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Geivison Ribeiro

Publications and source records attributed to Geivison Ribeiro.

16 recordsLinked to original sources
arXiv2026

Optimality in a Multilinear Extension of Kwapie\'n's Theorem

Bayart, Pellegrino and Rueda proved that every continuous $m$-linear operator from $(\ell_1)^m$ into $\ell_p$ is absolutely $(r_{m,p};1)$-summing, for explicit exponents $r_{m,p}$. The optimality of these exponents was known for $2\le p\le\infty$. We prove optimality in the remaining range $1\le p<2$. Our argument is finite-dimensional and is based on convolution on $\mathbb F_2^d$ and the Walsh character system. The same construction yields a local obstruction theorem for range spaces containing $\ell_p^n$ uniformly. As applications, we recover the sharp exponent $2/m$ for $L_1[0,1]$-valued mappings and show that cotype alone does not determine the optimal absolute $(r;1)$-coincidence exponent.

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arXiv2026

On the $(n,\aleph_0)$-Lineability of Nowhere Differentiable Functions

We establish a countable-dimensional extension property for the family of continuous nowhere differentiable functions, which we denote by $\ND[0,1]$. More precisely, we prove that every finite-dimensional subspace $F\subset C[0,1]$ whose nonzero elements are nowhere differentiable can be extended to a countably infinite-dimensional subspace $G\subset C[0,1]$ with the same property.

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arXiv2026

The super Alternative Daugavet property, unconditional bases and SCD geometry

We answer negatively the 1-unconditional part of Question 6.4, by Langemets, L\~oo, Mart\'in, Perreau and Rueda Zoca, and, more generally, prove that no infinite-dimensional Banach space with a $K$-unconditional basis, for $1\leq K<3/2$, can satisfy the super Alternative Daugavet property. We also address two recent questions posed by L\~oo and Perreau concerning weak topological structures and slicely countably determined phenomena in Banach spaces with unconditional bases. More precisely, we prove that the weak unit ball of every Banach space with a 1-unconditional basis admits a countable $\pi$-base, thereby giving a positive answer to Question 5.1. We further show that every bounded convex subset of a Banach space with a Schauder basis that is shrinking or boundedly complete admits a countable $\pi$-base for its relative weak topology. We complement this result with two permanence principles: one for shrinking Schauder decompositions whose finite partial sums have countable weak $\pi$-bases, and another for unconditional sums over boundedly complete Banach sequence spaces. Finally, we prove that, for every $k>1$, there exists a Banach space with a $k$-unconditional basis whose unit ball is not slicely countably determined, thereby giving a positive answer to Question 5.4.

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arXiv2026

Weak Minimizing Property and the Compact Perturbation Property for the Minimum Modulus

For an operator $T:X\to Y$, denote $m(T)=\inf\{\|Tx\|:x\in S_X\}$. A sequence $(x_n)$ in $S_X$ is said to be minimizing for $T$ if $\|Tx_n\|\to m(T)$. The weak minimizing property (WmP), introduced by Chakraborty, requires that every operator admitting a non-weakly null minimizing sequence attains its minimum modulus. More recently, Han~\cite{Han2026} introduced the Compact Perturbation Property for the minimum modulus (CPPm), which requires that for every operator $T:X\to Y$ that does not attain its minimum modulus, \[ \sup_{K\in\mathcal{K}(X,Y)} m(T+K)=m(T). \] In~\cite{Han2026}, it is shown that $(\ell_1,\ell_1)$ fails both properties, while $(c_0,c_0)$ fails the WmP. However, whether $(c_0,c_0)$ has the CPPm was left open (Problem~3.6). In this paper, we give a negative answer to this question by proving that $(c_0,c_0)$ does not have the CPPm. The proof is constructive, exhibiting a non-min-attaining operator whose minimum modulus is strictly increased by a rank-one compact perturbation. Moreover, we show that this phenomenon is not specific to $c_0$: if $X=\mathbb{K}\oplus_\infty Y$ with $Y$ non-reflexive, then the pair $(X,X)$ fails the CPPm.

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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.

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arXiv2026

Extension of hypercyclic and frequently hypercyclic subspaces

We focus on the existence of large linear structures within the sets of hypercyclic and frequently hypercyclic vectors. For operators $T$ satisfying Kitai's Criterion or the Frequent Hypercyclicity Criterion, we analyze the fundamental linear space $\{f(T)x | f \in H(\mathbb{C})\}$, studied by Herrero, Bourdon, B\`es, Wengenroth, and many others. We show that the set $\{f(T)x | f \in H(\mathbb{C})\}$ can be extended within $HC(T) \cup \{0\}$ or $FHC(T) \cup \{0\}$ if $x \in HC(T)$ or $x \in FHC(T)$, respectively. The extension is such that the quotient of the new space with $\{ f(T)x \mid f \in H(\mathbb{C}) \}$ has dimension $\mathfrak{c}$ (the cardinality of the continuum). Second, we prove that generically a finite-dimensional subspace contained in $HC(T) \cup \{0\}$ can be enlarged to a subspace of dimension $\mathfrak{c}$. Third, we establish sufficient conditions for extending arbitrary linear subspaces both from $HC(T) \cup \{0\}$ and $FHC(T) \cup \{0\}$ to larger subspaces of dimension $\mathfrak{c}$.

