{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:49:36Z","timestamp":1760150976291,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,31]],"date-time":"2022-01-31T00:00:00Z","timestamp":1643587200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["11801290","11701327"],"award-info":[{"award-number":["11801290","11701327"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Fujian Province of China","award":["2019J05122"],"award-info":[{"award-number":["2019J05122"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Generalizing bicommutant theorem to the higher-order commutator case is very useful for representation theory of Lie algebras, which plays an important role in symmetry analysis. In this paper, we mainly prove that for any spectral operator A on a complex Hilbert space whose radical part is locally nilpotent, if a bounded operator B lies in the k-centralizer of every bounded linear operator in the l-centralizer of A, where k and l are two arbitrary positive integers satisfying l\u2a7ek, then B must belong to the von Neumann algebra generated by A and the identity operator. This result generalizes a matrix commutator theorem proved by M. F. Smiley. To this aim, Smiley operators are defined and an example of a non-spectral Smiley operator is given by the unilateral shift, indicating that Smiley-type theorems might also hold for general spectral operators.<\/jats:p>","DOI":"10.3390\/sym14020283","type":"journal-article","created":{"date-parts":[[2022,1,31]],"date-time":"2022-01-31T01:46:21Z","timestamp":1643593581000},"page":"283","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Smiley Theorem for Spectral Operators Whose Radical Part Is Locally Nilpotent"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9952-9139","authenticated-orcid":false,"given":"Jianjian","family":"Jiang","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Ningde Normal University, Ningde 352100, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6755-5196","authenticated-orcid":false,"given":"Xiao","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University, Weihai 264209, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaolin","family":"Li","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Shandong University, Weihai 264209, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"709","DOI":"10.1016\/j.jmaa.2011.08.034","article-title":"Quasinilpotent operators in operator Lie algebras III","volume":"386","author":"Cao","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Wang, X., and Cao, P. (2020). Perturbation Theory for Quasinilpotents in Banach Algebras. Mathematics, 8.","DOI":"10.3390\/math8071163"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"353","DOI":"10.4153\/CJM-1961-030-x","article-title":"Matrix commutators","volume":"13","author":"Smiley","year":"1961","journal-title":"Canad. J. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"527","DOI":"10.4153\/CJM-1965-052-9","article-title":"On matrix commutators of higher order","volume":"17","author":"Robinson","year":"1965","journal-title":"Canad. J. Math."},{"key":"ref_5","first-page":"11","article-title":"From Pebbles to Commutators","volume":"16","author":"Robinson","year":"1976","journal-title":"BYU Stud. Q."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1016\/j.jalgebra.2016.06.022","article-title":"Partial classification of cuspidal simple modules for Virasoro-like algebra","volume":"464","author":"Jiang","year":"2016","journal-title":"J. Algebra"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1262","DOI":"10.22436\/jnsa.011.11.06","article-title":"Symmetry Lie algebra and exact solutions of some fourth-order difference equations","volume":"11","author":"Mnguni","year":"2018","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"241","DOI":"10.22436\/jmcs.020.03.07","article-title":"Semiconformal symmetry-A new symmetry of the space-time manifold of the general relativity","volume":"20","author":"Pundeer","year":"2020","journal-title":"J. Math. Comput. Sci."},{"key":"ref_9","first-page":"181","article-title":"Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra","volume":"14","author":"Mawed","year":"2021","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_10","unstructured":"Murphy, G.J. (1990). C*-Algebras and Operator Theory, Academic Press."},{"key":"ref_11","unstructured":"Dunford, N. (September, January 30). The reduction problem in spectral theory. Proceedings of the International Congress of Mathematicians, Cambridge, MA, USA."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Soltan, P. (2018). A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, Birkh\u00e4user\/Springer.","DOI":"10.1007\/978-3-319-92061-0"},{"key":"ref_13","unstructured":"Dunford, N., and Schwartz, J.T. (1988). Linear Operators. Part III. Spectral Operators, John Wiley and Sons, Inc.. Reprint of the 1971 Original, Wiley Classics Library, A Wiley-Interscience Publication."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1090\/S0002-9904-1958-10219-0","article-title":"A survey of the theory of spectral operators","volume":"64","author":"Dunford","year":"1958","journal-title":"Bull. Am. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"355","DOI":"10.2140\/pjm.1954.4.355","article-title":"Commuting spectral measures on Hilbert space","volume":"4","author":"Wermer","year":"1954","journal-title":"Pac. J. Math."},{"key":"ref_16","unstructured":"Humphreys, J.E. (1980). Introduction to Lie Algebras and Representation Theory, Springer. [3rd ed.]."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1006\/jfan.2000.3640","article-title":"Joint spectral radius, operator semigroups, and a problem of W. Wojtynski","volume":"177","author":"Shulman","year":"2000","journal-title":"J. Funct. Anal."},{"key":"ref_18","unstructured":"Conway, J.B. (1990). A Course in Functional Analysis, Springer. [2nd ed.]. Graduate Texts in Mathematics 96."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Halmos, P.R. (1982). A Hilbert Space Problem Book, Springer. [2nd ed.]. Graduate Texts in Mathematics 96.","DOI":"10.1007\/978-1-4684-9330-6"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/283\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:11:42Z","timestamp":1760134302000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/283"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,31]]},"references-count":19,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020283"],"URL":"https:\/\/doi.org\/10.3390\/sym14020283","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,31]]}}}