# Note on separately symmetric polynomials on the Cartesian product of $\ell_1$

Author
Vasyl Stefanyk Precarpathian National University Ivano-Frankivsk, Ukraine
Abstract
In the paper, we describe algebraic bases in algebras of separately symmetric polynomials which are defined on Cartesian products of $n$ copies of $\ell_1.$ Also, we describe spectra of algebras of entire functions, generated by these polynomials as Cartesian products of spectra of algebras of symmetric analytic functions of bounded type on $\ell_1.$ Finally, we consider algebras of separately symmetric analytic functions of bounded type on infinite direct sums of copies of $\ell_1.$ In particular, we show that there is a homomorphism from such algebra onto the algebra of all analytic functions of bounded type on a Banach space $X$ with an unconditional basis.
Keywords
polynomials and analytic functions on Banach spaces; symmetric analytic functions; block-symmetric polynomials; separately symmetric polynomials
DOI
doi:10.15330/ms.50.2.204-210
Reference
1. R. Alencar, R. Aron, P. Galindo, A. Zagorodnyuk, Algebras of symmetric holomorphic functions on .p, Bull. Lond. Math. Soc., 35 (2003), 55-64.

2. R.M. Aron, B.J. Cole, T.W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math., 415 (1991), 51-93.

3. R.M. Aron, J. Falco, D. Garca, M. Maestre, Algebras of symmetric holomorphic functions of several complex variables. Rev. Mat. Complut., 31 (2018), 651-672.

4. R. Aron, P. Galindo, D. Pinasco, I. Zalduendo, Group-symmetric holomorphic functions on a Banach space. Bull. Lond. Math. Soc., 48 (2016), ¹5, 779-796.

5. I. Chernega, P. Galindo, A. Zagorodnyuk, Some algebras of symmetric analytic functions and their spectra. Proc. Edinburgh Math. Soc., 55 (2012), 125-142.

6. I. Chernega, P. Galindo, A. Zagorodnyuk, The convolution operation on the spectra of algebras of symmetric analytic functions, J. Math. Anal. Appl., 395 (2012), 569-577.

7. I.Chernega, P. Galindo, A. Zagorodnyuk, A multiplicative convolution on the spectra of algebras of symmetric analytic functions, Rev. Mat. Complut., 27 (2014), ¹2, 575-585.

8. M. Gonzalez, R. Gonzalo, J. Jaramillo, Symmetric polynomials on rearrangement invariant function spaces, J. London Math. Soc., 59 (1999), ¹2, 681-697.

9. S. Dineen, Complex Analysis in Infinite Dimensional Spaces. London: Springer, 1999.

10. P. Galindo, T. Vasylyshyn, A. Zagorodnyuk, The algebra of symmetric analytic functions on L1. Proc. R. Soc. Edinb., Sect. A, Math., 147 (2017), ¹4, 743-761.

11. P. Galindo, T. Vasylyshyn, A. Zagorodnyuk, Symmetric and finitely symmetric polynomials on the spaces $\ell_\infty$ and $L_\infty[0,+\infty).$ Mathematische Nachrichten 291 (2018), 1712-1726.

12. P. A. MacMahon, Combinatory analysis, Chelsea Publishing Co. New York, 1960.

13. V. Kravtsiv, T. Vasylyshyn, A. Zagorodnyuk, On Algebraic Basis of the Algebra of Symmetric Polynomials on $\ell_p(\bf{C}^n)$, Journal of Function Spaces, 2017 (7), Article ID 4947925, 8 p.

14. V. V. Kravtsiv, A. V. Zagorodnyuk, Representation of spectra of algebras of block-symmetric analytic functions of bounded type, Carpath. Math. Publications, 8 (2016), ¹2, 263-271.

15. V. V. Kravtsiv, A. V. Zagorodnyuk, On algebraic bases of algebras of block-symmetric polynomials on Banach spaces, Mat. Stud., 37 (2012), ¹1, 109-112.

16. J. Lindestrauss, L. Tzafriri, Classical Banach spaces I. Sequence Spaces, Springer-Verlag, New York, 1977. 190 p.

17. J. Mujica, Complex Analysis in Banach Spaces, North-Holland, Amsterdam, New York, Oxford, 1986.

18. T. Vasylyshyn, Topology on the spectrum of the algebra of entire symmetric functions of bounded type on the complex L1. Carpathian Math. Publ., 9 (2017), ¹1, 22-27.

19. A. Zagorodnyuk, Spectra of Algebras of Entire Functions on Banach Spaces, Proc. Amer. Math. Soc., 134 (2006), 2559-2569.

Pages
204-210
Volume
50
Issue
2
Year
2018
Journal
Matematychni Studii
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