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arXiv2025

Operator Ranges and Spaceability: Extending a Result of Kitson and Timoney

We revisit the results of Kitson and Timoney \emph{[J.~Math.~Anal.~Appl.~\textbf{378} (2011), 680--686]} on the spaceability of complements of operator ranges, extending one of their main theorems to the general Fr\'echet setting. In particular, we provide an affirmative answer to the question posed in \emph{Remark~3.4} of that paper, showing that the conclusion remains valid when the operators act between Fr\'echet spaces. Moreover, we show that the same phenomenon occurs for arbitrary (possibly uncountable) families of operators. The arguments presented here follow the spirit of the original work.

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arXiv2025

The Exact $\varepsilon$-Hypercyclicity Threshold

In this paper we give an affirmative answer to the problem proposed by Bayart in [J. Math. Anal. Appl. \textbf{529} (2024), 127278]: given $\varepsilon\in(0,1)$, there exists an operator which is $\delta$-hypercyclic if and only if $\delta\in[\varepsilon,1)$?

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arXiv2025

Isometric embeddings of separable Banach spaces into $(\ell^\infty \setminus c)\cup\{0\}$

The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an embedding can be chosen so that its image intersects $c$ only at the origin. Moreover, we prove that any finite- or countable-dimensional, or more generally separable, subspace of $(\ell^\infty \setminus c)\cup\{0\}$ can be extended to a subspace containing an isometric copy of an arbitrary separable Banach space, while still avoiding $c$. We further establish that this extension property also holds for every subspace $D\subset \ell^\infty$ with $D\cap c=\{0\}$ and separable image in the quotient $\ell^\infty/c$.

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arXiv2024

Complements of Non-Minimal Subspaces: Characterization Results

Inspired by the work of L. Drewnowski in [Studia Math. 77 (1984) 373--391], our research reveals new insights and characterizes the notion of spaceability in the context of complements of subspaces (not necessarily closed) within the universe of F-spaces in terms of [S]-lineability.

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arXiv2024

On Sequences with at Most a Finite Number of Zero Coordinates

In this paper, we analyze the existence of algebraic and topological structures in the set of sequences that contain only a finite number of zero coordinates. Inspired by the work of Daniel Cariello and Juan B. Seoane-Sepúlveda, our research reveals new insights and complements their notable results beyond the classical \( \ell_p \) spaces for \( p \) in the interval from 1 to infinity, including the intriguing case where \( p \) is between 0 and 1. Our exploration employs notions such as S-lineability, pointwise lineability, and (alpha, beta)-spaceability. This investigation allowed us to verify, for instance, that the set \( F \setminus Z(F) \), where \( F \) is a closed subspace of \( \ell_p \) containing \( c_0 \), is (alpha, c)-spaceable if and only if alpha is finite.

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arXiv2024

A Quest for Convergence: Exploring Series in Non-Linear Environments

This note presents an extension of a result within the concept of [S]-lineability, originally developed in 2019 by L. Bernal-González, J.A. Conejero, M. Murillo-Arcila, and J.B. Seoane-Sepúlveda . Additionally, we provide a characterization in terms of lineability in the context of complements of unions of closed subspaces in F-spaces, and finally, we present a negative result in both normed spaces and p-Banach spaces. These findings contribute to the understanding of linearity in exotic settings in topological vector spaces.

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arXiv2023

Complements of unions: insights on spaceability and applications

This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the results of Kitson and Timoney in [J. Math. Anal. Appl. \textbf{378} (2011), 680-686]. This criterion extends and recovers some classical results in this theory. The second criterion establishes sufficient conditions for the complement of a union of Lebesgue spaces to be $\left(\alpha,\beta\right)$-spaceable, or not, even when they are not locally convex. We use this result to characterize the measurable subsets having positive measure. Armed with these results, we have improved existing results in environments such as: Lebesgue measurable function sets, spaces of continuous functions, sequence spaces, nowhere H\"{o}lder function sets, Sobolev spaces, non-absolutely summing operator spaces, and even sets of functions of bounded variation.

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arXiv2023

On the spaceability of the set of functions in the Lebesgue space $L_p$ which are in no other $L_q$

In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(α,\mathfrak{c})$-spaceable if, and only if, $α<\aleph_{0}$. Such a problem first appears in [V. Fávaro, D. Pellegrino, D. Tomaz, Bull. Braz. Math. Soc. \textbf{51} (2020) 27-46], where the authors get the $(1,\mathfrak{c})$-spaceability of $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ for $p>0$. The definitive answer to this problem continued to be sought by other authors, and some partial answers were obtained. The veracity of this result was expected, as a similar result is known for sequence spaces.

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