Manifolds of linear involutions

Kovarik, Zdislav V.

doi:10.1016/0024-3795(79)90164-2

“…Thus particular examples provided by functional spaces help to understand abstract results on the structure of geodesics of the Grassmann manifold previously obtained by Porta and Recht [37], Corach, Porta and Recht [16], Kovarik [30] and the first author [6,7]. Also it is interesting to point out here that the Grassmann manifold plays an essential role in the metric theory of general (infinite dimensional) homogeneous spaces arising on operator theory.…”

Section: Introductionmentioning

confidence: 68%

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Andruchow

¹

,

Chiumiento

²

,

Varela

³

2021

Journal of Mathematical Analysis and Applications

0

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Let H be a reproducing kernel Hilbert space of functions on a set X. We study the problem of finding a minimal geodesic of the Grassmann manifold of H that joins two subspaces consisting of functions which vanish on given finite subsets of X. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

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“…Thus particular examples provided by functional spaces help to understand abstract results on the structure of geodesics of the Grassmann manifold previously obtained by Porta and Recht [37], Corach, Porta and Recht [16], Kovarik [30] and the first author [6,7]. Also it is interesting to point out here that the Grassmann manifold plays an essential role in the metric theory of general (infinite dimensional) homogeneous spaces arising on operator theory.…”

Section: Introductionmentioning

confidence: 68%

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Andruchow¹,

Chiumiento²,

Varela³

2020

Preprint

0

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Let H be a reproducing kernel Hilbert space of functions on a set X. We study the problem of finding a minimal geodesic of the Grassmann manifold of H that joins two subspaces consisting of functions which vanish on given finite subsets of X. We establish a necessary and sufficient condition for existence and uniqueness of geodesics, and we then analyze it in examples. We discuss the relation of the geodesic distance with other known metrics when the mentioned finite subsets are singletons. We find estimates on the upper and lower eigenvalues of the unique self-adjoint operators which define the minimal geodesics, which can be made more precise when the underlying space is the Hardy space. Also for the Hardy space we discuss the existence of geodesics joining subspaces of functions vanishing on infinite subsets of the disk, and we investigate when the product of projections onto this type of subspaces is compact.

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“…As we shall see below, the connected components of R turn out to be a smooth homogeneous spaces of U (H). Thus, the present work can be seen as a contribution to the geometry of infinite dimensional homogeneous spaces arising in operator theory and operator algebras, which has been a subject of study in different settings, for instance Grassmann manifolds [1,6,9,16,21], Stiefel manifolds [4,8] and orbits of selfadjoint operators [2,7]. Also abstract homogeneous spaces are considered in [3,11,18]; for more examples and a detailed account we refer to the book [5].…”

Section: Introductionmentioning

confidence: 99%

“…As we shall see below, the connected components of R turn out to be a smooth homogeneous spaces of U (H). Thus, the present work can be seen as a contribution to the geometry of infinite dimensional homogeneous spaces arising in operator theory and operator algebras, which has been a subject of study in different settings, for instance Grassmann manifolds [1,6,9,16,21],…”

Section: Introductionmentioning

confidence: 99%

Canonical sphere bundles of the Grassmann manifold

Andruchow

¹

,

Chiumiento

²

,

Larotonda

³

2019

Geom Dedicata

0

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For a given Hilbert space H, consider the space of self-adjoint projections P(H). In this paper we study the differentiable structure of a canonical sphere bundle over P(H) given byWe establish the smooth action on R of the group of unitary operators of H, therefore R is an homogeneous space. Then we study the metric structure of R by endowing it first with the uniform quotient metric, which is a Finsler metric, and we establish minimality results for the geodesics. These are given by certain one-parameter groups of unitary operators, pushed into R by the natural action of the unitary group. Then we study the restricted bundle R + 2 given by considering only the projections in the restricted Grassmannian, locally modelled by Hilbert-Schmidt operators. Therefore we endow R + 2 with a natural Riemannian metric that can be obtained by declaring that the action of the group is a Riemannian submersion. We study the Levi-Civita connection of this metric and establish a Hopf-Rinow theorem for R + 2 , again obtaining a characterization of the geodesics as the image of certain one-parameter groups with special speeds. 1 1 2010 MSC: 22E65, 47B10, 58B20.

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Manifolds of linear involutions

Cited by 6 publications

References 2 publications

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Grassmann geometry of zero sets in reproducing kernel Hilbert spaces

Canonical sphere bundles of the Grassmann manifold

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