Metric geometry in infinite dimensional Stiefel manifolds

Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredho...

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Autor principal: Chiumiento, Eduardo Hernán
Formato: Articulo
Lenguaje:Inglés
Publicado: 2010
Materias:
Matemática
Banach ideal
Finsler metric
Minimal curves
Partial isometry
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/82430
Aporte de:
SEDICI (UNLP) de Universidad Nacional de La Plata
id I19-R120-10915-82430
record_format dspace
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
Banach ideal
Finsler metric
Minimal curves
Partial isometry
spellingShingle Matemática
Banach ideal
Finsler metric
Minimal curves
Partial isometry
Chiumiento, Eduardo Hernán
Metric geometry in infinite dimensional Stiefel manifolds
topic_facet Matemática
Banach ideal
Finsler metric
Minimal curves
Partial isometry
description Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.
format Articulo
Articulo
author Chiumiento, Eduardo Hernán
author_facet Chiumiento, Eduardo Hernán
author_sort Chiumiento, Eduardo Hernán
title Metric geometry in infinite dimensional Stiefel manifolds
title_short Metric geometry in infinite dimensional Stiefel manifolds
title_full Metric geometry in infinite dimensional Stiefel manifolds
title_fullStr Metric geometry in infinite dimensional Stiefel manifolds
title_full_unstemmed Metric geometry in infinite dimensional Stiefel manifolds
title_sort metric geometry in infinite dimensional stiefel manifolds
publishDate 2010
url http://sedici.unlp.edu.ar/handle/10915/82430
work_keys_str_mv AT chiumientoeduardohernan metricgeometryininfinitedimensionalstiefelmanifolds
bdutipo_str Repositorios
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