Searcharxiv⌕ Search

arXiv subjects

Vladimir Kadets

Publications and source records attributed to Vladimir Kadets.

At least 19 recordsLinked to original sources
arXiv2026

Optimality of no-dimensional bounds in Banach spaces

We discuss lower-bound constructions for several no-dimensional theorems of combinatorial geometry in Banach spaces. The common mechanism is the Maurey--Pisier theorem: the supremal Rademacher type of a Banach space forces finite-dimensional \(\ell_p\)-structures, and standard-coordinate configurations in these model spaces give lower bounds for the error terms. For the Helly approximation property the relevant type is the type of the dual space. For colorful Radon, colorful Tverberg, selection, and weak \(\varepsilon\)-net statements the relevant type is the type of the original space. We show that the powers appearing in the no-dimensional Helly, Radon, Tverberg, and selection estimates are optimal at the supremal-type exponent. If the supremal type is attained, the known upper estimates coming from the corresponding type inequalities have the best possible order. We also include endpoint statements for spaces of trivial type. In this case the error terms in the Helly, Radon, Tverberg, and selection statements cannot tend to zero. Finally, we prove an endpoint obstruction for no-dimensional weak \(\eps\)-nets in spaces of trivial type. For every fixed cardinality bound, one can find a finite set in the unit ball for which no approximate weak \(\eps\)-net of that size exists below a fixed positive radius. The proof combines the simplex example in \(\ell_1^N\), the Lov\'asz theorem on the chromatic number of Kneser's graph, and finite representability of \(\ell_1^N\) in spaces of trivial type.

math.FA↗
arXiv2026

Weak minimizing property and reflexivity

For an operator T from X to Y denote m(T) the infimum of $||Tx||$ on the unit sphere $S_X$ of X. A sequence $(x_n)$ in $S_X$ is said to be minimizing for T if $||Tx_n||$ tends to m(T). In 2020 U. S. Chakraborty introduced and studied the following weak minimizing property (WmP): a pair (X,Y) of Banach spaces is said to have the WmP if, for every bounded linear operator $T: X \to Y$ that admits a non-weakly null minimizing sequence, the function $x \mapsto \|Tx\|$ attains its minimum on the unit sphere. We present the following new results about the WmP for pairs of infinite-dimensional separable Banach spaces: (i) If (X,Y) has the WmP, then X is reflexive. (ii) If X is reflexive and Y does not contain isomorphic copies of X, then (X,Y) has the WmP. (iii) If X is reflexive and Y contains an isomorphic copy of X, then there is an equivalent norm on Y such that, for this equivalent norm, (X,Y) does not have the WmP. The first result extends to non-separable X if and only if X possesses a countable total set of functionals.

math.FA↗
arXiv2023

Generating operators between Banach spaces

We introduce and study the notion of generating operators as those norm-one operators $G\colon X\longrightarrow Y$ such that for every $0<δ<1$, the set $\{x\in X\colon \|x\|\leq 1,\ \|Gx\|>1-δ\}$ generates the unit ball of $X$ by closed convex hull. This class of operators includes isometric embeddings, spear operators (actually, operators with the alternative Daugavet property), and other examples like the natural inclusions of $\ell_1$ into $c_0$ and of $L_\infty[0,1]$ into $L_1[0,1]$. We first present a characterization in terms of the adjoint operator, make a discussion on the behaviour of diagonal generating operators on $c_0$-, $\ell_1$-, and $\ell_\infty$-sums, and present examples in some classical Banach spaces. Even though rank-one generating operators always attain their norm, there are generating operators, even of rank-two, which do not attain their norm. We discuss when a Banach space can be the domain of a generating operator which does not attain its norm in terms of the behaviour of some spear sets of the dual space. Finally, we study when the set of all generating operators between two Banach spaces $X$ and $Y$ generates all non-expansive operators by closed convex hull. We show that this is the case when $X=L_1(μ)$ and $Y$ has the Radon-Nikodým property with respect to $μ$. Therefore, when $X=\ell_1(Γ)$, this is the case for every target space $Y$. Conversely, we also show that a real finite-dimensional space $X$ satisfies that generating operators from $X$ to $Y$ generate all non-expansive operators by closed convex hull only in the case that $X$ is an $\ell_1$-space.

