Symmetric polynomials on the Cartesian power of the real Banach space $L_\infty[0,1]$

T. Vasylyshyn; A. Zagorodnyuk

doi:10.30970/ms.53.2.192-205

T. Vasylyshyn Vasyl Stefanyk Precarpathian National University, Ukraine
A. Zagorodnyuk Vasyl Stefanyk Precarpathian National University, Ukraine

DOI: https://doi.org/10.30970/ms.53.2.192-205

Keywords: Symmetric polynomial, Spectrum of the Fr\'{e}chet algebra

Abstract

We construct an algebraic basis of the algebra of symmetric (invariant under composition of the variable with any measure preserving bijection of $[0,1]$) continuous polynomials on the $n$th Cartesian power of the real Banach
space $L_^{(\mathbb{R})}\infty[0,1]$ of Lebesgue measurable essentially bounded real valued functions on $[0,1].$ Also we describe the spectrum of the Fr\'{e}chet algebra $A_s(L_^{(\mathbb{R})}\infty[0,1])$ of symmetric real-valued functions on the space $L_^{(\mathbb{R})}\infty[0,1]$, which is the completion of the algebra of symmetric continuous real-valued polynomials on $L_^{(\mathbb{R})}\infty[0,1]$ with respect to the family of norms of uniform convergence of complexifications of polynomials. We show that $A_s(L_^{(\mathbb{R})}\infty[0,1])$ contains not only analytic functions. Results of the paper can be used for investigations of algebras of symmetric functions on the $n$th Cartesian power of the Banach space $L_^{(\mathbb{R})}\infty[0,1]$.

Author Biographies

T. Vasylyshyn, Vasyl Stefanyk Precarpathian National University, Ukraine

Vasyl Stefanyk Precarpathian National University, Ukraine

A. Zagorodnyuk, Vasyl Stefanyk Precarpathian National University, Ukraine

Vasyl Stefanyk Precarpathian National University, Ukraine

References

M. Boiso, P. Hajek, Analytic approximations of uniformly continuous functions in real Banach spaces, Journal of Mathematical Analysis and Applications, 256 (2001), 80-98. doi:10.1006jmaa.2000.7291.

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.

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.

P. Galindo, T. Vasylyshyn, A. Zagorodnyuk, Analytic structure on the spectrum of the algebra of symmetric analytic functions on $L_infty$, RACSAM, 114 (2020), Article number 56.doi:10.1007/s13398-020-00791-w.

M. Gonz'{a}lez, 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.

V. Kravtsiv, Algebraic basis of the algebra of block-symmetric polynomials on $ell_1oplus ell_infty$, Carpathian Math. Publ., 11 (2019), №1, 89-95.

doi:10.15330/cmp.11.1.89-95.

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.

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

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.

T. Vasylyshyn, Point-evaluation functionals on algebras of symmetric functions on $(L_infty)^2$, Carpathian Math. Publ., 11 (2019), №2, 493-501. doi:10.15330/cmp.11.2.493-501.

T. Vasylyshyn, Symmetric polynomials on $(L_p)^n$, European Journal of Math., 6 (2020), №1, 164-178. doi:10.1007/s40879-018-0268-3.

Vasylyshyn T.V. The algebra of symmetric polynomials on $(L_infty)^n$, Mat. Stud., 52 (2019), №1, 71-85. doi:10.30970/ms.52.1.71-85

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.