The algebra of symmetric polynomials on $(L_\infty)^n$

T. V. Vasylyshyn

doi:10.30970/ms.52.1.71-85

The paper deals with continuous symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) complex-valued polynomials on the $n$th Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1].$ We construct an algebraic basis of the algebra of all such polynomials. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on the $n$th Cartesian power of $L_\infty.$

1. 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.

2. 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.

3. M. Gonzalez, 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.

4. V. Kravtsiv, Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1\oplus \ell_\infty$, Carpathian Math. Publ., 11 (2019), №1, 89.95. doi:10.15330/cmp.11.1.89-95.

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 p. 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, 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.

9. T. Vasylyshyn, Symmetric polynomials on $(L_p)^n$, European Journal of Math., doi:10.1007/s40879-018- 0268-3.

10. T.V. Vasylyshyn, Symmetric polynomials on the Cartesian power of $L_p$ on the semi-axis, Mat. Stud., 50 (2018), №1, 93.104.

11. T. Vasylyshyn, A. Zagorodnyuk, Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions, Methods of Functional Analysis and Topology, 24 (2018), №4, 381.398.

	The algebra of symmetric polynomials on $(L_\infty)^n$
Author	T. V. Vasylyshyn taras.v.vasylyshyn@gmail.com Vasyl Stefanyk Precarpathian National University, Ukraine
Abstract	The paper deals with continuous symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) complex-valued polynomials on the $n$th Cartesian power of the complex Banach space $L_\infty$ of all Lebesgue measurable essentially bounded complex-valued functions on $[0,1].$ We construct an algebraic basis of the algebra of all such polynomials. Results of the paper can be used for investigations of algebras of symmetric continuous polynomials and of symmetric analytic functions on the $n$th Cartesian power of $L_\infty.$
Keywords	polynomial; symmetric polynomial; algebraic combination; algebraic basis
DOI	doi:10.30970/ms.52.1.71-85
Reference	1. 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. 2. 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. 3. M. Gonzalez, 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. 4. V. Kravtsiv, Algebraic basis of the algebra of block-symmetric polynomials on $\ell_1\oplus \ell_\infty$, Carpathian Math. Publ., 11 (2019), №1, 89.95. doi:10.15330/cmp.11.1.89-95. 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 p. 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, 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. 9. T. Vasylyshyn, Symmetric polynomials on $(L_p)^n$, European Journal of Math., doi:10.1007/s40879-018- 0268-3. 10. T.V. Vasylyshyn, Symmetric polynomials on the Cartesian power of $L_p$ on the semi-axis, Mat. Stud., 50 (2018), №1, 93.104. 11. T. Vasylyshyn, A. Zagorodnyuk, Continuous symmetric 3-homogeneous polynomials on spaces of Lebesgue measurable essentially bounded functions, Methods of Functional Analysis and Topology, 24 (2018), №4, 381.398.
Pages	71-85
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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