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2022 Group-invariant separating polynomials on a Banach space
Javier Falcó, Domingo García, Manuel Maestre, Mingu Jung
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Publ. Mat. 66(1): 207-233 (2022). DOI: 10.5565/PUBLMAT6612209

Abstract

We study the group-invariant continuous polynomials on a Banach space X that separate a given set K in X and a point z outside K. We show that if X is a real Banach space, G is a compact group of L(X), K is a G-invariant set in X, and z is a point outside K that can be separated from K by a continuous polynomial Q, then z can also be separated from K by a G-invariant continuous polynomial P. It turns out that this result does not hold when X is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that X has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group-invariant polynomial from the closed unit ball.

Funding Statement

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The first, second, and fourth authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and fourth authors were also supported by Prometeo PROMETEO/2017/102.

Funding Statement

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The first, second, and fourth authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and fourth authors were also supported by Prometeo PROMETEO/2017/102.

Funding Statement

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1A2C1003857). The first, second, and fourth authors were supported by MINECO and FEDER Project MTM2017-83262-C2-1-P. The second and fourth authors were also supported by Prometeo PROMETEO/2017/102.

Citation

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Javier Falcó. Domingo García. Manuel Maestre. Mingu Jung. "Group-invariant separating polynomials on a Banach space." Publ. Mat. 66 (1) 207 - 233, 2022. https://doi.org/10.5565/PUBLMAT6612209

Information

Received: 20 April 2020; Accepted: 2 June 2020; Published: 2022
First available in Project Euclid: 4 January 2022

Digital Object Identifier: 10.5565/PUBLMAT6612209

Subjects:
Primary: 14L24 , 46G20

Keywords: Banach space , group-invariant , polynomials , separation theorem

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques

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Vol.66 • No. 1 • 2022
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