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Giovanni Parzanese - Denting Points of Convex Sets and Weak Property ($\pi$) of Cones in Locally Convex Spaces
Universidad de Alicante
+ info The notion of denting point goes back to the early studies of sets with the Radon-Nikodým property with applications to renorming theory and optimization. A remarkable result by L. Lin, P. K. Lin, and S. Troyanski in [2] states that the notions of denting point and point of continuity become equivalent at extreme points of closed, convex, and bounded subsets of Banach spaces. In [1], X. Gong asked if, in the context of general normed spaces, the vertex of a closed cone having a base is a denting point if and only if it is a point of continuity, question directly related to vector optimization and answered in the negative by Song in [3].
In this talk I will show a generalization of Lin-Lin-Troyanski's theorem for locally convex spaces. The corresponding version for normed spaces states that the class of cones in normed spaces having a pointed completion is the largest one for which the vertex is a denting point if and only if it is a point of continuity. I will also show some new characterizations of the weak property ($\pi$) of cones and the angle property in the context of locally convex spaces, both notions closely related to dentability and having applications to vector optimization.
These results are part of my doctoral thesis and also of a joint work (currently under review) with Fernando García-Castaño and M. A. Melguizo Padial. Department of Applied Mathematics. University of Alicante.
- X. H. Gong, Density of the Set of Positive Proper Minimal Points in the Set of Minimal Points, Journal of Optimization Theory and Applications, 86(3) (1995) 609-630.
- B. Lin, P. K. Lin, S. Troyanski, A characterization of denting points of a closed, bounded, convex set, Longhorn Notes, Y. T. Functional Analysis Seminar, The University of Texas, Austin, (1985-1986) 99-101.
- W. Song, Characterizations of some remarkable Classes of Cones, Journal of Mathematical Analysis and Applications 279 (1) (2003) 308-316.
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Mingu Jung - An improvement of a Theorem of Holub and Mujica
Pohang University Of Science and Technology
+ info In this talk we provide necessary and sufficient conditions for the existence of non-norm-attaining operators in $\mathcal{L}(E, F)$. By using a theorem due to Pfitzner on James boundaries, we show that if there exists a relatively compact set $K$ of $\mathcal{L}(E, F)$ (in the weak operator topology) such that $0$ is an element of its closure (in the weak operator topology) but it is not in its norm closed convex hull, then we can guarantee the existence of an operator which does not attain its norm. This allows us to provide the following generalization of results due to Holub and Mujica. If $E$ is a reflexive space, $F$ is an arbitrary Banach space, and the pair $(E, F)$ has the bounded compact approximation property, then the following are equivalent:- (i)$\mathcal{K}(E, F) = \mathcal{L}(E, F)$;
- (ii) Every operator from $E$ into $F$ attains its norm;
- (iii) $(\mathcal{L}(E,F), \tau_c)^* = (\mathcal{L}(E, F), \Vert\cdot\Vert)^*$;
where $\tau_c$ denotes the topology of compact convergence. We conclude the talk presenting a characterization of the Schur property in terms of norm-attaining operators.
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María Luisa Castillo Godoy - Espacios $L^p$ no conmutativos y algunos problemas relacionados con la ortogonalidad
Universidad de Granada
+ info Es relativamente sencillo definir espacios $L^p$ asociados a un Álgebra de von Neumann semifinita, ya que el hecho de que sea semifinita implica la existencia de una aplicación con propiedades similares a las de una integral. Cuando el Álgebra no es semifinita, la definición de estos espacios se debe a Haagerup y es bastante más complicada.
En esta charla hablaremos de estos dos tipos de espacios $L^p$ no conmutativos. En este contexto, resolveremos parcialmente dos problemas: la representación lineal de polinomios ortogonalmente aditivos y la reflexividad o hiperreflexividad del espacio de los homomorfismos de módulos.
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Óscar Roldán Blay - Norm attaining tensors and nuclear operators
Universitat de València
+ info Let $X$ and $Y$ be Banach spaces. We introduce and study a concept of norm-attainment in the space of nuclear operators $\mathcal N(X,Y)$ and in the projective tensor product space $X\widehat{\otimes}_\pi Y$. We provide examples where those norm-attainments hold as well as negative examples. We also study the problem of whether or not the class of elements which attain their norms in $\mathcal N(X,Y)$ or in $X\widehat{\otimes}_\pi Y$, is dense. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces $X$ and $Y$ which, in particular, covers all classical Banach spaces (for instance, $L_p$ spaces, $L_1$ predual spaces, Banach spaces with a monotone Schauder basis, etc.). Nevertheless, we present Banach spaces $X$ and $Y$ such that the class of elements in $X\widehat{\otimes}_\pi Y$ which attain their projective norms is not dense. This talk is based on a joint work with Sheldon Dantas, Mingu Jung and Abraham Rueda Zoca.
