# Sebastián Tapia GarcíaTU Wien | TU Wien · Institute of Mathematical Methods in Economics

Sebastián Tapia García

PhD at Universidad de Chile and at Université de Bordeaux

Main research topics: Functional analysis, operator theory, variational analysis

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21

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Introduction

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## Publications

Publications (21)

We construct a differentiable locally Lipschitz function in with the property that for every convex body there exists such that coincides with the set of limits of derivatives of sequences converging to . The technique can be further refined to recover all compact connected subsets with nonempty interior, disclosing an important difference between...

Eikonal equations in metric spaces have strong connections with the local slope operator (or the De Giorgi slope). In this manuscript, we explore and delve into an analogous model based on the global slope operator, expressed as λu + G[u] = ℓ, where λ ≥ 0. In strong contrast with the classical theory, the global slope operator relies neither on the...

A classical result of variational analysis, known as Attouch theorem, establishes an equivalence between epigraphical convergence of a sequence of proper convex lower semicontinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we sho...

We construct a differentiable locally Lipschitz function f in R N with the property that for every convex body K ⊂ R N there existsx ∈ R N such that K coincides with the set ∂ L f (x) of limits of derivatives {Df (x n)} n≥1 of sequences {x n } n≥1 converging tox. The technique can be further refined to recover all compact connected subsets with non...

A classical result of variational analysis, known as Attouch theorem, establishes the equivalence between epigraphical convergence of a sequence of proper convex lower semicon-tinuous functions and graphical convergence of the corresponding subdifferential maps up to a normalization condition which fixes the integration constant. In this work, we s...

Daniilidis and Drusviatskiy, in 2017, extended the celebrated Kurdyka–Łojasiewicz inequality from definable functions to definable multivalued maps by establishing that the coderivative mapping admits a desingularization around every critical value. As was the case in the gradient dynamics, this desingularization yields a uniform control of the len...

We investigate dynamical properties of linear operators that are obtained as the linearization of Lipschitz self-maps defined on a pointed metric space. These operators are known as Lipschitz operators. More concretely, for a Lipschitz operator $\widehat{f}$, we study the set of recurrent vectors and the set of vectors $\mu$ such that the sequence...

In this work we provide a characterization of distinct types of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact,...

In this paper we provide a full characterization of cyclic composition operators defined on the d-dimensional Fock space $\mathcal F(\mathbb C^d)$ in terms of their symbol. Also, we study the supercyclicity and convex-cyclicity of this type of operators. We end this work by computing the approximation numbers of compact composition operators define...

Savin [‘ $\mathcal {C}^{1}$ regularity for infinity harmonic functions in two dimensions’, Arch. Ration. Mech. Anal. 3 (176) (2005), 351–361] proved that every planar absolutely minimizing Lipschitz (AML) function is continuously differentiable whenever the ambient space is Euclidean. More recently, Peng et al . [‘Regularity of absolute minimizers...

This thesis deals with three topics related to linear operators defined on infinite dimensional spaces and two topics of real analysis and variational analysis in finite dimensional spaces.The first chapter contains preliminaries on Banach space theory which will be relevant for the three topics related to linear operators.The second chapter is a c...

In [9], the celebrated K{\L}-inequality has been extended from definable functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R} $ to definable multivalued maps $S:\mathbb{R}\rightrightarrows\mathbb{R}^{n}$, by establishing that the co-derivative mapping $D^{\ast}S$ admits a desingularization around every critical value. As was the case in the gradient dy...

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological and non-linear framework. Restricted to the linear case, we can apply our results to compact, weakly-compact,...

Let T : Y → X be a bounded linear operator between two real normed spaces. We characterize compactness of T in terms of differentiability of the Lipschitz functions defined on X with values in another normed space Z. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider class...

We provide a criterion for $\varepsilon$-hypercyclicity. Also, we extend the ideas of Badea, Grivaux, M\"uller and Bayart to construct $\varepsilon$-hypercyclic operators which are not hypercyclic in a wider class of separable Banach spaces, including several classical Banach spaces. For instance, our result can be applied to separable infinite dim...

Let X X be a separable infinite dimensional (real or complex) Banach space. Hájek and Smith, in 2010, constructed a linear bounded operator T T on X X such that A T ≔ { x ∈ X : ‖ T n x ‖ → ∞ } A_T≔\{x\in X:~ \|T^nx\|\to \infty \} is not dense and has nonempty interior, whenever X X admits a symmetric basis. Augé, in 2012, extended the previous cons...

We study the notion of finitely determined functions defined on a topological vector space E equipped with a biorthogonal system. We prove that, for real-valued convex functions defined on a Banach space with a Schauder basis, the notion of finitely determined function coincides with the classical continuity but outside the convex case there are ma...

Let $T:Y\to X$ be a bounded linear operator between two normed spaces. We characterize compactness of $T$ in terms of differentiability of the Lipschitz functions defined on $X$ with values in another normed space $Z$. Furthermore, using a similar technique we can also characterize finite rank operators in terms of differentiability of a wider clas...

We introduce the notions of extended topological vector spaces and extended seminormed spaces, following the main ideas of extended normed spaces, which were introduced by G. Beer and J. Vanderwerff. We provide a topological study of such structures, giving a unifying theory with main applications in the study of spaces of continuous functions. We...