Group invariant separating polynomials on a Banach space

  1. Falcó, Javier 1
  2. García, Domingo 1
  3. Maestre, Manuel 1
  4. Jung, Mingu 2
  1. 1 Universidad de Valencia. Departamento de Análisis Matemático
  2. 2 POSTECH. Department of Mathematics.
Revista:
Publicacions matematiques

ISSN: 0214-1493

Año de publicación: 2022

Volumen: 66

Número: 1

Páginas: 207-233

Tipo: Artículo

DIALNET GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Publicacions matematiques

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.

Referencias bibliográficas

  • 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.
Fuente de los datos: Dialnet