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A pseudospectral mapping theorem
Author(s):
S.-H.
Lui.
Journal:
Math. Comp.
72
(2003),
1841-1854.
MSC (2000):
Primary 15A18, 15A60, 65F15
Posted:
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.
References:
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- 1.
- M. Embree and L. N. Trefethen.
Generalizing eigenvalue theorems to pseudospectra theorems.
SIAM J. Sci. Comput., 23:583-590, 2001. MR 2002k:15019 - 2.
- L. N. Trefethen.
Pseudospectra of linear operators.
SIAM Rev., 39:383-406, 1997. MR 98i:47004 - 3.
- L. N. Trefethen.
Spectra and pseudospectra: The behavior of nonnormal matrices and operators.
In M. Ainsworth, J. Levesley, and M. Marletta, editors, The Graduate Student's Guide to Numerical Analysis, pages 217-250. Springer-Verlag, Berlin, 1998. - 4.
- L. N. Trefethen.
Computation of pseudospectra.
Acta Numer., 8:247-295, 1999. MR 2002b:65062 - 5.
- T. G. Wright and L. N. Trefethen.
Large-scale computation of pseudospectra using ARPACK and Eigs.
SIAM J. Sci. Comput., 23:591-605, 2001. MR 2002h:65061
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Additional Information:
S.-H.
Lui
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
Email:
luish@cc.umanitoba.ca
DOI:
10.1090/S0025-5718-03-01542-4
PII:
S 0025-5718(03)01542-4
Keywords:
Pseudospectra,
eigenvalues,
spectral mapping theorem
Received by editor(s):
October 11, 2001
Received by editor(s) in revised form:
March 29, 2002
Posted:
May 20, 2003
Additional Notes:
This work was supported in part by a grant from NSERC
Copyright of article:
Copyright
2003,
American Mathematical Society
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