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Daniel L. Rodriguez-Vidanes - Lineability on the non-Archimedean setting
Universidad Complutense de Madrid
+ info The branch of mathematics known as lineability focuses on the search of large linear spaces within nonlinear subsets of given linear spaces. In the last two decades, the main line of research of lineability has been done on sets defined over the fields of real or complex numbers endowed with the classical absolute value, that is, over Archimedean fields. However, in the last year, various authors have started to search for linear spaces on sets defined over other fields such as the field of $p$-adic numbers (a non-Archimedean field). In this talk, we will analyze the study of lineability when we consider non-Archimedean fields. -
Jesús Oliva - Range spaces of Hardy-Hausdorff operators of real and complex variable as reproducing kernel Hilbert spaces
Universidad de Zaragoza
+ info The study of Hardy-Hausdorff operators sets off back to the Hausdorff summability method. Later on, this family of operators was studied by G. H. Hardy on the $L^p(\mathbb{R}^+)$ spaces from a different viewpoint, and it has been recently studied on the holomorphic Hardy spaces on the half-plane $H^p(\mathbb{C}^+)$.
In this talk, I present our recent work on the range spaces of Hardy-Hausdorff operators on classical Hilbertian $L^2(\mathbb{R}^+)$ and Hardy $H^2(\mathbb{C}^+)$ spaces. All these operators can be described as integral operators given by Hardy kernels, which entail an algebraic $L^1$-structure which allows us to study the range spaces of those operators as reproducing kernel Hilbert spaces.
In our work, we are able to obtain the reproducing kernels of these spaces, which turn out to be Hardy kernels as well in the $L^2(\mathbb{R}^+)$ case. In the $H^2(\mathbb{C}^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results of this research are a theorem of Paley-Wiener type, and some connections with the semi-infinite Hilbert transform.
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María de Nazaret Cueto Avellaneda - Isometries on remarkable subsets of the unit sphere
Universidad de Granada
+ info The Mazur-Ulam theorem, stated in 1932, assures that every surjective isometry between two real normed spaces is affine. This statement can be reformulated by affirming that the algebraic structure of a real normed space is determined by its underlying metric space. Among the subsequent generalisations derived from this theorem, we highlight the research line initiated by P. Mankiewicz in 1972, who focused his attention on whether an isometry $T:U\to Y$ from a subset $U\subseteq X$ of a real normed space $X$ into a real normed space $Y$ admits an extension to an isometry from $X$ onto $Y$. This talk is intended to review the state-of-the-art of this isometric extension problem and its different variations, seeking to optimise the subset $U$. -
Christian Cobollo Gómez - A note on Hahn–Banach extensions: uniqueness and renormings
Universidad Politécnica de Valencia
+ info Sullivan (1970) and more recently, Oja, Viil, and Werner (2019), provided a renorming of a Banach space $X$ in order to improve the property of uniqueness of Hahn--Banach ---i.e., norm-preserving--- extension to its bidual $X^{**}$ of linear functionals defined on $X$.
Following Phelps (1960), we say that a linear subspace $M$ has property U in $X$ if every continuous linear functional $f:M\longrightarrow \mathbb{R}$ has a unique Hahn--Banach extension $\tilde f : X \longrightarrow \mathbb{R}$. Later, Sullivan and Smith (1977) defined the property of $X$ being a Hahn--Banach Smooth (for short, HBS) space as having property U in $X^{**}$. A strictly stronger property (Taylor, 1939, Foguel, 1958) ---called Totally Smoothness (for short, TS)---, asks for {\em every} subspace of $X$ having property U in $X^{**}$. The aforementioned results of Sullivan on one side and of Oja, Viil and Werner on the other, show that, under some topological conditions on $X$ (separability and weakly compact generation, respectively), a HBS Banach space has a TS renorming.
Along this talk, we will study the relation between these unique extension properties and the differentiability and convexity of the norm, and we shall provide an improvement of the quoted renorming results by showing that {\em no extra assumptions on the space} are needed for such a renorming besides the existence of an HBS norm. This is a joint work with A.J. Guirao and V. Montesinos.
