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Geometry of $ \mathfrak{I}$-Stiefel manifolds


Author: Eduardo Chiumiento
Journal: Proc. Amer. Math. Soc. 138 (2010), 341-353
MSC (2000): Primary 22E65; Secondary 47B10, 58B20
DOI: https://doi.org/10.1090/S0002-9939-09-10080-1
Published electronically: August 28, 2009
MathSciNet review: 2550200
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Abstract: Let $ \mathfrak{I}$ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space $ \mathcal{H}$ and $ \mathcal{U}(\mathcal{H})_{\mathfrak{I}}$ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in $ \mathfrak{I}$. In this paper we study the geometry of the unitary orbits

$\displaystyle \{ U V : U \in \mathcal{U}(\mathcal{H})_{\mathfrak{I}}\}$

and

$\displaystyle \{ U V W^* : U,W \in \mathcal{U}(\mathcal{H})_{\mathfrak{I}}\},$

where $ V$ is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in $ V + \mathfrak{I}$, and while the first one consists of partial isometries with the same kernel of $ V$, the second is given by partial isometries such that their initial projections and $ V^*V$ have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space $ V + \mathfrak{I}$ and homogeneous reductive spaces of $ \mathcal{U}(\mathcal{H})_{\mathfrak{I}}$ and $ \mathcal{U}(\mathcal{H})_{\mathfrak{I}} \times \mathcal{U}(\mathcal{H})_{\mathfrak{I}}$ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of $ \mathcal{U}(\mathcal{H})_{\mathfrak{I}}$ (or $ \mathcal{U}(\mathcal{H})_{\mathfrak{I}} \times\mathcal{U}(\mathcal{H})_{\mathfrak{I}}$) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.


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Additional Information

Eduardo Chiumiento
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Calles 50 y 115, (1900) La Plata, Argentina
Email: eduardo@mate.unlp.edu.ar

DOI: https://doi.org/10.1090/S0002-9939-09-10080-1
Keywords: Partial isometry, Banach ideal, Finsler metric
Received by editor(s): September 18, 2008
Received by editor(s) in revised form: April 22, 2009
Published electronically: August 28, 2009
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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