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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Finsler geometry and actions of the $ p$-Schatten unitary groups


Authors: Esteban Andruchow, Gabriel Larotonda and Lázaro Recht
Journal: Trans. Amer. Math. Soc. 362 (2010), 319-344
MSC (2000): Primary 22E65; Secondary 58B20, 58E50
Published electronically: July 27, 2009
MathSciNet review: 2550153
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Abstract: Let $ p$ be an even positive integer and $ U_p(\mathcal{H})$ the Banach-Lie group of unitary operators $ u$ which verify that $ u-1$ belongs to the $ p$-Schatten ideal $ \mathcal{B}_p(\mathcal{H})$. Let $ \mathcal{O}$ be a smooth manifold on which $ U_p(\mathcal{H})$ acts transitively and smoothly. Then one can endow $ \mathcal{O}$ with a natural Finsler metric in terms of the $ p$-Schatten norm and the action of $ U_p(\mathcal{H})$. Our main result establishes that for any pair of given initial conditions

$\displaystyle x\in\mathcal{O}\hbox{ and } X\in(T \mathcal{O})_x $

there exists a curve $ \delta(t)=e^{tz}\cdot x$ in $ \mathcal{O}$, with $ z$ a skew-hermitian element in the $ p$-Schatten class, such that

$\displaystyle \delta(0)=x \hbox{ and } \dot{\delta}(0)=X, $

which remains minimal as long as $ t\Vert z\Vert _p\le \pi/4$. Moreover, $ \delta$ is unique with these properties. We also show that the metric space $ (\mathcal{O},d)$ (where $ d$ is the rectifiable distance) is complete. In the process we establish minimality results in the groups $ U_p(\mathcal{H})$ and a convexity property for the rectifiable distance. As an example of these spaces, we treat the case of the unitary orbit

$\displaystyle \mathcal{O}=\{uAu^*: u\in U_p(\mathcal{H})\} $

of a self-adjoint operator $ A\in\mathcal{B}(\mathcal{H})$.


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Additional Information

Esteban Andruchow
Affiliation: Instituto de Ciencias, J. M. Gutierrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina
Email: eandruch@ungs.edu.ar

Gabriel Larotonda
Affiliation: Instituto de Ciencias, J. M. Gutierrez 1150, (1613) Los Polvorines, Buenos Aires, Argentina
Email: glaroton@ungs.edu.ar

Lázaro Recht
Affiliation: Departamento de Matemática P y A, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela
Email: recht@usb.ve

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04877-6
PII: S 0002-9947(09)04877-6
Received by editor(s): December 19, 2007
Published electronically: July 27, 2009
Additional Notes: This work was partially supported by IAM-CONICET
Dedicated: In memory of A. R. Larotonda (1939-2005)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.