Óscar Roldán

• University of Valencia

In this article, we study the ccs-Daugavet, ccs-$\Delta$, super-Daugavet, super-$\Delta$, Daugavet, $\Delta$, and $\nabla$ points in the unit balls of vector-valued function spaces $C_0(L, X)$, $A(K, X)$, $L_\infty(\mu, X)$, and $L_1(\mu, X)$. To partially or fully characterize these diametral points, we first provide improvements of several stabil...

We study the set MA(X,Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text {MA}}(X,Y)$$\end{document} of operators between Banach spaces X and Y that attain their m...

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 ℓ ∞ , and moreover, those functions can be chosen not to attain their n...

Motivated by the result [15] that there exist metric spaces for which the set of strongly norm-attaining Lipschitz functions does not contain an isometric copy of c_0, we introduce and study a weaker notion of norm-attainment for Lipschitz functions called the pointwise norm-attainment. As a main result, we show that for every infinite metric space...

Given a pointed metric space M , we study when there exist n -dimensional linear subspaces of $$\mathrm {Lip}_0(M)$$ Lip 0 ( M ) consisting of strongly norm-attaining Lipschitz functionals, for $$n\in {\mathbb {N}}$$ n ∈ N . We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study...

In this paper, we provide an infinite metric space M such that the set SNA(M) of strongly norm-attaining Lipschitz functions does not contain a subspace which is isometric to c_0. This answers a question posed by Antonio Avilés, Gonzalo Martínez Cervantes, Abraham Rueda Zoca, and Pedro Tradacete. On the other hand, we prove that SNA(M) contains an...

In this survey, we provide an overview from 2008 to 2021 about the Bishop–Phelps–Bollobás theorem.KeywordsNorm attaining operatorsBishop–Phelps theoremBishop–Phelps–Bollobás property

Given a pointed metric space $M$, we study when we can form non-trivial linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. As a side effect, we obtain the existence of isometric copi...

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators N(X,Y) and in the projective tensor product space X⊗^πY. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N(X...

En el presente trabajo se describen los resultados de un estudio exploratorio cuyo objetivo es identificar características de estudiantes de talento matemático cuando se enfrentan a una secuencia de problemas de solución múltiple. Para abordar este propósito, se han seleccionado dos muestras de alumnos: una formada por estudiantes de entre 15 y 17...

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 pa...

In this paper, we are interested in studying the set \(\mathcal {A}_{\Vert \cdot \Vert }(X, Y)\) of all norm-attaining operators T from X into Y satisfying the following: given \(\varepsilon >0\), there exists \(\eta \) such that if \(\Vert Tx\Vert > 1 - \eta \), then there is \(x_0\) such that \(\Vert x_0 - x\Vert < \varepsilon \) and T itself att...

We study the Bishop-Phelps-Bollob\'as 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...

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators N(X,Y) and in the projective tensor product space X\pten Y. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in...

In this paper, we are interested in studying the set $\Ano$ of all norm-attaining operators $T$ from $X$ into $Y$ satisfying the following: given $\e>0$, there exists $\eta$ such that if $\|Tx\| > 1 - \eta$, then there is $x_0$ such that $\| x_0 - x\| < \e$ and $T$ itself attains its norm at $x_0$. We show that every norm one functional on $c_0$ wh...