Nonpositive curvature: A geometrical approach to Hilbert–Schmidt operators

By G. Larotonda. 2007

51 citations recorded by the source corpus.

Abstract 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 G M , the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, G M admits a polar decomposition relative to M, namely G M ≃ M × K as Hilbert manifolds (here K is the isotropy of p = 1 for the action I g : p ↦ g p g ∗ ), and also G M / 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 Π M : Σ → M , a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism N M ≃ Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed e a , we obtain e a = e x e v e x with e x ∈ 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 ( T 1 M ⊥ ) × K .

Subfields: Applied Mathematics, Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/89171eda11fe9865e0aee9f698f746f4325bec83

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In Applied Mathematics

1. Miklós Pálfia
2. Yongdo Lim
3. Marco San-Biagio
4. Vittorio Murino
5. Loris Bazzani
6. Ha Quang Minh

In Astronomy and Astrophysics

1. Marco San-Biagio
2. Vittorio Murino
3. Loris Bazzani
4. Ha Quang Minh
5. Miklós Pálfia
6. Yongdo Lim

Citations in the corpus

• Weighted deterministic walks for the least squares mean on Hadamard spaces (2014, background): “The tangent space is identified with HSR = {λ + a : λ ∈ R, a = a∗ is a Hilbert–Schmidt operator} [7, 11].”
• Approximate Log-Hilbert-Schmidt Distances between Covariance Operators for Image Classification (2016, methodology): “We wish to emphasize that the Log-Euclidean metric is itself a Riemannian metric on Sym ++ ( n ) , a so-called bi-invariant metric , which is a fact not always discussed in recent theoretical work in computer vision, e.g. [13].”
• Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces (2014, background): “These operators have recently been shown to form an infinite-dimensional Riemann-Hilbert manifold by [14, 1, 15], who formulated the infinite-dimensional version of the affine-invariant metric from a purely mathematical viewpoint.”