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Uncomplemented subspaces in operator and polynomial ideals

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Generalizing a number of known linear and nonlinear results, we establish conditions for the space of linear operators belonging to a given operator ideal not to be complemented in the space of bounded linear operators, and for the space of n-homogeneous polynomials belonging to a given polynomial ideal not to be complemented in the space of continuous n-homogeneous polynomials. Illustrative examples are provided.

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The authors thank the referees for their valuable suggestions that improved the final version of the manuscript. This paper was written while Sergio A. Pérez was a Postdoctoral Fellow in the School of Mathematics and Statistics at Universidad Pedagógica y Tecnológica de Colombia. He thanks Professor Richard Alexander De la Cruz Guerrero for being his advisor in this work. Sergio A. Pérez also thanks Fondo Nacional De Financiamiento Para La Ciencia, La Tecnología Y La Innovación “Francisco José De Caldas”- Minciencias and the Vicerrectoría de Investigación y Extensión (VIE) of the Universidad Pedagógica y Tecnológica de Colombia for supporting this project with SGI 2954. Finally this work will be part of the Grupo de Investigación de Ecuaciones Diferenciales, Modelación y Simulación (GEDMYS)-UPTC.

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Correspondence to Sergio A. Pérez.

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G. Botelho: Supported by CNPq Grant 304262/2018-8 and Fapemig Grant PPM-00450-17.

V.V. Fávaro: Supported by CNPq 310500/2017-6 and FAPEMIG Grant PPM-00217-18.

S. A. Pérez: Supported by Fondo Nacional De Financiamiento Para La Ciencia, La Tecnología Y La Innovación “Francisco José De Caldas” -Minciencias Convocatoria 848 de 2019 Grant 80740-081-2020-Uptc VIE SGI 2954.

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Botelho, G., Fávaro, V.V. & Pérez, S.A. Uncomplemented subspaces in operator and polynomial ideals. Rev Mat Complut 35, 851–869 (2022).

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