{"title":"\\(A\\)群代数中卷积算子的遍历性","authors":"H. Mustafaev, A. Huseynli","doi":"10.1134/S0016266322020046","DOIUrl":null,"url":null,"abstract":"<p> Let <span>\\(G\\)</span> be a locally compact Abelian group with dual group <span>\\(\\Gamma \\)</span>, let <span>\\(\\mu\\)</span> be a power bounded measure on <span>\\(G\\)</span>, and let <span>\\(A=[ a_{n,k}]_{n,k=0}^{\\infty}\\)</span> be a strongly regular matrix. We show that the sequence <span>\\(\\{\\sum_{k=0}^{\\infty}a_{n,k}\\mu^{k}\\ast f\\}_{n=0}^{\\infty}\\)</span> converges in the <span>\\(L^{1}\\)</span>-norm for every <span>\\(f\\in L^{1}(G)\\)</span> if and only if <span>\\(\\mathcal{F}_{\\mu}:=\\{\\gamma \\in \\Gamma:\\widehat{\\mu}(\\gamma) =1\\} \\)</span> is clopen in <span>\\(\\Gamma \\)</span>, where <span>\\(\\widehat{\\mu}\\)</span> is the Fourier–Stieltjes transform of <span>\\(\\mu \\)</span>. If <span>\\(\\mu \\)</span> is a probability measure, then <span>\\(\\mathcal{F}_{\\mu}\\)</span> is clopen in <span>\\(\\Gamma \\)</span> if and only if the closed subgroup generated by the support of <span>\\(\\mu \\)</span> is compact. </p>","PeriodicalId":575,"journal":{"name":"Functional Analysis and Its Applications","volume":"56 2","pages":"110 - 115"},"PeriodicalIF":0.6000,"publicationDate":"2022-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"\\\\(A\\\\)-Ergodicity of Convolution Operators in Group Algebras\",\"authors\":\"H. Mustafaev, A. Huseynli\",\"doi\":\"10.1134/S0016266322020046\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> Let <span>\\\\(G\\\\)</span> be a locally compact Abelian group with dual group <span>\\\\(\\\\Gamma \\\\)</span>, let <span>\\\\(\\\\mu\\\\)</span> be a power bounded measure on <span>\\\\(G\\\\)</span>, and let <span>\\\\(A=[ a_{n,k}]_{n,k=0}^{\\\\infty}\\\\)</span> be a strongly regular matrix. We show that the sequence <span>\\\\(\\\\{\\\\sum_{k=0}^{\\\\infty}a_{n,k}\\\\mu^{k}\\\\ast f\\\\}_{n=0}^{\\\\infty}\\\\)</span> converges in the <span>\\\\(L^{1}\\\\)</span>-norm for every <span>\\\\(f\\\\in L^{1}(G)\\\\)</span> if and only if <span>\\\\(\\\\mathcal{F}_{\\\\mu}:=\\\\{\\\\gamma \\\\in \\\\Gamma:\\\\widehat{\\\\mu}(\\\\gamma) =1\\\\} \\\\)</span> is clopen in <span>\\\\(\\\\Gamma \\\\)</span>, where <span>\\\\(\\\\widehat{\\\\mu}\\\\)</span> is the Fourier–Stieltjes transform of <span>\\\\(\\\\mu \\\\)</span>. If <span>\\\\(\\\\mu \\\\)</span> is a probability measure, then <span>\\\\(\\\\mathcal{F}_{\\\\mu}\\\\)</span> is clopen in <span>\\\\(\\\\Gamma \\\\)</span> if and only if the closed subgroup generated by the support of <span>\\\\(\\\\mu \\\\)</span> is compact. </p>\",\"PeriodicalId\":575,\"journal\":{\"name\":\"Functional Analysis and Its Applications\",\"volume\":\"56 2\",\"pages\":\"110 - 115\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2022-10-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Functional Analysis and Its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S0016266322020046\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Functional Analysis and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1134/S0016266322020046","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
\(A\)-Ergodicity of Convolution Operators in Group Algebras
Let \(G\) be a locally compact Abelian group with dual group \(\Gamma \), let \(\mu\) be a power bounded measure on \(G\), and let \(A=[ a_{n,k}]_{n,k=0}^{\infty}\) be a strongly regular matrix. We show that the sequence \(\{\sum_{k=0}^{\infty}a_{n,k}\mu^{k}\ast f\}_{n=0}^{\infty}\) converges in the \(L^{1}\)-norm for every \(f\in L^{1}(G)\) if and only if \(\mathcal{F}_{\mu}:=\{\gamma \in \Gamma:\widehat{\mu}(\gamma) =1\} \) is clopen in \(\Gamma \), where \(\widehat{\mu}\) is the Fourier–Stieltjes transform of \(\mu \). If \(\mu \) is a probability measure, then \(\mathcal{F}_{\mu}\) is clopen in \(\Gamma \) if and only if the closed subgroup generated by the support of \(\mu \) is compact.
期刊介绍:
Functional Analysis and Its Applications publishes current problems of functional analysis, including representation theory, theory of abstract and functional spaces, theory of operators, spectral theory, theory of operator equations, and the theory of normed rings. The journal also covers the most important applications of functional analysis in mathematics, mechanics, and theoretical physics.