{"title":"通过双盘哈代空间上的通用托普利兹算子论不变子空间问题","authors":"João Marcos R. do Carmo, Marcos S. Ferreira","doi":"10.1007/s00574-024-00386-8","DOIUrl":null,"url":null,"abstract":"<p>The <i>invariant subspace problem</i> (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for <span>\\(T^{*}_{\\varphi }|_{M}\\)</span> to have a non-trivial subspace where <span>\\(M\\subset H^{2}({\\mathbb {D}}^{2})\\)</span> is an invariant subspace of the Toeplitz operator <span>\\(T_{\\varphi }^{*}\\)</span> on the Hardy space over the bidisk <span>\\(H^{2}({\\mathbb {D}}^{2})\\)</span> induced by the symbol <span>\\(\\varphi \\in H^{\\infty }({\\mathbb {D}})\\)</span>. We then use this fact to obtain sufficient conditions for the ISP to be true.</p>","PeriodicalId":501417,"journal":{"name":"Bulletin of the Brazilian Mathematical Society, New Series","volume":"17 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Invariant Subspace Problem via Universal Toeplitz Operators on the Hardy Space Over the Bidisk\",\"authors\":\"João Marcos R. do Carmo, Marcos S. Ferreira\",\"doi\":\"10.1007/s00574-024-00386-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The <i>invariant subspace problem</i> (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for <span>\\\\(T^{*}_{\\\\varphi }|_{M}\\\\)</span> to have a non-trivial subspace where <span>\\\\(M\\\\subset H^{2}({\\\\mathbb {D}}^{2})\\\\)</span> is an invariant subspace of the Toeplitz operator <span>\\\\(T_{\\\\varphi }^{*}\\\\)</span> on the Hardy space over the bidisk <span>\\\\(H^{2}({\\\\mathbb {D}}^{2})\\\\)</span> induced by the symbol <span>\\\\(\\\\varphi \\\\in H^{\\\\infty }({\\\\mathbb {D}})\\\\)</span>. We then use this fact to obtain sufficient conditions for the ISP to be true.</p>\",\"PeriodicalId\":501417,\"journal\":{\"name\":\"Bulletin of the Brazilian Mathematical Society, New Series\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the Brazilian Mathematical Society, New Series\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00574-024-00386-8\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Brazilian Mathematical Society, New Series","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00574-024-00386-8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the Invariant Subspace Problem via Universal Toeplitz Operators on the Hardy Space Over the Bidisk
The invariant subspace problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this work, we obtain conditions for \(T^{*}_{\varphi }|_{M}\) to have a non-trivial subspace where \(M\subset H^{2}({\mathbb {D}}^{2})\) is an invariant subspace of the Toeplitz operator \(T_{\varphi }^{*}\) on the Hardy space over the bidisk \(H^{2}({\mathbb {D}}^{2})\) induced by the symbol \(\varphi \in H^{\infty }({\mathbb {D}})\). We then use this fact to obtain sufficient conditions for the ISP to be true.