Metrics in the set of partial isometries with finite rank

Esteban Andruchow; Gustavo Corach

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2005)

  • Volume: 16, Issue: 1, page 31-44
  • ISSN: 1120-6330

Abstract

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Let I be the set of partial isometries with finite rank of an infinite dimensional Hilbert space H . We show that I is a smooth submanifold of the Hilbert space B 2 H of Hilbert-Schmidt operators of H and that each connected component is the set I N , which consists of all partial isometries of rank N < . Furthermore, I is a homogeneous space of U × U , where U is the classical Banach-Lie group of unitary operators of H , which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics in I : one, via the ambient inner product of B 2 H , the other, by means of the group action. We show that both metrics are equivalent and complete.

How to cite

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Andruchow, Esteban, and Corach, Gustavo. "Metrics in the set of partial isometries with finite rank." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.1 (2005): 31-44. <http://eudml.org/doc/252394>.

@article{Andruchow2005,
abstract = {Let $\mathcal\{I\}_\{(\infty)\}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal\{H\}$. We show that $\mathcal\{I\}_\{(\infty)\}$ is a smooth submanifold of the Hilbert space $\mathcal\{B\}_\{2\}(\mathcal\{H\})$ of Hilbert-Schmidt operators of $\mathcal\{H\}$ and that each connected component is the set $\mathcal\{I\}_\{N\}$, which consists of all partial isometries of rank $N < \infty$. Furthermore, $\mathcal\{I\}_\{(\infty)\}$ is a homogeneous space of $\mathcal\{U\}_\{(\infty)\} \times \mathcal\{U\}_\{(\infty)\}$, where $\mathcal\{U\}_\{(\infty)\}$ is the classical Banach-Lie group of unitary operators of $\mathcal\{H\}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics in $\mathcal\{I\}_\{(\infty)\}$: one, via the ambient inner product of $\mathcal\{B\}_\{2\}(\mathcal\{H\})$, the other, by means of the group action. We show that both metrics are equivalent and complete.},
author = {Andruchow, Esteban, Corach, Gustavo},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Partial isometry; Projection; Riemannian metric; partial isometry; projection},
language = {eng},
month = {3},
number = {1},
pages = {31-44},
publisher = {Accademia Nazionale dei Lincei},
title = {Metrics in the set of partial isometries with finite rank},
url = {http://eudml.org/doc/252394},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Andruchow, Esteban
AU - Corach, Gustavo
TI - Metrics in the set of partial isometries with finite rank
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/3//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 1
SP - 31
EP - 44
AB - Let $\mathcal{I}_{(\infty)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that $\mathcal{I}_{(\infty)}$ is a smooth submanifold of the Hilbert space $\mathcal{B}_{2}(\mathcal{H})$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set $\mathcal{I}_{N}$, which consists of all partial isometries of rank $N < \infty$. Furthermore, $\mathcal{I}_{(\infty)}$ is a homogeneous space of $\mathcal{U}_{(\infty)} \times \mathcal{U}_{(\infty)}$, where $\mathcal{U}_{(\infty)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics in $\mathcal{I}_{(\infty)}$: one, via the ambient inner product of $\mathcal{B}_{2}(\mathcal{H})$, the other, by means of the group action. We show that both metrics are equivalent and complete.
LA - eng
KW - Partial isometry; Projection; Riemannian metric; partial isometry; projection
UR - http://eudml.org/doc/252394
ER -

References

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  2. ANDRUCHOW, E. - CORACH, G. - MBEKHTA, M., On the geometry of generalized inverses. Math. Nacht., to appear. Zbl1086.46037MR2141955DOI10.1002/mana.200310270
  3. CORACH, G. - PORTA, H. - RECHT, L., The geometry of spaces of projections in C * -algebras. Adv. Math., 101, 1993, 59-77. Zbl0799.46067MR1239452DOI10.1006/aima.1993.1041
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  5. GRAMSCH, B., Relative Inversion in der Störungstheorie von Operatoren und ψ -Algebren. Math. Ann., 269, 1984, 27-71. Zbl0661.47037MR756775DOI10.1007/BF01455995
  6. HALMOS, P.R. - MC LAUGHLIN, J.E., Partial isometries. Pacific J. Math., 13, 1963, 585-596. Zbl0189.13402MR157241
  7. KUIPER, N., The homotopy type of the unitary group of Hilbert space. Topology, 3, 1965, 19-30. Zbl0129.38901MR179792
  8. LANG, S., Introduction to differentiable manifolds. Second edition, Universitext, Springer-Verlag, New York2002. Zbl0103.15101MR1931083
  9. MBEKHTA, M. - STRATILA, S., Homotopy classes of partial isometries in von Neumann algebras. Acta Sci. Math. (Szeged), 68, 2002, 271-277. Zbl1027.46073MR1916580
  10. RAEBURN, I., The relationship between a commutative Banach algebra and its maximal ideal space. J. Funct. Anal., 25, 1977, 366-390. Zbl0353.46041MR458180

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