arXiv subjects

Oscar Roldan

Publications and source records attributed to Oscar Roldan.

3 recordsLinked to original sources

arXiv2026

The Bishop-Phelps-Bollob\'as property for the numerical radius: a Zizler-type approach

We investigate the Bishop-Phelps-Bollob\'as property for the numerical radius (BPBp-nu) through a Zizler-type perspective on the classical Bishop-Phelps-Bollob\'as property (BPBp). This approach allows us to establish two new results: the real Banach space $\ell_\infty$ satisfies the BPBp-nu, while the complex space $\ell_1 \oplus_\infty c_0$ does not. Note that the latter provides the first natural example (constructed without renorming techniques) of a Banach space where the numerical radius attaining operators are dense but the BPBp-nu fails. Along the way, we strengthen the main results of the paper [Kim et al, On the Bishop-Phelps-Bollob\'as theorem for operators and numerical radius, Studia Math., 2016] concerning the interplay between the BPBp for the pair $(X,Y)$ and the BPBp-nu for a direct sum $X\oplus Y$ of Banach spaces. We further explore the validity of the Zizler-type BPBp across different pairs of Banach spaces, and how this property relates to the classical BPBp and the BPBp-nu. Finally, we specialize our analysis to the framework of compact operators.

math.FA↗

arXiv2024

Linear structures of norm-attaining Lipschitz functions and their complements

We solve two main questions on linear structures of (non-)norm-attaining Lipschitz functions. First, we show that for every infinite metric space $M$, the set consisting of Lipschitz functions on $M$ which do not strongly attain their norm and the zero contains an isometric copy of $\ell_\infty$, and moreover, those functions can be chosen not to attain their norm as functionals on the Lipschitz-free space over $M$. Second, we prove that for every infinite metric space $M$, neither the set of strongly norm-attaining Lipschitz functions on $M$ nor the union of its complement with zero is ever a linear space. Furthermore, we observe that the set consisting of Lipschitz functions which cannot be approximated by strongly norm-attaining ones and the zero element contains $\ell_\infty$ isometrically in all the known cases. Some natural observations and spaceability results are also investigated for Lipschitz functions that attain their norm in one way but do not in another, for several norm-attainment notions considered in the literature.

math.FA↗

arXiv2021

On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius

We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\ell_1$ have the BPBp-nu for compact operators.

math.FA↗