A Fixed-dimension Characterization of Entrywise Positivity Preservers [r-libre/4271]

Bouthat, Ludovick ORCID: https://orcid.org/0000-0002-2094-3028, Mashreghi, Javad et Morneau-Guérin, Frédéric ORCID: https://orcid.org/0000-0001-7610-4648 (12 oct. 2026). A Fixed-dimension Characterization of Entrywise Positivity Preservers. Communication (sur invitation) présentée au Discrete Mathematics Seminar, Newark, Delaware.

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Catégorie de document :	Communications à des congrès/colloques et conférences (non publiées)
Évaluation par un comité de lecture :	Non
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Résumé :	Fix an interval from zero to rho, where rho may be finite or infinite, and consider positive semidefinite matrices of fixed size whose entries lie in this interval. A longstanding problem in matrix theory, originating in work of Horn and Loewner from 1969, asks which real-valued functions preserve positive semidefiniteness when applied entrywise to every such matrix. We give a complete characterization of these functions. Under a finite smoothness assumption, positivity preservation is equivalent to two conditions: the first several derivatives of the function must be nonnegative, and a certain matrix built from repeated applications of the Euler differential operator must itself be positive semidefinite at every point of the interval. A regularization argument then removes the initial smoothness assumption. In particular, for matrices of size at least two, every positivity preserver automatically has substantial regularity, and the same characterization remains valid when the final derivatives are interpreted in the sense of distributions. If time permits, we will also explain how this characterization recovers several classical results, yields optimal constants in entrywise domination inequalities under finite regularity assumptions, and answers a question of Khare and Tao concerning whether positivity preservation can be detected using only a finite collection of test matrices.
Déposant:	Morneau-Guérin, Frédéric
Responsable :	Frédéric Morneau-Guérin
Dépôt :	13 oct. 2026 18:07
Dernière modification :	13 oct. 2026 18:07

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