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 $\mathcal {H}$. 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 $V_1$.

- E. Andruchow, G. Corach, and M. Mbekhta,
*On the geometry of generalized inverses*, Math. Nachr.**278**(2005), no. 7-8, 756–770. MR**2141955**, DOI https://doi.org/10.1002/mana.200310270 - C. J. Atkin,
*The Finsler geometry of groups of isometries of Hilbert space*, J. Austral. Math. Soc. Ser. A**42**(1987), no. 2, 196–222. MR**869747** - Chandler Davis, W. M. Kahan, and H. F. Weinberger,
*Norm-preserving dilations and their applications to optimal error bounds*, SIAM J. Numer. Anal.**19**(1982), no. 3, 445–469. MR**656462**, DOI https://doi.org/10.1137/0719029 - Carlos E. Durán, Luis E. Mata-Lorenzo, and Lázaro Recht,
*Metric geometry in homogeneous spaces of the unitary group of a $C^*$-algebra. I. Minimal curves*, Adv. Math.**184**(2004), no. 2, 342–366. MR**2054019**, DOI https://doi.org/10.1016/S0001-8708%2803%2900148-8 - Carlos E. Durán, Luis E. Mata-Lorenzo, and Lázaro Recht,
*Metric geometry in homogeneous spaces of the unitary group of a $C^\ast $-algebra. II. Geodesics joining fixed endpoints*, Integral Equations Operator Theory**53**(2005), no. 1, 33–50. MR**2183595**, DOI https://doi.org/10.1007/s00020-003-1305-1 - Shoshichi Kobayashi and Katsumi Nomizu,
*Foundations of differential geometry. Vol. I*, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1996. Reprint of the 1963 original; A Wiley-Interscience Publication. MR**1393940** - M. G. Kreĭn,
*The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. II*, Mat. Sbornik N.S.**21(63)**(1947), 365–404 (Russian). MR**0024575** - P. R. Halmos and J. E. McLaughlin,
*Partial isometries*, Pacific J. Math.**13**(1963), 585–596. MR**157241** - Luis E. Mata-Lorenzo and Lázaro Recht,
*Infinite-dimensional homogeneous reductive spaces*, Acta Cient. Venezolana**43**(1992), no. 2, 76–90 (English, with English and Spanish summaries). MR**1185114** - Horacio Porta and Lázaro Recht,
*Minimality of geodesics in Grassmann manifolds*, Proc. Amer. Math. Soc.**100**(1987), no. 3, 464–466. MR**891146**, DOI https://doi.org/10.1090/S0002-9939-1987-0891146-6 - Frigyes Riesz and Béla Sz.-Nagy,
*Functional analysis*, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR**0071727**

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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

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.