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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

On a subspace perturbation problem


Authors: Vadim Kostrykin, Konstantin A. Makarov and Alexander K. Motovilov
Journal: Proc. Amer. Math. Soc. 131 (2003), 3469-3476
MSC (2000): Primary 47A55, 47A15; Secondary 47B15
Published electronically: February 14, 2003
MathSciNet review: 1991758
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Abstract: We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let $A$ and $V$be bounded self-adjoint operators. Assume that the spectrum of $A$ consists of two disjoint parts $\sigma$ and $\Sigma$ such that $d=\text{dist}(\sigma, \Sigma)>0$. We show that the norm of the difference of the spectral projections

\begin{displaymath}\mathsf{E}_A(\sigma)\quad \text{and} \quad \mathsf{E}_{A+V}\b... ... \,\, {\ensuremath{\mathrm{dist}} }(\lambda, \sigma)<d/2\}\big)\end{displaymath}

for $A$ and $A+V$ is less than one whenever either (i) $\Vert V\Vert<\frac{2}{2+\pi}d$ or (ii) $\Vert V\Vert<\frac{1}{2}d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.


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Additional Information

Vadim Kostrykin
Affiliation: Fraunhofer-Institut für Lasertechnik, Steinbachstraße 15, D-52074, Aachen, Germany
Email: kostrykin@ilt.fhg.de, kostrykin@t-online.de

Konstantin A. Makarov
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: makarov@math.missouri.edu

Alexander K. Motovilov
Affiliation: Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia
Address at time of publication: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: motovilv@thsun1.jinr.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06917-X
PII: S 0002-9939(03)06917-X
Keywords: Perturbation theory, spectral subspaces
Received by editor(s): March 29, 2002
Received by editor(s) in revised form: May 30, 2002
Published electronically: February 14, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 by the authors