math.FA↗
arXiv2023

Plastic pairs of metric spaces

We address pairs $(X, Y)$ of metric spaces with the following property: for every mapping $f: X \to Y$ the existence of points $x, y \in X$ with $d(f(x),f(y)) > d(x,y)$ implies the existence of $\widetilde{x}, \widetilde{y}\in X$ for which $d(f(\widetilde{x}),f(\widetilde{y})) < d(\widetilde{x},\widetilde{y})$. We give sufficient conditions for this property and for its uniform version in terms of finite $\varepsilon$-nets and finite $\varepsilon$-separated subsets.

math.FA↗
arXiv2022

Some remarks on orthogonally additive operators on Riesz spaces

We study orthogonally additive operators between Riesz spaces without the Dedekind completeness assumption on the range space. Our first result gives necessary and sufficient conditions on a pair of Riesz spaces $(E,F)$ for which every orthogonally additive operator from $E$ to $F$ is laterally-to-order bounded. Second result provides sufficient conditions on a pair of orthogonally additive operators $S$ and $T$ to have $S \vee T$, as well as to have $S \wedge T$, and consequently, for an orthogonally additive operator $T$ to have $T^+$, $T^-$ or $|T|$ without any assumption on the domain and range spaces. Finally we prove an analogue of Meyer's theorem on the existence of modules of disjointness preserving operator for the setting of orthogonally additive operators.

math.FA↗
arXiv2022

Closed linear spaces consisting of strongly norm attaining Lipschitz mappings

Given a pointed metric space $M$, we study when there exist $n$-dimensional linear subspaces of $\operatorname{Lip}_0(M)$ consisting of strongly norm-attaining Lipschitz functionals, for $n\in\mathbb{N}$. We show that this is always the case for infinite metric spaces, providing a definitive answer to the question. We also study the possible sizes of such infinite-dimensional closed linear subspaces $Y$, as well as the inverse question, that is, the possible sizes of the metric space $M$ given that such a subspace $Y$ exists. We also show that if the metric space $M$ is $σ$-precompact, then the aforementioned subspaces $Y$ need to be always separable and isomorphically polyhedral, and we show that for spaces containing $[0,1]$ isometrically, they can be infinite-dimensional.

math.FA↗
arXiv2022

Banach actions preserving unconditional convergence

Let $A,X,Y$ be Banach spaces and $A\times X\to Y$, $(a,x)\mapsto ax$, be a continuous bilinear function, called a *Banach action*. We say that this action *preserves unconditional convergence* if for every bounded sequence $(a_n)_{n\inω}$ in $A$ and unconditionally convergent series $\sum_{n\inω}x_n$ in $X$ the series $\sum_{n\inω}a_nx_n$ is unconditionally convergent. We prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if and only if for any linear functional $y^*\in Y^*$ the operator $D_{y^*}:X\to A^*$, $D_{y^*}(x)(a)=y^*(ax)$, is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from $\ell_1$ to $\ell_2$, we prove that a Banach action $A\times X\to Y$ preserves unconditional convergence if $A$ is a Hilbert space possessing an orthonormal basis $(e_n)_{n\inω}$ such that for every $x\in X$ the series $\sum_{n\inω}e_nx$ is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers $p,q,r\in[1,\infty]$ with $\frac1r\le\frac1p+\frac1q$, the coordinatewise multplication $\ell_p\times\ell_q\to\ell_r$ preserves unconditional convergence if and only if one of the following conditions holds: (i) $p\le 2$ and $q\le r$, (ii) $2<p<q\le r$, (iii) $2<p=q<r$, (iv) $r=\infty$, (v) $2\le q<p\le r$, (vi) $q<2<p$ and $\frac1p+\frac1q\ge\frac1r+\frac12$.

