{"title":"On unitary equivalence to a self-adjoint or doubly–positive Hankel operator","authors":"R.T.W. Martin","doi":"10.1515/conop-2022-0132","DOIUrl":null,"url":null,"abstract":"Abstract Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"114 - 126"},"PeriodicalIF":0.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Concrete Operators","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/conop-2022-0132","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Abstract Let A be a bounded, injective and self-adjoint linear operator on a complex separable Hilbert space. We prove that there is a pure isometry, V, so that AV > 0 and A is Hankel with respect to V, i.e. V*A = AV, if and only if A is not invertible. The isometry V can be chosen to be isomorphic to N ∈ ℕ ∪ {+∞} copies of the unilateral shift if A has spectral multiplicity at most N. We further show that the set of all isometries, V, so that A is Hankel with respect to V, are in bijection with the set of all closed, symmetric restrictions of A−1.