Sphere bundle over the set of inner products in a Hilbert space

IF 0.6 4区 数学 Q3 MATHEMATICS Differential Geometry and its Applications Pub Date : 2023-12-14 DOI:10.1016/j.difgeo.2023.102092
E. Andruchow , M.E. Di Iorio y Lucero
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引用次数: 0

Abstract

Let (H,,) be a complex Hilbert space and B(H) the space of bounded linear operators in H. Any other equivalent inner product in H is of the form f,gA=Af,g (f,gH) for some positive invertible operator AB(H). In this paper we study the bundle M which consist of the unit sphere {fH:f,fA=1} over each (equivalent) inner product ,A, which due to the observation above can be definedM={(A,f)B(H)×H:A is positive and invertible and Af,f=1}. We prove that M is a complemented submanifold of the Banach space B(H)×H and a homogeneous space of the Banach-Lie group G(H)B(H) of invertible operators. We introduce a reductive structure in M, and study properties of the geodesics of the linear connection induced by this reductive structure. We consider certain submanifolds of M, for instance, the one obtained when the positive elements A describing the inner products lie in a prescribed C-algebra AB(H).

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希尔伯特空间内积集合上的球体束
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。
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来源期刊
CiteScore
1.00
自引率
20.00%
发文量
81
审稿时长
6-12 weeks
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.
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