{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:21:15Z","timestamp":1787329275022,"version":"build-2736575974"},"reference-count":14,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber\u2013Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352\u2013362]. Motivated by the description by J. Liesen and Z. Strako\u0161, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber\u2013Manteuffel theorem for matrices.<\/jats:p>","DOI":"10.1137\/060678087","type":"journal-article","created":{"date-parts":[[2008,3,26]],"date-time":"2008-03-26T18:05:19Z","timestamp":1206554719000},"page":"1323-1337","source":"Crossref","is-referenced-by-count":10,"title":["The Faber\u2013Manteuffel Theorem for Linear Operators"],"prefix":"10.1137","volume":"46","author":[{"given":"V.","family":"Faber","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J.","family":"Liesen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"P.","family":"Tich\u00fd","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,3,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/42792"},{"key":"R2","unstructured":"E. W. Cheney,\n                      Introduction to Approximation Theory\n                      , McGraw-Hill, New York, 1966."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(98)10161-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0721026"},{"key":"R5","unstructured":"D. K. Faddeev and V. N. Faddeeva,\n                      Computational Methods of Linear Algebra\n                      , W. H. Freeman, San Francisco, 1963."},{"key":"R6","unstructured":"F. R. Gantmacher,\n                      The Theory of Matrices\n                      . Vols. 1, 2, Chelsea, New York, 1959."},{"key":"R7","doi-asserted-by":"crossref","unstructured":"A. Greenbaum,\n                      Iterative Methods for Solving Linear Systems\n                      , Frontiers in Appl. Math. 17, SIAM, Philadelphia, 1997.","DOI":"10.1137\/1.9781611970937"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(87)90168-6"},{"key":"R9","unstructured":"K. Hoffman and R. Kunze,\n                      Linear Algebra\n                      , 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1971."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01203029"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903393372"},{"key":"R12","unstructured":"J. Liesen and Z. Strako\u0161,\n                      On optimal short recurrences for generating orthogonal Krylov subspace bases\n                      , SIAM Rev., to appear."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Iterative Methods for Sparse Linear Systems\n                      , 2nd ed., SIAM, Philadelphia, 2003.","DOI":"10.1137\/1.9780898718003"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(97)82725-5"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060678087","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:13:36Z","timestamp":1787325216000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060678087"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":14,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060678087"],"URL":"https:\/\/doi.org\/10.1137\/060678087","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}