Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on \([0,1]\)
type: Item
instance of: scholarly article
title: Symmetric analytic functions on the Cartesian power of the complex Banach space of Lebesgue measurable essentially bounded functions on \([0,1]\)
zbMATH Open document ID: 1494.46040
published in: Journal of Mathematical Analysis and Applications
publication date: 2022-02-14T00:00:00Z
review text: The author characterises the spectrum of the algebra of bounded symmetric holomorphic functions on the \(n\)-fold Cartesian product of \(L_\infty[0,1]\). The author uses \(\Xi\) to denote the group of all measurable bijections \(\sigma \colon [0,1]\to [0,1]\) such that both \(\sigma\) and \(\sigma^{-1}\) preserve Lebesgue measure and \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) to denote the space of all holomorphic functions \(f\) on \((L_\infty[0,1])^n\) which are bounded on bounded subsets of \((L_\infty[0,1])^n\) and which have the property that \[ f(y_1,\ldots,y_n)=f(y_1\circ\sigma,\ldots,y_n\circ\sigma) \] for all \((y_1,\ldots, y_n)\) in \((L_\infty[0,1])^n\) and all \(\sigma\) in \(\Xi\). Using the result [\textit{T.~V. Vasylyshyn}, Mat. Stud. 52, No.~1, 71--85 (2019; Zbl 1441.46033)] that every \(N\)-homogeneous polynomial \(P\) in \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) has a unique representation of the form \[ P(y)=\sum_{\substack{l\colon M_N\to \mathbb{Z}^+\\ \sum_{k\in M_N}|\mathbf{k}|l(\mathbf{k})=N}} \alpha_l\prod_{\substack{\mathbf{k}\in M_N,\\ l(k)>0}}(R_{\mathbf{k}}(y))^{l(\mathbf{k})} , \] where \(M_N=\{\mathbf{k}\in \mathbb Z_+^n:1\le |\mathbf{k}|\le N\}\) and \[ R_{\mathbf{k}}(y)=\int_{[0,1]}\prod_{s=1,k_s>0}^n(y_s(t))^{k_s}\, dt , \] and the corresponding result for the algebra of symmetric functions \(\mathcal{H}_ {bs}(L_\infty[0,1])\), the author shows that every non-zero, multiplicative linear functional on \(\mathcal{H}_{bs}((L_\infty[0,1])^n)\) is an evaluation at a point of \((L_\infty[0, 1])^n\). The main result used in establishing this characterisation of the spectrum is that for each \(n\) in \(\mathbb{N}\) there is \(K_n>0\) such that for every function \(c\colon \mathbb{Z}^n _+\setminus\{(0,\ldots,0)\}\to \mathbb{C}\) with \(\sup_{\mathbf{k}\in \mathbf{Z}^n _+\setminus\{(0,\ldots,0)\}}|c(\mathbf{k})|^{1/|\mathbf{k}|}<\infty\) there is \(x_c\) in \((L_\infty[0,1])^n\) such that \(R_{\mathbf{k}}(x_c)=c(\mathbf{k})\) for all \(\mathbf{k} \…
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 7472996
zbMATH Keywords: symmetric polynomial
zbMATH Keywords: symmetric analytic function
zbMATH Keywords: spectrum of a Fréchet algebra
zbMATH Keywords: Lebesgue measurable essentially bounded function
author: T. V. Vasylyshyn
MaRDI profile type: Publication
full work available at URL: j.jmaa.2021.125977
OpenAlex ID: W4200618569
cites work: Q5317928
cites work: ALGEBRAS OF SYMMETRIC HOLOMORPHIC FUNCTIONS ON ${\cal L}_p$
cites work: Group-symmetric holomorphic functions on a Banach space
cites work: Some algebras of symmetric analytic functions and their spectra
cites work: Continuity and hypercyclicity of composition operators on algebras of symmetric analytic functions on Banach spaces
cites work: The algebra of symmetric analytic functions on L∞
cites work: Symmetric and finitely symmetric polynomials on the spaces and
cites work: Analytic structure on the spectrum of the algebra of symmetric analytic functions on \(L_\infty \)
cites work: Q4354270
cites work: Spectra of some algebras of entire functions of bounded type, generated by a sequence of polynomials
cites work: Q3687398
cites work: On algebraic basis of the algebra of symmetric polynomials on \(\ell_p(\mathbb{C}^n)\)
cites work: Q4188284
cites work: Q3711155
cites work: ON POLYNOMIAL APPROXIMATION OF FUNCTIONS ON HILBERT SPACE
cites work: Symmetric polynomials on the Cartesian power of \(L_p\) on the semi-axis
cites work: The algebra of symmetric polynomials on \((L_\infty)^n\)
cites work: Point-evaluation functionals on algebras of symmetric functions on $(L_\infty)^2$
cites work: Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$
cites work: Symmetric polynomials on \((L_p)^n\)
cites work: Q5204156
cites work: Symmetric polynomials on the Cartesian power of the real Banach space \(L_\infty[0,1]\)
DOI: 10.1016/J.JMAA.2021.125977
description: scientific article; zbMATH DE number 7472996