math.FA↗
arXiv2021

Completeness in topological vector spaces and filters on N

We study completeness of a topological vector space with respect to different filters on the set N of all naturals. In the metrizable case all these kinds of completeness are the same, but in non-metrizable case the situation changes. For example, a space may be complete with respect to one ultrafilter on N, but incomplete with respect to another. Our study was motivated by [Aizpuru, Listán-García and Rambla-Barreno; Quaest. Math., 2014] and [Listán-García; Bull. Belg. Math. Soc. Simon Stevin, 2016] where for normed spaces the equivalence of the ordinary completeness and completeness with respect to f-statistical convergence was established.

math.FA↗
arXiv2021

Connection between the Riemann integrability of a multi-valued function and of its convex hull

For a Banach space $X$ we demonstrate the equivalence of the following two properties: (1) $X$ is B-convex (that is, possesses a nontrivial infratype), and (2) if ${F: [0,1] \to 2^{X} \setminus \{\varnothing\}}$ is a {multifunction}, $\mathrm{conv} F$ denotes the mapping $t \mapsto \mathrm{conv} F(t)$, then the Riemann integrability of $\mathrm{conv} F$ is equivalent to the Riemann integrability of $F$. For multifunctions with compact values the Riemann integrability of $\mathrm{conv} F$ is equivalent to the Riemann integrability of $F$ without any restrictions on the Banach space $X$.

math.FA↗
arXiv2020

Conglomerated filters, statistical measures, and representations by ultrafilters

Using a new concept of conglomerated filter we demonstrate in a purely combinatorial way that none of Erdös-Ulam filters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as intersections of countable families of ulrafilters. Minimal families of ultrafilters and their intersections are studied and several open questions are discussed.

math.FA↗
arXiv2019

Norm attaining operators of finite rank

We provide sufficient conditions on a Banach space $X$ in order that there exist norm attaining operators of rank at least two from $X$ into any Banach space of dimension at least two. For example, a rather weak such condition is the existence of a non-trivial cone consisting of norm attaining functionals on $X$. We go on to discuss density of norm attaining operators of finite rank among all operators of finite rank, which holds for instance when there is a dense linear subspace consisting of norm attaining functionals on $X$. In particular, we consider the case of Hilbert space valued operators where we obtain a complete characterization of these properties. In the final section we offer a candidate for a counterexample to the complex Bishop-Phelps theorem on $c_0$, the first such counterexample on a certain complex Banach space being due to V. Lomonosov.

math.FA↗
arXiv2019

On Banach spaces whose group of isometries acts micro-transitively on the unit sphere

We study Banach spaces whose group of isometries acts micro-transitively on the unit sphere. We introduce a weaker property, which one-complemented subspaces inherit, that we call uniform micro-semitransitivity. We prove a number of results about both micro-transitive and uniformly micro-semitransitive spaces, including that they are uniformly convex and uniformly smooth, and that they form a self-dual class. To this end, we relate the fact that the group of isometries acts micro-transitively with a property of operators called the pointwise Bishop-Phelps-Bollobás property and use some known results on it. Besides, we show that if there is a non-Hilbertian non-separable Banach space with uniform micro-semitransitive (or micro-transitive) norm, then there is a non-Hilbertian separable one. Finally, we show that an $L_p(μ)$ space is micro-transitive or uniformly micro-semitransitive only when $p=2$.

