{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:30:57Z","timestamp":1787329857113,"version":"build-2736575974"},"reference-count":15,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We extend the proof in [M. Crouzeix and C. Palencia, SIAM J. Matrix Anal. Appl., 38 (2017), pp. 649--655] to show that other regions in the complex plane are $K$-spectral sets. In particular, we show that various annular regions are $(1 + \\sqrt{2} )$-spectral sets and that a more general convex region with a circular hole or cutout is a $(3 + 2 \\sqrt{3} )$-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylov subspace methods for approximating $f(A)b$, where $A$ is a square matrix, $b$ is a given vector, and $f$ is a function that can be uniformly approximated on such a region by rational functions with poles outside the region.<\/jats:p>","DOI":"10.1137\/18m1198417","type":"journal-article","created":{"date-parts":[[2019,9,12]],"date-time":"2019-09-12T11:52:21Z","timestamp":1568289141000},"page":"1087-1101","source":"Crossref","is-referenced-by-count":17,"title":["Spectral Sets: Numerical Range and Beyond"],"prefix":"10.1137","volume":"40","author":[{"given":"Michel","family":"Crouzeix","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3783-3466","authenticated-orcid":true,"given":"Anne","family":"Greenbaum","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,9,12]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2009.8.37"},{"key":"atypb2","first-page":"219","volume":"6","author":"Beckermann B.","year":"2014","journal-title":"Jaen J. Approx."},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/080741744"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/17M1140832"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/140961742"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.3934\/cpaa.2012.11.2291"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/17M1116672"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144596305582"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895292400"},{"key":"atypb11","unstructured":"G. B. Folland,\n                      Introduction to Partial Differential Equations\n                      , Princeton University Press, Princeton, NJ, 1995."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201310002"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01446535"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/17M1143757"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"A. L. Shields,\n                      Weighted Shift Operators and Analytic Function Theory\n                      , in Topics in Operator Theory Math. Surveys 13, American Mathematical Society, Providence, RI, 1974, pp. 49-128.","DOI":"10.1090\/surv\/013\/02"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1070\/RM1967v022n06ABEH003763"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1198417","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:36:20Z","timestamp":1787326580000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1198417"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":15,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1198417"],"URL":"https:\/\/doi.org\/10.1137\/18m1198417","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}