{"title":"Quasi-isometric liftings for operators similar to contractions","authors":"Laurian Suciu, Andra-Maria Stoica","doi":"10.1016/j.laa.2025.01.006","DOIUrl":null,"url":null,"abstract":"<div><div>A class of quasi-isometric liftings for the operators <em>T</em> similar to contractions in Hilbert spaces <span><math><mi>H</mi></math></span> is studied. These liftings are isometric operators on their ranges, and are naturally induced by <em>T</em> and an invertible intertwiner of <em>T</em> with a contraction. In the case when <em>T</em> is a quasicontraction, meaning that <em>T</em> is contractive on its range, we obtain a quasi-isometric lifting on a space <span><math><mi>K</mi><mo>⊃</mo><mi>H</mi></math></span>, which is isometric on <span><math><mi>K</mi><mo>⊖</mo><mi>H</mi></math></span>. Some liftings with closed ranges, or even similar to quasinormal partial isometries are mentioned. Additionally, we study the isomorphic minimal quasi-isometric liftings for <em>T</em>, as well as the uniqueness property of such liftings. Our results show similarities with those from the isometric dilation theory for contractions, although our context is more general than that of the latter.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"709 ","pages":"Pages 40-57"},"PeriodicalIF":1.1000,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379525000060","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2025/1/10 0:00:00","PubModel":"Epub","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
A class of quasi-isometric liftings for the operators T similar to contractions in Hilbert spaces is studied. These liftings are isometric operators on their ranges, and are naturally induced by T and an invertible intertwiner of T with a contraction. In the case when T is a quasicontraction, meaning that T is contractive on its range, we obtain a quasi-isometric lifting on a space , which is isometric on . Some liftings with closed ranges, or even similar to quasinormal partial isometries are mentioned. Additionally, we study the isomorphic minimal quasi-isometric liftings for T, as well as the uniqueness property of such liftings. Our results show similarities with those from the isometric dilation theory for contractions, although our context is more general than that of the latter.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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