Group invariant separating polynomials on a Banach space

• Falcó, Javier ^[1] ; García, Domingo ^[1] ; Maestre, Manuel ^[1] ; Jung, Mingu ^[2]
  1. [1] Universitat de València

     Universitat de València

     Valencia, España

     
  2. [2] POSTECH. Department of Mathematics.
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 1, 2022, págs. 207-233
• Idioma: catalán
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• Resumen

  • 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.