Abstract
We study the group-invariant continuous polynomials on a Banach space that separate a given set in and a point outside . We show that if is a real Banach space, is a compact group of , is a -invariant set in , and is a point outside that can be separated from by a continuous polynomial , then can also be separated from by a -invariant continuous polynomial . It turns out that this result does not hold when 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 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
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
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