{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:47:58Z","timestamp":1776728878532,"version":"3.51.2"},"reference-count":5,"publisher":"American Mathematical Society (AMS)","issue":"244","license":[{"start":{"date-parts":[[2004,5,20]],"date-time":"2004-05-20T00:00:00Z","timestamp":1085011200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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.<\/p>","DOI":"10.1090\/s0025-5718-03-01542-4","type":"journal-article","created":{"date-parts":[[2003,6,20]],"date-time":"2003-06-20T10:23:27Z","timestamp":1056104607000},"page":"1841-1854","source":"Crossref","is-referenced-by-count":9,"title":["A pseudospectral mapping theorem"],"prefix":"10.1090","volume":"72","author":[{"given":"S.-H.","family":"Lui","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,5,20]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1137\/S1064827500373012","article-title":"Generalizing eigenvalue theorems to pseudospectra theorems","volume":"23","author":"Embree, Mark","year":"2001","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"383","DOI":"10.1137\/S0036144595295284","article-title":"Pseudospectra of linear operators","volume":"39","author":"Trefethen, Lloyd N.","year":"1997","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"3","unstructured":"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\u2019s Guide to Numerical Analysis, pages 217\u2013250. Springer-Verlag, Berlin, 1998."},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1017\/S0962492900002932","article-title":"Computation of pseudospectra","author":"Trefethen, Lloyd N.","year":"1999","ISBN":"https:\/\/id.crossref.org\/isbn\/0521770882"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1137\/S106482750037322X","article-title":"Large-scale computation of pseudospectra using ARPACK and eigs","volume":"23","author":"Wright, Thomas G.","year":"2001","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01542-4\/S0025-5718-03-01542-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01542-4\/S0025-5718-03-01542-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:25:33Z","timestamp":1776727533000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-244\/S0025-5718-03-01542-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,5,20]]},"references-count":5,"journal-issue":{"issue":"244","published-print":{"date-parts":[[2003,10]]}},"alternative-id":["S0025-5718-03-01542-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01542-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2003,5,20]]}}}