Bhatia, Rajendra ; Davis, Chandler ; McIntosh, Alan
(1983)
*Perturbation of spectral subspaces and solution of linear operator equations*
Linear Algebra and its Applications, 52-53
.
pp. 45-67.
ISSN 0024-3795

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0024-3795(83)80007-X

## Abstract

Let A and B be normal operators on a Hilbert space. Let K_{A} and K_{B} be subsets of the complex plane, at distance at least δ from each other; let E be the spectral projector for A belonging to K_{A}, and let F be the spectral projector for B belonging to K_{B}. Our main results are estimates of the form δ||EF||< c||E(A − B)F||; in some special situations, the constant c is as low as 1. As an application, we prove, for an absolute constant d, that if the space is finite-dimensional and if A and B are normal with ||;A-B||≤ε/d, then the spectrum of B can be obtained from that of A (multiplicities counted) by moving each eigenvalue by at most ϵ. Our main results have equivalent formulations as statements about the operator equation AQ - QB = S. Let A and B be normal operators on perhaps different Hilbert spaces. Assume σ(A)K_{A} and σ(B) K_{B}, where K_{A}, K_{B}, and δ are as before. Then we give estimates of the forms δ||Q||⩽c||AQ − QB||.

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ID Code: | 97446 |

Deposited On: | 11 Feb 2013 05:32 |

Last Modified: | 11 Feb 2013 05:32 |

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