A pseudospectral mapping theorem
HTML articles powered by AMS MathViewer
- by S.-H. Lui;
- Math. Comp. 72 (2003), 1841-1854
- DOI: https://doi.org/10.1090/S0025-5718-03-01542-4
- Published electronically: May 20, 2003
- PDF | Request permission
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
- Mark Embree and Lloyd N. Trefethen, Generalizing eigenvalue theorems to pseudospectra theorems, SIAM J. Sci. Comput. 23 (2001), no. 2, 583–590. Copper Mountain Conference (2000). MR 1861266, DOI 10.1137/S1064827500373012
- Lloyd N. Trefethen, Pseudospectra of linear operators, SIAM Rev. 39 (1997), no. 3, 383–406. MR 1469941, DOI 10.1137/S0036144595295284
- 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.
- Lloyd N. Trefethen, Computation of pseudospectra, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp. 247–295. MR 1819647, DOI 10.1017/S0962492900002932
- Thomas G. Wright and Lloyd N. Trefethen, Large-scale computation of pseudospectra using ARPACK and eigs, SIAM J. Sci. Comput. 23 (2001), no. 2, 591–605. Copper Mountain Conference (2000). MR 1861267, DOI 10.1137/S106482750037322X
Bibliographic Information
- S.-H. Lui
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
- Email: luish@cc.umanitoba.ca
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Math. Comp. 72 (2003), 1841-1854
- MSC (2000): Primary 15A18, 15A60, 65F15
- DOI: https://doi.org/10.1090/S0025-5718-03-01542-4
- MathSciNet review: 1986807