Author:
García-Pacheco Francisco Javier
Abstract
AbstractThe intersection of all zero-neighborhoods in a topological module over a topological ring is a bounded and closed submodule whose inherited topology is the trivial topology. In this manuscript, we prove that this is the smallest closed submodule and thus replaces the null submodule in the Hausdorff setting. This fact motivates to introduce a new notion in operator theory called topological kernel. Another new concept is also defined that of Pareto optimal element for a family of continuous linear operators between topological modules. It is then proved that topological kernels have a strong influence on the existence of Pareto optimal elements. This work is strongly motivated by the ongoing search for a consistent operator theory on topological modules over general topological rings.
Funder
Consejería de Universidad, Investigación e Innovación de la Junta de Andalucía
Universidad de Cadiz
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics,Analysis
Reference21 articles.
1. Monographs and Textbooks in Pure and Applied Mathematics;V.I. Arnautov,1996
2. Del-Vecchio, R.R., Pombo, D.P. Jr., Vinagre, C.T.M.: Topics on topological modules, Instituto de Matemática. Relatório de Pesquisa [Institute of Mathematics. Research Report], vol 3. Editora da Universidade Federal Fluminense, Niterói (2007)
3. García-Pacheco, F.: Abstract Calculus: A Categorical Approach, 1st edn. Chapman and Hall/CRC, London (2021). https://doi.org/10.1201/9781003166559
4. García-Pacheco, F.J.: Regularity in topological modules. Mathematics 8(9), 1580 (2020). https://doi.org/10.3390/math8091580
5. García-Pacheco, F.J., Miralles, A., Murillo-Arcila, M.: Invertibles in topological rings: a new approach. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 116(1), Paper No. 38 (2022). https://doi.org/10.1007/s13398-021-01183-4