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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Metric geodesics of isometries in a Hilbert space and the extension problem
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by Esteban Andruchow, Lázaro Recht and Alejandro Varela PDF
Proc. Amer. Math. Soc. 135 (2007), 2527-2537 Request permission

Abstract:

We consider the problem of finding short smooth curves of isometries in a Hilbert space $\mathcal {H}$. The length of a smooth curve $\gamma (t)$, $t\in [0,1]$, is measured by means of $\int _0^1 \|\dot {\gamma }(t)\|\ dt$, where $\|\ \|$ denotes the usual norm of operators. The initial value problem is solved: for any isometry $V_0$ and each tangent vector at $V_0$ (which is an operator of the form $iXV_0$ with $X^*=X$) with norm less than or equal to $\pi$, there exist curves of the form $e^{itZ}V_0$, with initial velocity $iZV_0=iXV_0$, 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 \[ X_0|_{R(V_0)}:R(V_0)\to \mathcal {H}, \] find all possible $Z^*=Z$ extending $X_0|_{R(V_0)}$ to all $\mathcal {H}$, with $\|Z\|=\|X_0\|$. We also consider the problem of finding metric geodesics joining two given isometries $V_0$ and $V_1$. It is well known that if there exists a continuous path joining $V_0$ and $V_1$, then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining $V_0$ and $V_1$.
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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
  • MR Author ID: 26110
  • 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
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2302573