Metrics in the set of partial isometries with finite rank

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 -