Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces
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 such algebras are studied. In this work we investigate the properties of the topological algebras of entire functions, ge...
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Ivan Franko National University of Lviv
2021-10-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/196 |
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| author | S.I. Halushchak |
| author_facet | S.I. Halushchak |
| author_sort | S.I. Halushchak |
| collection | DOAJ |
| description | 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 such
algebras 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 $\ell_{\infty}.$ |
| format | Article |
| id | doaj-art-0ac2e0030fee46639c73e1a52c9e8a96 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2021-10-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-0ac2e0030fee46639c73e1a52c9e8a962025-08-20T02:41:33ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202021-10-0156110711210.30970/ms.56.1.107-112196Isomorphisms of some algebras of analytic functions of bounded type on Banach spacesS.I. Halushchak0Vasyl Stefanyk Precarpathian National UniversityThe theory of analytic functions is an important section of nonlinear functional analysis. In many modern investigations topological algebras of analytic functions and spectra of such algebras 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 $\ell_{\infty}.$http://matstud.org.ua/ojs/index.php/matstud/article/view/196homogeneous polynomials on banach spaces;symmetric analytic functions;spectra of algebras of analytic functions |
| spellingShingle | S.I. Halushchak Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces Математичні Студії homogeneous polynomials on banach spaces; symmetric analytic functions; spectra of algebras of analytic functions |
| title | Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces |
| title_full | Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces |
| title_fullStr | Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces |
| title_full_unstemmed | Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces |
| title_short | Isomorphisms of some algebras of analytic functions of bounded type on Banach spaces |
| title_sort | isomorphisms of some algebras of analytic functions of bounded type on banach spaces |
| topic | homogeneous polynomials on banach spaces; symmetric analytic functions; spectra of algebras of analytic functions |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/196 |
| work_keys_str_mv | AT sihalushchak isomorphismsofsomealgebrasofanalyticfunctionsofboundedtypeonbanachspaces |