Symmetric polynomials on the Cartesian power of $L_p$ on the semi-axis

T. V. Vasylyshyn>
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doi:10.15330/ms.50.1.93-104

The paper deals with polynomials in the complex Banach space $(L_p[0,+\infty))^n,$ which are the $n$th Cartesian power of the complex Banach space of Lebesgue measurable integrable in a power $p$ complex-valued functions on $[0,+\infty),$ where $1\leq p \le +\infty.$ It is proved that if $p$ is an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ can be uniquely represented as an algebraic combination of some ``elementary'' $p$-homogeneous symmetric polynomials. It is also proved that if $p$ is not an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ is constant. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on $(L_p[0,+\infty))^n.$

1. I. Chernega, Symmetric polynomials and holomorphic functions on infinite dimensional spaces, Journal of Vasyl Stefanyk Precarpathian National University, 2 (2015), №4, 23-49. doi:10.15330/jpnu.2.4.23-49.

2. P. Galindo, T. Vasylyshyn, A. Zagorodnyuk, The algebra of symmetric analytic functions on $L_\infty$, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 147 (2017), №4, 743-761. doi:10.1017/S0308210516000287.

3. 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), №11-12, 1712-1726. doi:10.1002/mana.201700314.

4. M. GonzЃLalez, R. Gonzalo, J. A. Jaramillo, Symmetric polynomials on rearrangement invariant function spaces, J. London Math. Soc., 59 (1999), №2, 681-697. doi:10.1112/S0024610799007164.

5. V. Kravtsiv, T. Vasylyshyn, A. Zagorodnyuk, On algebraic basis of the algebra of symmetric polynomials on $\ell_p(\mathbb{C}^n)$, Journal of Function Spaces, 2017 (2017), Article ID 4947925, 8 pages. doi:10.1155/2017/4947925.

6. J. Mujica, Complex Analysis in Banach Spaces, North Holland, 1986.

7. A. S. Nemirovskii, S. M. Semenov, On polynomial approximation of functions on Hilbert space, Mat. USSR Sbornik, 21 (1973), №2, 255-277. doi:10.1070/SM1973v021n02ABEH002016

8. T. Vasylyshyn, Symmetric continuous linear functionals on complex space $L_\infty$, Carpathian Math. Publ., 6 (2014), №1, 8-10. doi:10.15330/cmp.6.1.8-10.

9. T. Vasylyshyn, Continuous block-symmetric polynomials of degree at most two on the space $(L_\infty)^2$, Carpathian Math. Publ., 8 (2016), №1, 38-43. doi:10.15330/cmp.8.1.38-43.

10. 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), №1, 22-27. doi:10.15330/cmp.9.1.22-27.

11. T. Vasylyshyn, Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$, Carpathian Math. Publ., 9 (2017), №2, 198-201. doi:10.15330/cmp.9.2.198-201.

12. T. Vasylyshyn, Symmetric polynomials on (Lp)n, European Journal of Math., doi:10.1007/s40879-018- 0268-3.

13. T. Vasylyshyn, A. Zagorodnyuk, Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions, Methods of Functional Analysis and Topology (to appear).

	Symmetric polynomials on the Cartesian power of $L_p$ on the semi-axis
Author	T. V. Vasylyshyn taras.v.vasylyshyn@gmail.com Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk, Ukraine
Abstract	The paper deals with polynomials in the complex Banach space $(L_p[0,+\infty))^n,$ which are the $n$th Cartesian power of the complex Banach space of Lebesgue measurable integrable in a power $p$ complex-valued functions on $[0,+\infty),$ where $1\leq p \le +\infty.$ It is proved that if $p$ is an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ can be uniquely represented as an algebraic combination of some ``elementary'' $p$-homogeneous symmetric polynomials. It is also proved that if $p$ is not an integer, then every continuous symmetric polynomial on $(L_p[0,+\infty))^n$ is constant. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on $(L_p[0,+\infty))^n.$
Keywords	polynomial; symmetric polynomial; algebraic combination
DOI	doi:10.15330/ms.50.1.93-104
Reference	1. I. Chernega, Symmetric polynomials and holomorphic functions on infinite dimensional spaces, Journal of Vasyl Stefanyk Precarpathian National University, 2 (2015), №4, 23-49. doi:10.15330/jpnu.2.4.23-49. 2. P. Galindo, T. Vasylyshyn, A. Zagorodnyuk, The algebra of symmetric analytic functions on $L_\infty$, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 147 (2017), №4, 743-761. doi:10.1017/S0308210516000287. 3. 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), №11-12, 1712-1726. doi:10.1002/mana.201700314. 4. M. GonzЃLalez, R. Gonzalo, J. A. Jaramillo, Symmetric polynomials on rearrangement invariant function spaces, J. London Math. Soc., 59 (1999), №2, 681-697. doi:10.1112/S0024610799007164. 5. V. Kravtsiv, T. Vasylyshyn, A. Zagorodnyuk, On algebraic basis of the algebra of symmetric polynomials on $\ell_p(\mathbb{C}^n)$, Journal of Function Spaces, 2017 (2017), Article ID 4947925, 8 pages. doi:10.1155/2017/4947925. 6. J. Mujica, Complex Analysis in Banach Spaces, North Holland, 1986. 7. A. S. Nemirovskii, S. M. Semenov, On polynomial approximation of functions on Hilbert space, Mat. USSR Sbornik, 21 (1973), №2, 255-277. doi:10.1070/SM1973v021n02ABEH002016 8. T. Vasylyshyn, Symmetric continuous linear functionals on complex space $L_\infty$, Carpathian Math. Publ., 6 (2014), №1, 8-10. doi:10.15330/cmp.6.1.8-10. 9. T. Vasylyshyn, Continuous block-symmetric polynomials of degree at most two on the space $(L_\infty)^2$, Carpathian Math. Publ., 8 (2016), №1, 38-43. doi:10.15330/cmp.8.1.38-43. 10. 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), №1, 22-27. doi:10.15330/cmp.9.1.22-27. 11. T. Vasylyshyn, Metric on the spectrum of the algebra of entire symmetric functions of bounded type on the complex $L_\infty$, Carpathian Math. Publ., 9 (2017), №2, 198-201. doi:10.15330/cmp.9.2.198-201. 12. T. Vasylyshyn, Symmetric polynomials on (Lp)n, European Journal of Math., doi:10.1007/s40879-018- 0268-3. 13. T. Vasylyshyn, A. Zagorodnyuk, Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions, Methods of Functional Analysis and Topology (to appear).
Pages	93-104
Volume	50
Issue	1
Year	2018
Journal	Matematychni Studii
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