{"title":"单元算子的共轭，I","authors":"Javad Mashreghi, Marek Ptak, William T. Ross","doi":"10.1007/s13324-024-00924-z","DOIUrl":null,"url":null,"abstract":"<div><p>If <i>U</i> is a unitary operator on a separable complex Hilbert space <span>\\(\\mathcal {H}\\)</span>, an application of the spectral theorem says there is a conjugation <i>C</i> on <span>\\(\\mathcal {H}\\)</span> (an antilinear, involutive, isometry on <span>\\(\\mathcal {H}\\)</span>) for which <span>\\( C U C = U^{*}.\\)</span> In this paper, we fix a unitary operator <i>U</i> and describe <i>all</i> of the conjugations <i>C</i> which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for <i>U</i> if and only if it is invariant for any conjugation <i>C</i> for which <span>\\(CUC = U^{*}\\)</span>.\n</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"14 3","pages":""},"PeriodicalIF":1.4000,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13324-024-00924-z.pdf","citationCount":"0","resultStr":"{\"title\":\"Conjugations of unitary operators, I\",\"authors\":\"Javad Mashreghi, Marek Ptak, William T. Ross\",\"doi\":\"10.1007/s13324-024-00924-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>If <i>U</i> is a unitary operator on a separable complex Hilbert space <span>\\\\(\\\\mathcal {H}\\\\)</span>, an application of the spectral theorem says there is a conjugation <i>C</i> on <span>\\\\(\\\\mathcal {H}\\\\)</span> (an antilinear, involutive, isometry on <span>\\\\(\\\\mathcal {H}\\\\)</span>) for which <span>\\\\( C U C = U^{*}.\\\\)</span> In this paper, we fix a unitary operator <i>U</i> and describe <i>all</i> of the conjugations <i>C</i> which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for <i>U</i> if and only if it is invariant for any conjugation <i>C</i> for which <span>\\\\(CUC = U^{*}\\\\)</span>.\\n</p></div>\",\"PeriodicalId\":48860,\"journal\":{\"name\":\"Analysis and Mathematical Physics\",\"volume\":\"14 3\",\"pages\":\"\"},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2024-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s13324-024-00924-z.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Mathematical Physics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s13324-024-00924-z\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Analysis and Mathematical Physics","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s13324-024-00924-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
如果 U 是可分离复希尔伯特空间上的一个单元算子，那么应用谱定理就可以知道在 \(\mathcal {H}\) 上存在一个共轭 C（\(\mathcal {H}/\)上的一个反线性、非卷积、等势），对于这个共轭 C，\( C U C = U^{*}.\在本文中，我们将固定一个单元算子 U，并描述所有满足这一性质的共轭 C。我们的结果表明，当且仅当一个子空间对于任意共轭 C 都是不变的（\(CUC = U^{*}\)），那么这个子空间对于 U 就是超不变的。
If U is a unitary operator on a separable complex Hilbert space \(\mathcal {H}\), an application of the spectral theorem says there is a conjugation C on \(\mathcal {H}\) (an antilinear, involutive, isometry on \(\mathcal {H}\)) for which \( C U C = U^{*}.\) In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which \(CUC = U^{*}\).
期刊介绍：
Analysis and Mathematical Physics (AMP) publishes current research results as well as selected high-quality survey articles in real, complex, harmonic; and geometric analysis originating and or having applications in mathematical physics. The journal promotes dialog among specialists in these areas.
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