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Anselmo Raposo Jr.

Publications and source records attributed to Anselmo Raposo Jr..

5 recordsLinked to original sources
arXiv2026

Bombieri--Weyl Contractivity and Rigidity for Homogeneous Polynomials

We determine the dimension-free threshold for the comparison between the Bombieri--Weyl norm and the supremum norm of complex homogeneous polynomials on $\ell_p^n$. For $m$-homogeneous polynomials, the critical scale is $p=2m$: below this threshold no dimension-free comparison is possible, while at $p=2m$ we obtain $\|P\|_{\mathrm{BW}}\le (1+C/m)\|P\|_{2m}$. Moreover, there exist absolute constants $A>0$ and $m_0\in\mathbb N$ such that, for every $m\ge m_0$ and $p\ge 2m+A$, $\|P\|_{\mathrm{BW}}\le\|P\|_p$, with optimal constant one. Equality holds precisely for coordinate pure powers. We also obtain a quantitative stability statement for near-extremizers. The proof is based on a decomposition by multiplicity patterns, contractive orbit projections, Hardy--Littlewood estimates for reduced multilinear forms, and Wiener-type slice estimates. Finally, we show that the corresponding contractive phenomenon fails over the real scalar field.

math.FA↗
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.

math.FA↗
arXiv2024

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 $\left(α,β\right)$-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 Hölder function sets, Sobolev spaces, non-absolutely summing operator spaces, and even sets of functions of bounded variation.

math.FA↗
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.

math.FA↗