{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T17:47:11Z","timestamp":1780508831608,"version":"3.54.1"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,8,28]],"date-time":"2023-08-28T00:00:00Z","timestamp":1693180800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Jouf University","award":["DSR-2021-03-0341"],"award-info":[{"award-number":["DSR-2021-03-0341"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Our aim in this study is to consider a generalization of the concept of m-complex symmetric transformations to n-quasi-m-complex symmetric transformations. A map S\u2208B(Y) is said to be an n-quasi-m-complex symmetric transformation if there exists a conjugation C on Y such that S satisfies the condition S*n\u22110\u2264k\u2264m(\u22121)m\u2212kmkS*kCSm\u2212kCSn=0, for some positive integers n and m. This class of transformation contains the class of m-complex symmetric transformations as a proper subset. Some basic structural properties of n-quasi-m-complex symmetric linear transformations are established with the help of transformation matrix representation. In particular, we obtain that a power of an n-quasi-m-complex symmetric is again an n-quasi-m-complex symmetric operator. Moreover, if T and S are such that T is an n1-quasi-m1-complex symmetric and S is an n2-quasi-m2-complex symmetric, their product TS is an max{n1,n2}-quasi-(m1+m2\u22121)-complex symmetric under suitable conditions. We examine the stability of n-quasi-m-complex symmetric operators under perturbation by nilpotent operators.<\/jats:p>","DOI":"10.3390\/sym15091662","type":"journal-article","created":{"date-parts":[[2023,8,29]],"date-time":"2023-08-29T08:26:52Z","timestamp":1693297612000},"page":"1662","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["n-Quasi-m-Complex Symmetric Transformations"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0678-6775","authenticated-orcid":false,"given":"Abeer A.","family":"Al-Dohiman","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6891-7849","authenticated-orcid":false,"given":"Sid Ahmed Ould Ahmed","family":"Mahmoud","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jouf University, Sakaka P.O. Box 2014, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8608-8063","authenticated-orcid":false,"given":"Basem Aref","family":"Frasin","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Al al-Bayt University, Mafraq 25113, Jordan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,28]]},"reference":[{"key":"ref_1","unstructured":"Helton, J.W. (1970). Operators with a Representation as Multiplication by x on a Sobolev Space, Colloquia Mathematical Society Janos Bolyai 5, Hilbert Space Operator."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1007\/BF01222016","article-title":"m-Isometric transformations of Hilbert space. I","volume":"21","author":"Agler","year":"1995","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01261201","article-title":"m-Isometric transformations of Hilbert space. II","volume":"23","author":"Agler","year":"1995","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/BF01191619","article-title":"m-Isometric transformations of Hilbert space. III","volume":"24","author":"Agler","year":"1996","journal-title":"Integral Equ. Oper. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1016\/j.jmaa.2013.05.043","article-title":"An isometry plus a nilpotent operator is an m-isometry. Applications","volume":"407","author":"Noda","year":"2013","journal-title":"J. Math. Anal. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.laa.2012.07.011","article-title":"Products of m-isometries","volume":"438","author":"Noda","year":"2013","journal-title":"Linear Algebra Its Appl."},{"key":"ref_7","first-page":"141","article-title":"2-isometric operators","volume":"37","author":"Patel","year":"2002","journal-title":"Glas. Mat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1285","DOI":"10.1090\/S0002-9947-05-03742-6","article-title":"Complex symmetric operators and applications","volume":"358","author":"Garcia","year":"2006","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"353001","DOI":"10.1088\/1751-8113\/47\/35\/353001","article-title":"Mathematical and physical aspects of complex symmetric operators","volume":"47","author":"Garcia","year":"2014","journal-title":"J. Phys. A Math. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1679","DOI":"10.1007\/s11785-016-0549-0","article-title":"On (m,C)-Isometric Operators","volume":"10","author":"Ko","year":"2016","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_11","first-page":"307","article-title":"A note on quasi-isometries","volume":"35","author":"Patel","year":"2000","journal-title":"Glas. Mat."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"111","DOI":"10.3336\/gm.38.1.09","article-title":"A note on quasi-isometries. II","volume":"38","author":"Patel","year":"2003","journal-title":"Glas. Mat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1080\/03081087.2017.1335283","article-title":"On quasi-2-isometric operators","volume":"66","author":"Mecheri","year":"2017","journal-title":"Linear Multlinear Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.15672\/hujms.532964","article-title":"Some results on higher orders quasi-isometries","volume":"49","author":"Ahmed","year":"2020","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1650073","DOI":"10.1142\/S179355711650073X","article-title":"On n-quasi-m-isometric operators","volume":"9","author":"Mecheri","year":"2016","journal-title":"Asian-Eur. J. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1001","DOI":"10.1080\/03081087.2018.1524437","article-title":"On n-quasi-(m,C)-isomtric operators","volume":"68","author":"Ahmed","year":"2020","journal-title":"Linear Multilinear Algebra"},{"key":"ref_17","first-page":"145157","article-title":"Classes of Operators related to m-isometric operatros","volume":"14","author":"Mechri","year":"2020","journal-title":"Oper. Matrices"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1300","DOI":"10.3906\/mat-2102-11","article-title":"A class of operators related to m-symmetric operators","volume":"45","author":"Zuo","year":"2021","journal-title":"Turk. J. Math."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1088\/0305-4470\/39\/2\/009","article-title":"Norm estimates of complex symmetric operators applied to quantum systems","volume":"39","author":"Garcia","year":"2006","journal-title":"J. Phys. A"},{"key":"ref_20","first-page":"393","article-title":"Stability on parametric strong symmetric quasi-equilibrium problems via nonlinear scalarization","volume":"6","author":"Peng","year":"2022","journal-title":"J. Nonlinear Var. Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"233","DOI":"10.24193\/subbmath.2017.2.09","article-title":"Properties of m-complex symmetric operators","volume":"62","author":"Ko","year":"2017","journal-title":"Stud. Univ. Babes-Bolyai Math."},{"key":"ref_22","first-page":"35","article-title":"On \u221e-Complex symmetric operators","volume":"60","author":"Ko","year":"2017","journal-title":"Glasg. Math. J. Trust"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2025","DOI":"10.1007\/s00009-015-0597-0","article-title":"On m-complex symmetric operators","volume":"13","author":"Ko","year":"2016","journal-title":"Mediterr. J. Math."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2975","DOI":"10.2298\/FIL1911577K","article-title":"Skew m-Complex Symmetric Operators","volume":"33","author":"Ko","year":"2019","journal-title":"Filomat"},{"key":"ref_25","first-page":"468","article-title":"Skew [m,C]-Symmetic operatros","volume":"2","author":"Nacevska","year":"2017","journal-title":"Adv. Oper. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1090\/S0002-9939-99-04965-5","article-title":"Invertible completions of 2 \u00d7 2 upper triangular operator matrices","volume":"128","author":"Han","year":"1999","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1215\/S0012-7094-56-02324-9","article-title":"On the operator equation BX \u2212 XA = Q","volume":"23","author":"Rosenblum","year":"1956","journal-title":"Duke Math. J."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.2298\/FIL1707073C","article-title":"On [m,C]-Isometric Operators","volume":"31","author":"Lee","year":"2017","journal-title":"Filomat"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1662\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:41:12Z","timestamp":1760128872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1662"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,28]]},"references-count":28,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091662"],"URL":"https:\/\/doi.org\/10.3390\/sym15091662","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,28]]}}}