Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces

Q3 Mathematics Matematychni Studii Pub Date : 2021-10-23 DOI:10.30970/ms.56.1.106-112
S. Halushchak
{"title":"Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces","authors":"S. Halushchak","doi":"10.30970/ms.56.1.106-112","DOIUrl":null,"url":null,"abstract":"The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces. Let $X$ and $Y$ be complex Banach spaces. Let $\\mathbb{A}= \\{A_1, A_2, \\ldots, A_n, \\ldots\\}$ and $\\mathbb{P}=\\{P_1, P_2,$ \\ldots, $P_n, \\ldots \\}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\\|A_n\\|_1=\\|P_n\\|_1=1$ and $\\deg A_n=\\deg P_n=n,$ $n\\in \\mathbb{N}.$ We consider the subalgebras $H_{b\\mathbb{A}}(X)$ and $H_{b\\mathbb{P}}(Y)$ of the Fr\\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\\mathbb{A}$ and $\\mathbb{P}$, respectively. It is easy to see that $H_{b\\mathbb{A}}(X)$ and $H_{b\\mathbb{P}}(Y)$ are the Fr\\'{e}chet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\\mathbb{A}}(X)$ and $H_{b\\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\\infty})$ of entire functions of bounded type on $L_{\\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\\ell_{\\infty}.$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.56.1.106-112","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 7

Abstract

The theory of analytic functions is an important section of nonlinear functional analysis.In many modern investigations topological algebras of analytic functions and spectra of suchalgebras are studied. In this work we investigate the properties of the topological algebras of entire functions,generated by countable sets of homogeneous polynomials on complex Banach spaces. Let $X$ and $Y$ be complex Banach spaces. Let $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ and $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ be sequences of continuous algebraically independent homogeneous polynomials on spaces $X$ and $Y$, respectively, such that $\|A_n\|_1=\|P_n\|_1=1$ and $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ We consider the subalgebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ of the Fr\'{e}chet algebras $H_b(X)$ and $H_b(Y)$ of entire functions of bounded type, generated by the sets $\mathbb{A}$ and $\mathbb{P}$, respectively. It is easy to see that $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y)$ are the Fr\'{e}chet algebras as well. In this paper we investigate conditions of isomorphism of the topological algebras $H_{b\mathbb{A}}(X)$ and $H_{b\mathbb{P}}(Y).$ We also present some applications for algebras of symmetric analytic functions of bounded type. In particular, we consider the subalgebra $H_{bs}(L_{\infty})$ of entire functions of bounded type on $L_{\infty}[0,1]$ which are symmetric, i.e. invariant with respect to measurable bijections of $[0,1]$ that preserve the measure. We prove that$H_{bs}(L_{\infty})$ is isomorphic to the algebra of all entire functions of bounded type, generated by countable set of homogeneous polynomials on complex Banach space $\ell_{\infty}.$
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Banach空间上一些有界型解析函数代数的同构
解析函数理论是非线性泛函分析的一个重要分支。在许多现代研究中,研究了解析函数的拓扑代数和这种代数的谱。本文研究了复巴拿赫空间上由齐次多项式的可数集生成的完整函数的拓扑代数的性质。让 $X$ 和 $Y$ 是复巴拿赫空间。让 $\mathbb{A}= \{A_1, A_2, \ldots, A_n, \ldots\}$ 和 $\mathbb{P}=\{P_1, P_2,$ \ldots, $P_n, \ldots \}$ 是空间上连续代数无关齐次多项式的序列 $X$ 和 $Y$,分别,这样 $\|A_n\|_1=\|P_n\|_1=1$ 和 $\deg A_n=\deg P_n=n,$ $n\in \mathbb{N}.$ 我们考虑子代数 $H_{b\mathbb{A}}(X)$ 和 $H_{b\mathbb{P}}(Y)$ fr日新月异的代数 $H_b(X)$ 和 $H_b(Y)$ 由集合生成的有界类型的整个函数 $\mathbb{A}$ 和 $\mathbb{P}$,分别。这一点很容易看出 $H_{b\mathbb{A}}(X)$ 和 $H_{b\mathbb{P}}(Y)$ 也是fracimet代数。本文研究了拓扑代数同构的条件 $H_{b\mathbb{A}}(X)$ 和 $H_{b\mathbb{P}}(Y).$ 给出了有界型对称解析函数代数的一些应用。特别地,我们考虑子代数 $H_{bs}(L_{\infty})$ 上有界类型的整个函数 $L_{\infty}[0,1]$ 哪些是对称的,也就是说,对于的可测双射是不变的 $[0,1]$ 这就保留了度量。我们证明$H_{bs}(L_{\infty})$ 是否同构于由复巴拿赫空间上齐次多项式的可数集合生成的所有有界型函数的代数 $\ell_{\infty}.$
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1