Geometric structures in banach spaces: differentiability, operators and their dynamics

• Autores: Christian Cobollo Gómez
• Directores de la Tesis: Antonio José Guirao Sánchez (dir. tes.) , Alfredo Peris Manguillot (dir. tes.) , Vicente Montesinos Santalucía (dir. tes.)
• Lectura: En la Universitat Politècnica de València ( España ) en 2025
• Idioma: inglés
• Tribunal Calificador de la Tesis: José Antonio Bonet Solves (presid.) , Javier Merí (secret.) , Elisabetta M. Mangino (voc.)
• Enlaces
  • Tesis en acceso abierto en: RiuNet
• Resumen

  • The present thesis, Geometric Structures in Banach Spaces: Differentiability, Operators, and their Dynamics, aims to study specific interactions among the geometric, topological, and linear structures inherent to Banach spaces in order to address a selection of problems and research directions. Specifically, these techniques allow us to tackle questions about norm differentiability, the existence of certain types of operators and their behavior, with particular interest in these situations within Lipschitz-free spaces.

    According to the regulations established by the Doctoral School of the Universitat Politècnica de València, this dissertation has been written as a compendium of articles, following the structure outlined below:

    - Introduction: This preliminary chapter serves multiple purposes, aiming to contextualize and motivate the reader while presenting the other chapters, functioning as a narrative thread for the work.

    1. Some remarks on Phelps property U of a Banach space into C(K)spaces adapts the author version of the article [CGM24a]. Motivated by R. R. Phelps's work and our earlier contributions [Cob20; CGM20], this chapter introduces U- and wU-embeddings as isometries whose range satisfies Phelps's property U or its weaker form, wU. A systematic study is initiated, with a subsequent focus on the existence of such operators when the range is a C(K) space.

    2. Octahedrality and Gâteaux smoothness adapts the authors' version of the article [CH25], which addresses an open renorming problem from the 1990s: obtaining norms that are both Gâteaux differentiable and octahedral simultaneously. This question has been posed several times for separable spaces. Through a new geometric construction, this renorming is achieved for any space admitting an equivalent Gâteaux smooth norm and containing a complemented copy of l1.

    3. On the strongly subdifferentiable points in Lipschitz-free spaces adapts the au- thor's version of the article [Cob+25], which studies the strong subdifferentiability (SSD) property of the norm in Lipschitz-free spaces. Sufficient conditions are obtained, depending on the underlying metric M, for the space F(M) to have SSD points and to ensure the denseness of the set of all SSD points under suitable conditions.

    4. The numerical index of 2-dimensional Lipschitz-free spaces is an adaptation of the authors' version of the article [CGM24b], where a systematic study of the numerical index of Lipschitz-free spaces begins, focusing on the 2-dimensional case. An explicit formula for its calculation in terms of the underlying metric M is obtained, along with a construction of operators achieving the minimum possible numerical radius. Among its applications, previous results by Martín and Merí [MM07] are extended, providing a method to calculate the numerical index of any hexagonal norm in the plane.

    5. On disjoint dynamical properties and Lipschitz-free spaces adapts the authors' version of the article [CP25]. The chapter explores the consequences of linear operator dynamics by studying the underlying nonlinear dynamical systems derived from restrict- ing the former. The notion of disjoint A-transitivity is introduced and studied, offering methods to derive this property from nonlinear systems. Specifically, results are provided for obtaining (or inheriting) these properties in Lipschitz-free operators Tf , based on the behavior of the Lipschitz functions f.

    - General discussion of the results: The results obtained are discussed, along with their implications and the future research directions they suggest.

    - Conclusions: The conclusions drawn from this doctoral thesis are synthesized.

    Additionally, the dissertation includes a General Bibliography, an Index of Symbols, a Term Index, and a List of Figures.