Searcharxiv⌕ Search

arXiv subjects

Christian Bargetz

Publications and source records attributed to Christian Bargetz.

At least 19 recordsLinked to original sources
arXiv2026

Some observations on transitivity of Lipschitz operators

The universal property of the Lipschitz-free spaces allows for the linearisation of a base point preserving Lipschitz map $f\colon M\to M$ to a bounded linear operator $T_f\colon \mathcal{F}(M) \to \mathcal{F}(M)$. While it is known that the operator $T_f$ is weakly mixing whenever $f$ has this property, it is open whether a similar inheritance result for topological transitivity holds. Motivated by this question we investigate properties of $T_f$ under the assumption that $f$ is topologically transitive. In particular we prove Kitai's theorem for Lipschitz operators on connected spaces, i.e. we show that every connected component of the complex variant of $T_f$ intersects the unit circle if $f$ is topologically transitive.

math.FA↗
arXiv2025

On extremal nonexpansive mappings

We study the extremality of nonexpansive mappings on a nonempty bounded closed and convex subset of a normed space (therein specific Banach spaces). We show that surjective isometries are extremal in this sense for many Banach spaces, including Banach spaces with the Radon-Nikodym property and all $C(K)$-spaces for compact Hausdorff $K$. We also conclude that the typical, in the sense of Baire category, nonexpansive mapping is close to being extremal.

math.FA↗
arXiv2024

Continuous operators from spaces of Lipschitz functions

We study the existence of continuous (linear) operators from the Banach spaces $\mbox{Lip}_0(M)$ of Lipschitz functions on infinite metric spaces $M$ vanishing at a distinguished point and from their predual spaces $\mathcal{F}(M)$ onto certain Banach spaces, including $C(K)$-spaces and the spaces $c_0$ and $\ell_1$. For pairs of spaces $\mbox{Lip}_0(M)$ and $C(K)$ we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. It is also showed that if a metric space $M$ contains a bilipschitz copy of the unit sphere $S_{c_0}$ of the space $c_0$, then $\mbox{Lip}_0(M)$ admits a continuous operator onto $\ell_1$ and hence onto $c_0$. Using this, we provide several conditions for a space $M$ implying that $\mbox{Lip}_0(M)$ is not a Grothendieck space. Finally, we obtain a new characterization of the Schur property for Lipschitz-free spaces: a space $\mathcal{F}(M)$ has the Schur property if and only if for every complete discrete metric space $N$ with cardinality $d(M)$ the spaces $\mathcal{F}(M)$ and $\mathcal{F}(N)$ are weakly sequentially homeomorphic.

math.FA↗
arXiv2024

Homogeneous isosceles-free spaces

We study homogeneity aspects of metric spaces in which all triples of distinct points admit pairwise different distances; such spaces are called isosceles-free. In particular, we characterize all homogeneous isosceles-free spaces up to isometry as vector spaces over the two-element field, endowed with an injective norm. Using isosceles-free decompositions, we provide bounds on the maximal number of distances in arbitrary homogeneous finite metric spaces.

math.LO↗
arXiv2024

Tilings of the Hyperbolic Space and Lipschitz Functions

We use a special tiling for the hyperbolic $d$-space $\mathbb{H}^d$ for $d=2,3,4$ to construct an (almost) explicit isomorphism between the Lipschitz-free space $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(P)\oplus\mathcal{F}(\mathcal{N})$ where $P$ is a polytope in $\mathbb{R}^d$ and $\mathcal{N}$ a net in $\mathbb{H}^d$ coming from the tiling. This implies that the spaces $\mathcal{F}(\mathbb{H}^d)$ and $\mathcal{F}(\mathbb{R}^{d})\oplus \mathcal{F}(\mathcal{M})$ are isomorphic for every net $\mathcal{M}$ in $\mathbb{H}^d$. In particular, we obtain that, for $d=2,3,4$, $\mathcal{F}(\mathbb{H}^d)$ has a Schauder basis. Moreover, using a similar method, we also give an explicit isomorphism between $\mathrm{Lip}(\mathbb{H}^{d})$ and $\mathrm{Lip}(\mathbb{R}^d)$.

math.FA↗
arXiv2023

On complemented copies of the space $c_0$ in spaces $C_p(X,E)$

We study the question for which Tychonoff spaces $X$ and locally convex spaces $E$ the space $C_p(X,E)$ of continuous $E$-valued functions on $X$ contains a complemented copy of the space $(c_0)_p=\{x\in\mathbb{R}^ω\colon x(n)\to0\}$, both endowed with the pointwise topology. We provide a positive answer for a vast class of spaces, extending classical theorems of Cembranos, Freniche, and Domański and Drewnowski, proved for the case of Banach and Fréchet spaces $C_k(X,E)$. Also, for given infinite Tychonoff spaces $X$ and $Y$, we show that $C_p(X,C_p(Y))$ contains a complemented copy of $(c_0)_p$ if and only if any of the spaces $C_p(X)$ and $C_p(Y)$ contains such a subspace.