math.FA↗
arXiv2019

On the numerical index with respect to an operator

Given Banach spaces $X$ and $Y$, and a norm-one operator $G\in \mathcal{L}(X,Y)$, the numerical index with respect to $G$, $n_G(X,Y)$, is the greatest constant $k\geq 0$ such that $$\max_{|w|=1}\|G+wT\|\geq 1 + k \|T\|$$ for all $T\in \mathcal{L}(X,Y)$. We present some results on the set $\mathcal{N}(\mathcal{L}(X,Y))$ of the values of the numerical indices with respect to all norm-one operators on $\mathcal{L}(X,Y)$. We show that $\mathcal{N}(\mathcal{L}(X,Y))=\{0\}$ when $X$ or $Y$ is a real Hilbert space of dimension greater than one and also when $X$ or $Y$ is the space of bounded or compact operators on an infinite-dimensional real Hilbert space. For complex Hilbert spaces $H_1$, $H_2$ of dimension greater than one, we show that $\mathcal{N}(\mathcal{L}(H_1,H_2))\subseteq \{0,1/2\}$ and the value $1/2$ is taken if and only if $H_1$ and $H_2$ are isometrically isomorphic. Besides, $\mathcal{N}(\mathcal{L}(X,H))\subseteq [0,1/2]$ and $\mathcal{N}(\mathcal{L}(H,Y))\subseteq [0,1/2]$ when $H$ is a complex infinite-dimensional Hilbert space and $X$ and $Y$ are arbitrary complex Banach spaces. We also show that $\mathcal{N}(\mathcal{L}(L_1(μ_1),L_1(μ_2)))\subseteq \{0,1\}$ and $\mathcal{N}(\mathcal{L}(L_\infty(μ_1),L_\infty(μ_2)))\subseteq \{0,1\}$ for arbitrary $σ$-finite measures $μ_1$ and $μ_2$, in both the real and the complex cases. Also, we show that the Lipschitz numerical range of Lipschitz maps can be viewed as the numerical range of convenient bounded linear operators with respect to a bounded linear operator. Further, we provide some results which show the behaviour of the value of the numerical index when we apply some Banach space operations, as constructing diagonal operators between $c_0$-, $\ell_1$-, or $\ell_\infty$-sums of Banach spaces, composition operators on some vector-valued function spaces, and taking the adjoint to an operator.

math.FA↗
arXiv2019

Generalized-lush spaces revisited

We study geometric properties of GL-spaces. We demonstrate that every finite-dimensional GL-space is polyhedral; that in dimension 2 there are only two, up to isometry, GL-spaces, namely the space whose unit sphere is a square (like $\ell_\infty^2$ or $\ell_1^2$) and the space whose unit sphere is an equilateral hexagon. Finally, we address the question what are the spaces $E = (\R^n, \|\cdot\|_E)$ with absolute norm such that for every collection $X_1, \ldots, X_n$ of GL-spaces their $E$-sum is a GL-space.

math.FA↗
arXiv2018

There is no operatorwise version of the Bishop-Phelps-Bollobás property

Given two real Banach spaces $X$ and $Y$ with dimensions greater than one, it is shown that there is a sequence $\{T_n\}_{n\in \mathbb{N}}$ of norm attaining norm-one operators from $X$ to $Y$ and a point $x_0\in X$ with $\|x_0\|=1$, such that $\|T_n(x_0)\|\longrightarrow 1$ but $\inf_{n \in \mathbb{N}} \{\mbox{dist} (x_0,\,\{x\in X: \|T_n(x)\|=\|x\|=1\})\} >0.$ This shows that a version of the Bishop-Phelps-Bollobás property in which the operator is not changed is possible only if one of the involved Banach spaces is one-dimensional.

math.FA↗
arXiv2018

On the pointwise Bishop--Phelps--Bollobás property for operators

We study approximation of operators between Banach spaces $X$ and $Y$ that nearly attain their norms in a given point by operators that attain their norms at the same point. When such approximations exist, we say that the pair $(X, Y)$ has the pointwise Bishop-Phelps-Bollobás property (pointwise BPB property for short). In this paper we mostly concentrate on those $X$, called universal pointwise BPB domain spaces, such that $(X, Y)$ possesses pointwise BPB property for every $Y$, and on those $Y$, called universal pointwise BPB range spaces, such that $(X, Y)$ enjoys pointwise BPB property for every uniformly smooth $X$. We show that every universal pointwise BPB domain space is uniformly convex and that $L_p(μ)$ spaces fail to have this property when $p>2$. For universal pointwise BPB range space, we show that every simultaneously uniformly convex and uniformly smooth Banach space fails it if its dimension is greater than one. We also discuss a version of the pointwise BPB property for compact operators.

math.FA↗
arXiv2018

Non-expansive bijections, uniformities and polyhedral faces

We extend the result of B. Cascales at al. about expand-contract plasticity of the unit ball of strictly convex Banach space to those spaces whose unit ball is the union of all its finite-dimensional polyhedral extreme subsets. We also extend the definition of expand-contract plasticity to uniform spaces and generalize the theorem on expand-contract plasticity of totally bounded metric spaces to this new setting.

math.FA↗