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Mathematics of Computation

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A pseudospectral mapping theorem

Author: S.-H. Lui
Journal: Math. Comp. 72 (2003), 1841-1854
MSC (2000): Primary 15A18, 15A60, 65F15
Published electronically: May 20, 2003
MathSciNet review: 1986807
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Abstract: The pseudospectrum has become an important quantity for analyzing stability of nonnormal systems. In this paper, we prove a mapping theorem for pseudospectra, extending an earlier result of Trefethen. Our result consists of two relations that are sharp and contains the spectral mapping theorem as a special case. Necessary and sufficient conditions for these relations to collapse to an equality are demonstrated. The theory is valid for bounded linear operators on Banach spaces. For normal matrices, a special version of the pseudospectral mapping theorem is also shown to be sharp. Some numerical examples illustrate the theory.

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Additional Information

S.-H. Lui
Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Keywords: Pseudospectra, eigenvalues, spectral mapping theorem
Received by editor(s): October 11, 2001
Received by editor(s) in revised form: March 29, 2002
Published electronically: May 20, 2003
Additional Notes: This work was supported in part by a grant from NSERC
Article copyright: © Copyright 2003 American Mathematical Society