Javier Falcó
580 California St., Suite 400
San Francisco, CA, 94104
Group invariant separating polynomials on a Banach space
2022, Publicacions Matemàtiques
https://doi.org/10.5565/PUBLMAT6612209
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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.
FAQs
AI
What are the criteria for polynomials to be G-invariant in Banach spaces?
A polynomial P on a Banach space X is G-invariant under a group G if P(z) = P(γ(z)) for all z in X and γ in G, emphasizing continuity in its definition.
How does the symmetry of polynomials affect separation in Banach spaces?
The paper demonstrates that symmetric polynomials can achieve separations that non-symmetric counterparts may not, particularly in the context of compact topological groups acting on Banach spaces.
What implications does the Hahn-Banach theorem have on invariant polynomial separations?
Theorem 2.1 establishes that Hahn-Banach conditions enable the construction of G-invariant continuous polynomials that separate points from closed convex balanced sets under group actions.
How do G-invariant holomorphic functions relate to polynomials in this study?
The mapping S_G from H(U) to H_G(U) illustrates the continuity and projection properties of G-invariant holomorphic functions, offering a significant connection to polynomial behaviors.
What limitations arise when separating points in complex Banach spaces?
The study reveals that in complex Banach spaces, unlike real ones, certain necessary conditions for separation by G-invariant polynomials may not always hold, hindering general results.
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