Metric geodesics of isometries in a Hilbert space and the extension problem

Authors:
Esteban Andruchow, Lázaro Recht and Alejandro Varela

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2527-2537

MSC (2000):
Primary 47A05, 47B15, 58B20

DOI:
https://doi.org/10.1090/S0002-9939-07-08753-9

Published electronically:
March 21, 2007

MathSciNet review:
2302573

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the problem of finding *short* smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

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Additional Information

**Esteban Andruchow**

Affiliation:
Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J. M. Gutiérrez 1150, (1613) Los Polvorines, Argentina

Email:
eandruch@ungs.edu.ar

**Lázaro Recht**

Affiliation:
Departamento de Matemática P y A, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela

Email:
recht@usb.ve

**Alejandro Varela**

Affiliation:
Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J. M. Gutiérrez 1150, (1613) Los Polvorines, Argentina

Email:
avarela@ungs.edu.ar

DOI:
https://doi.org/10.1090/S0002-9939-07-08753-9

Keywords:
Isometries,
geodesics

Received by editor(s):
April 22, 2005

Received by editor(s) in revised form:
April 11, 2006

Published electronically:
March 21, 2007

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.