## On a subspace perturbation problem

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- by Vadim Kostrykin, Konstantin A. Makarov and Alexander K. Motovilov
- Proc. Amer. Math. Soc.
**131**(2003), 3469-3476 - DOI: https://doi.org/10.1090/S0002-9939-03-06917-X
- Published electronically: February 14, 2003

## 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 \[ \mathsf {E}_A(\sigma )\quad \text {and} \quad \mathsf {E}_{A+V}\big (\{\lambda | \mathrm {dist}(\lambda , \sigma )<d/2\}\big )\] for $A$ and $A+V$ is less than one whenever either (i) $\|V\|<\frac {2}{2+\pi }d$ or (ii) $\|V\|<\frac {1}{2}d$ and certain assumptions on the mutual disposition of the sets $\sigma$ and $\Sigma$ are satisfied.## References

- Vadim M. Adamjan and Heinz Langer,
*Spectral properties of a class of rational operator valued functions*, J. Operator Theory**33**(1995), no. 2, 259–277. MR**1354980** - N. I. Akhiezer and I. M. Glazman,
*Theory of linear operators in Hilbert space*, Dover Publications, Inc., New York, 1993. Translated from the Russian and with a preface by Merlynd Nestell; Reprint of the 1961 and 1963 translations; Two volumes bound as one. MR**1255973** - J. Avron, R. Seiler, and B. Simon,
*The index of a pair of projections*, J. Funct. Anal.**120**(1994), no. 1, 220–237. MR**1262254**, DOI 10.1006/jfan.1994.1031 - Rajendra Bhatia, Chandler Davis, and Alan McIntosh,
*Perturbation of spectral subspaces and solution of linear operator equations*, Linear Algebra Appl.**52/53**(1983), 45–67. MR**709344**, DOI 10.1016/0024-3795(83)80007-X - Rajendra Bhatia, Chandler Davis, and Paul Koosis,
*An extremal problem in Fourier analysis with applications to operator theory*, J. Funct. Anal.**82**(1989), no. 1, 138–150. MR**976316**, DOI 10.1016/0022-1236(89)90095-5 - Chandler Davis,
*Separation of two linear subspaces*, Acta Sci. Math. (Szeged)**19**(1958), 172–187. MR**98980** - H. Davenport,
*On Waring’s problem for cubes*, Acta Math.**71**(1939), 123–143. MR**26**, DOI 10.1007/BF02547752 - Chandler Davis and W. M. Kahan,
*The rotation of eigenvectors by a perturbation. III*, SIAM J. Numer. Anal.**7**(1970), 1–46. MR**264450**, DOI 10.1137/0707001 - R. McEachin,
*Closing the gap in a subspace perturbation bound*, Linear Algebra Appl.**180**(1993), 7–15. MR**1206407**, DOI 10.1016/0024-3795(93)90522-P - Tosio Kato,
*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473**

## Bibliographic 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
- 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
- © Copyright 2003 by the authors
- Journal: Proc. Amer. Math. Soc.
**131**(2003), 3469-3476 - MSC (2000): Primary 47A55, 47A15; Secondary 47B15
- DOI: https://doi.org/10.1090/S0002-9939-03-06917-X
- MathSciNet review: 1991758