Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators
Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators
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非正曲率：希尔伯特-施密特算子的几何方法
DOI：
10.1016/j.difgeo.2007.06.016
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发表时间：
2007
影响因子：
0.5
通讯作者：
G. Larotonda
中科院分区：
文献类型：
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作者：
G. Larotonda
我们通过迹内积为正可逆单位化希尔伯特-施密特算子集合 Σ 给出黎曼结构。该度量使 Σ 成为非正弯曲、单连通且度量完备的希尔伯特流形。流形 Σ 是非紧型对称空间的通用模型：任何此类空间都可以等距嵌入到 Σ 中。我们给出了凸闭子流形 M 的内在代数表征。我们研究了此类子流形的等距群：我们证明了 GM（由 M 生成的 Banach-Lie 群）对 M 具有等距且传递的作用。此外，GM 允许相对于 M 进行极分解，即 GM≃M×K 作为希尔伯特流形（这里 K 是作用于 p=1 的各向同性） Ig:p↦gpg*)，并且 GM/K≃M 所以 M 是齐次空间。我们通过测地凸子流形 M 获得了几个分解定理。这些分解是通过非线性但解析的正交投影 ΠM:Σ→M 获得的，该映射是测地距离的收缩。作为副产品，我们证明了同构 NM≃Σ（这里 NM 代表凸闭子流形 M 的法丛）。写下固定 ea 的因式分解，我们得到 ea=exevex，其中 ex∈M 且 v 在 p=1 处与 M 正交。作为推论，我们获得了全组可逆元素 G≃M×exp(T1M⊥)×K 的分解。
We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, GMadmits a polar decomposition relative to M, namely GM≃M×K as Hilbert manifolds (here K is the isotropy of p=1 for the action Ig:p↦gpg∗), and also GM/K≃M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection ΠM:Σ→M, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM≃Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea=exevexwith ex∈M and v orthogonal to M at p=1. As a corollary we obtain decompositions for the full group of invertible elements G≃M×exp(T1M⊥)×K.