Group invariant separating polynomials on a Banach space

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Javier Falcó
Universidad de Valencia. Departamento de Análisis Matemático
https://orcid.org/0000-0001-5435-3053
Domingo García
Universidad de Valencia. Departamento de Análisis Matemático
https://orcid.org/0000-0002-2193-3497
Manuel Maestre
Universidad de Valencia. Departamento de Análisis Matemático
https://orcid.org/0000-0001-5291-6705
Mingu Jung
POSTECH. Department of Mathematics.
https://orcid.org/0000-0003-2240-2855

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.

Paraules clau
group-invariant, separation theorem, polynomials, Banach space

Article Details

Com citar
Falcó, Javier et al. «Group invariant separating polynomials on a Banach space». Publicacions Matemàtiques, 2022, vol.VOL 66, núm. 1, p. 207-33, https://raco.cat/index.php/PublicacionsMatematiques/article/view/396446.
Referències
R. Alencar, R. Aron, P. Galindo, and A. Zagorodnyuk, Algebras of symmetric holomorphic functions on lp, Bull. London Math. Soc. 35(1) (2003),
55–64. DOI: 10.1112/S0024609302001431.

R. M. Aron, J. Falco, D. García, and M. Maestre, Algebras of symmetric
holomorphic functions of several complex variables, Rev. Mat. Complut. 31(3)
(2018), 651–672. DOI: 10.1007/s13163-018-0261-x.

R. M. Aron, J. Falcó, and M. Maestre, Separation theorems for group
invariant polynomials, J. Geom. Anal. 28(1) (2018), 393–404. DOI: 10.1007/
s12220-017-9825-0.

R. Aron, P. Galindo, D. Pinasco, and I. Zalduendo, Group-symmetric holomorphic functions on a Banach space, Bull. Lond. Math. Soc. 48(5) (2016),
779–796. DOI: 10.1112/blms/bdw043.

D. Carando, M. Mazzitelli, and P. Sevilla-Peris, A note on the symmetry of
sequence spaces, Math. Notes 110 (2021), 26–40. DOI: 10.1134/S000143462107
0038.

I. Chernega, P. Galindo, and A. Zagorodnyuk, Some algebras of symmetric
analytic functions and their spectra, Proc. Edinb. Math. Soc. (2) 55(1) (2012),
125–142. DOI: 10.1017/S0013091509001655.

I. Chernega, P. Galindo, and A. Zagorodnyuk, The convolution operation
on the spectra of algebras of symmetric analytic functions, J. Math. Anal. Appl.
395(2) (2012), 569–577. DOI: 10.1016/j.jmaa.2012.04.087.

I. Chernega, P. Galindo, and A. Zagorodnyuk, A multiplicative convolution
on the spectra of algebras of symmetric analytic functions, Rev. Mat. Complut.
27(2) (2014), 575–585. DOI: 10.1007/s13163-013-0128-0.

A. Defant, D. García, M. Maestre, and P. Sevilla-Peris, “Dirichlet Series and Holomorphic Functions in High Dimensions”, New Mathematical
Monographs 37, Cambridge University Press, Cambridge, 2019. DOI: 10.1017/
9781108691611.

S. Dineen, “Complex Analysis on Infinite Dimensional Spaces”, Springer
Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999.
DOI: 10.1007/978-1-4471-0869-6.

P. Galindo, T. Vasylyshyn, and A. Zagorodnyuk, Symmetric and finitely
symmetric polynomials on the spaces `∞ and L∞[0, +∞), Math. Nachr.
291(11–12) (2018), 1712–1726. DOI: 10.1002/mana.201700314.

D. J. H. Garling, On symmetric sequence spaces, Proc. London Math. Soc. (3)
16(1) (1966), 85–106. DOI: 10.1112/plms/s3-16.1.85.

M. González, R. Gonzalo, and J. A. Jaramillo, Symmetric polynomials
on rearrangement-invariant function spaces, J. London Math. Soc. (2) 59(2)
(1999), 681–697. DOI: 10.1112/S0024610799007164.

F. Jawad and A. Zagorodnyuk, Supersymmetric polynomials on the space
of absolutely convergent series, Symmetry 11(9) (2019), 1111. DOI: 10.3390/
sym11091111.

V. Kravtsiv, T. Vasylyshyn, and A. Zagorodnyuk, On algebraic basis of the
algebra of symmetric polynomials on `p(Cn), J. Funct. Spaces, Art. ID 4947925
(2017), 8 pp. DOI: 10.1155/2017/4947925.

A. S. Nemirovski˘ı and S. M. Semenov, The polynomial approximation of functions on Hilbert space (Russian), Mat. Sb. (N.S.) 92(134) (1973), 257–281, 344.

A. N. Sergeev, On rings of supersymmetric polynomials, J. Algebra 517 (2019),
336–364. DOI: 10.1016/j.jalgebra.2018.10.003.

J. R. Stembridge, A characterization of supersymmetric polynomials, J. Algebra 95(2) (1985), 439–444.

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