{"title":"Sphere bundle over the set of inner products in a Hilbert space","authors":"E. Andruchow , M.E. Di Iorio y Lucero","doi":"10.1016/j.difgeo.2023.102092","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mo>(</mo><mi>H</mi><mo>,</mo><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo><mo>)</mo></math></span><span> be a complex Hilbert space and </span><span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span><span> the space of bounded linear operators in </span><span><math><mi>H</mi></math></span>. Any other equivalent inner product in <span><math><mi>H</mi></math></span> is of the form <span><math><msub><mrow><mo>〈</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo>〉</mo></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><mo>〈</mo><mi>A</mi><mi>f</mi><mo>,</mo><mi>g</mi><mo>〉</mo></math></span> (<span><math><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>H</mi></math></span>) for some positive invertible operator <span><math><mi>A</mi><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>. In this paper we study the bundle <span><math><mi>M</mi></math></span> which consist of the unit sphere <span><math><mo>{</mo><mi>f</mi><mo>∈</mo><mi>H</mi><mo>:</mo><msub><mrow><mo>〈</mo><mi>f</mi><mo>,</mo><mi>f</mi><mo>〉</mo></mrow><mrow><mi>A</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>}</mo></math></span> over each (equivalent) inner product <span><math><msub><mrow><mo>〈</mo><mspace></mspace><mo>,</mo><mspace></mspace><mo>〉</mo></mrow><mrow><mi>A</mi></mrow></msub></math></span>, which due to the observation above can be defined<span><span><span><math><mi>M</mi><mo>=</mo><mo>{</mo><mo>(</mo><mi>A</mi><mo>,</mo><mi>f</mi><mo>)</mo><mo>∈</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>×</mo><mi>H</mi><mo>:</mo><mi>A</mi><mtext> is positive and invertible and </mtext><mo>〈</mo><mi>A</mi><mi>f</mi><mo>,</mo><mi>f</mi><mo>〉</mo><mo>=</mo><mn>1</mn><mo>}</mo><mo>.</mo></math></span></span></span> We prove that <span><math><mi>M</mi></math></span><span><span> is a complemented submanifold of the </span>Banach space </span><span><math><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>×</mo><mi>H</mi></math></span><span> and a homogeneous space of the Banach-Lie group </span><span><math><mi>G</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>⊂</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> of invertible operators. We introduce a reductive structure in <span><math><mi>M</mi></math></span><span>, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of </span><span><math><mi>M</mi></math></span>, for instance, the one obtained when the positive elements <em>A</em> describing the inner products lie in a prescribed C<sup>⁎</sup>-algebra <span><math><mi>A</mi><mo>⊂</mo><mi>B</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224523001183","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a complex Hilbert space and the space of bounded linear operators in . Any other equivalent inner product in is of the form () for some positive invertible operator . In this paper we study the bundle which consist of the unit sphere over each (equivalent) inner product , which due to the observation above can be defined We prove that is a complemented submanifold of the Banach space and a homogeneous space of the Banach-Lie group of invertible operators. We introduce a reductive structure in , and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of , for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C⁎-algebra .
设(H,〈,〉)为复希尔伯特空间,B(H)为 H 中的有界线性算子空间。对于某个正向可逆算子 A∈B(H),H 中任何其他等价内积的形式为〈f,g〉A=〈Af,g〉 (f,g∈H)。本文研究由单位球{f∈H:〈f,f〉A=1}在每个(等价)内积〈,〉A上构成的束 M,根据上述观察,可以定义M={(A,f)∈B(H)×H:A为正且可逆且〈Af,f〉=1}。我们证明 M 是巴纳赫空间 B(H)×H 的补集子漫空间,也是可反算子的巴纳赫-李群 G(H)⊂B(H) 的同调空间。我们在 M 中引入了还原结构,并研究了该还原结构诱导的线性连接的大地线性质。我们考虑 M 的某些子曲面,例如,当描述内积的正元素 A 位于规定的 C⁎-代数 A⊂B(H)中时得到的曲面。
期刊介绍:
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.