-nu and the strong subdifferentiability of the norm of the space X. We show that finite dimensional spaces and c0 are examples of Banach spaces satisfying the local BPBpp-nu, and we exhibit an example of a Banach space with strongly subdifferentiable norm failing it. We finish the paper by showing that finite dimensional spaces satisfy the local BPBop-nu and that, if X has strictly positive numerical index and has the approximation property, this property is equivalent to finite dimensionality.