{"title":"Indeterminate Jacobi operators II","authors":"Christian Berg, Ryszard Szwarc","doi":"10.1007/s43036-026-00537-2","DOIUrl":null,"url":null,"abstract":"<div><p>We consider the Jacobi operator (<i>T</i>, <i>D</i>(<i>T</i>)) associated with an indeterminate Hamburger moment problem, and present countable subsets <i>S</i> of the domain <i>D</i>(<i>T</i>) such that <span>\\({{\\,\\textrm{span}\\,}}(S)\\)</span> is dense in <span>\\(\\ell ^2.\\)</span> As an example we have <span>\\(S=\\{(p_n(u))+B(u)(p_n(0)) \\mid D(u)=0, u \\ne 0\\},\\)</span> where <span>\\((p_n)\\)</span> denotes the orthonormal polynomials of the moment problem and <i>B</i>, <i>D</i> are two of the Nevanlinna functions. It is also proved that sets like <i>S</i> are optimal in the sense that if one vector is removed, then the span is no longer dense.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"11 4","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2026-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-026-00537-2.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Operator Theory","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s43036-026-00537-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
We consider the Jacobi operator (T, D(T)) associated with an indeterminate Hamburger moment problem, and present countable subsets S of the domain D(T) such that \({{\,\textrm{span}\,}}(S)\) is dense in \(\ell ^2.\) As an example we have \(S=\{(p_n(u))+B(u)(p_n(0)) \mid D(u)=0, u \ne 0\},\) where \((p_n)\) denotes the orthonormal polynomials of the moment problem and B, D are two of the Nevanlinna functions. It is also proved that sets like S are optimal in the sense that if one vector is removed, then the span is no longer dense.
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