math.FA↗
arXiv2023

Observations on the metric projection in finite dimensional Banach spaces

We consider the method of alternating (metric) projections for pairs of linear subspaces of finite dimensional Banach spaces. We investigate the size of the set of points for which this method converges to the metric projection onto the intersection of these subspaces. In addition we give a characterisation of the pairs of subspaces for which the alternating projection method converges to the projection onto the intersection for every initial point. We provide a characterisation of the linear subspaces of $\ell_p^n$, $1<p<\infty$, $p\neq 2$, which admit a linear metric projection and use this characterisation to show that in $\ell_p^3$, $1<p<\infty$, $p\neq 2$, the set of pairs of subspaces for which the alternating projection method converges to the projection onto the intersection is small in a probabilistic sense.

math.FA↗
arXiv2023

Generic properties of nonexpansive mappings on unbounded domains

We investigate typical properties of nonexpansive mappings on unbounded complete hyperbolic metric spaces. For two families of metrics of uniform convergence on bounded sets, we show that the typical nonexpansive mapping is a Rakotch contraction on every bounded subset and that there is a bounded set which is mapped into itself by this mapping. In particular, we obtain that the typical nonexpansive mapping in this setting has a unique fixed point which can be reached by iterating the mapping. Nevertheless, it turns out that the typical mapping is not a Rakotch contraction on the whole space and that it has the maximal possible Lipschitz constant of one on a residual subset of its domain. By typical we mean that the complement of the set of mappings with this property is $σ$-$ϕ$-porous, that is, small in a metric sense. For a metric of pointwise convergence, we show that the set of Rakotch contractions is meagre.

math.FA↗
arXiv2023

On generic convergence of successive approximations of mappings with convex and compact point images

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

math.FA↗
arXiv2022

Sequence space representations for spaces of smooth functions and distributions via Wilson bases

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram.

math.FA↗
arXiv2022

Projective descriptions of spaces of functions and distributions

We present projective descriptions of classical spaces of functions and distributions. More precisely, we provide descriptions of these spaces by semi-norms which are defined by a combination of classical norms and multiplication or convolution with certain functions. These seminorms are simpler than the ones given by a supremum over bounded or compact sets.

math.FA↗
arXiv2021

On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature

We show that the typical nonexpansive mapping on a small enough subset of a CAT($κ$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $σ$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.

math.FA↗
arXiv2020

On angles, projections and iterations

We investigate connections between the geometry of linear subspaces and the convergence of the alternating projection method for linear projections. The aim of this article is twofold: in the first part, we show that even in Euclidean spaces the convergence of the alternating method is not determined by the principal angles between the subspaces involved. In the second part, we investigate the properties of the Oppenheim angle between two linear projections. We discuss, in particular, the question of existence and uniqueness of "consistency projections" in this context.

math.FA↗
arXiv2019

On the rate of convergence of iterated Bregman projections and of the alternating algorithm

We study the alternating algorithm for the computation of the metric projection onto the closed sum of two closed subspaces in uniformly convex and uniformly smooth Banach spaces. For Banach spaces which are convex and smooth of power type, we exhibit a condition which implies linear convergence of this method. We show these convergence results for iterates of Bregman projections onto closed linear subspaces. Using an intimate connection between the metric projection onto a closed linear subspace and the Bregman projection onto its annihilator, we deduce the convergence rate results for the alternating algorithm from the corresponding results for the iterated Bregman projection method.

math.FA↗
arXiv2018

A simpler description of the $κ$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$, $\mathscr{M}^1$

The $κ$-topologies on the spaces $\mathscr{D}_{L^p}$, $L^p$ and $\mathscr{M}^1$ are defined by a neighbourhood basis consisting of polars of absolutely convex and compact subsets of their (pre-)dual spaces. In many cases it is more convenient to work with a description of the topology by means of a family of semi-norms defined by multiplication and/or convolution with functions and by classical norms. We give such families of semi-norms generating the $κ$-topologies on the above spaces of functions and measures defined by integrability properties. In addition, we present a sequence-space representation of the spaces $\mathscr{D}_{L^p}$ equipped with the $κ$-topology, which complements a result of J.~Bonet and M.~Maestre. As a byproduct, we give a characterisation of the compact subsets of the spaces $\mathscr{D}'_{L^p}$, $L^p$ and $\mathscr{M}^1$.

math.FA↗
arXiv2018

Linear Convergence Rates for Extrapolated Fixed Point Algorithms

We establish linear convergence rates for a certain class of extrapolated fixed point algorithms which are based on dynamic string-averaging methods in a real Hilbert space. This applies, in particular, to the extrapolated simultaneous and cyclic cutter methods. Our analysis covers the cases of both metric and subgradient projections.

math.OC↗
arXiv2017

Pivot duality of universal interpolation and extrapolation spaces

It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the shift semigroup acting on L^2(R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff, already complete, and can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space.

math.FA↗