{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:41:08Z","timestamp":1760186468866,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2019,1,8]],"date-time":"2019-01-08T00:00:00Z","timestamp":1546905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>    T \u2208 B ( H )     is said to be     ( n , k )    -quasi-\u2217-paranormal operator if, for non-negative integers k and n,      \u2225   T \u2217   (  T k  x )    \u2225   ( 1 + n    ) \u2264 \u2225   T  ( 1 + n )    (  T k  x )   \u2225 \u2225   T k    x \u2225  n     ; for all     x \u2208 H    . In this paper, the asymmetric Putnam-Fuglede theorem for the pair     ( A , B )     of power-bounded operators is proved when (i) A and     B \u2217     are n-\u2217-paranormal operators (ii) A is a     ( n , k )    -quasi-\u2217-paranormal operator with reduced kernel and     B \u2217     is n-\u2217-paranormal operator. The class of     ( n , k )    -quasi-\u2217-paranormal operators properly contains the classes of n-\u2217-paranormal operators,     ( 1 , k )    -quasi-\u2217-paranormal operators and k-quasi-\u2217-class A operators. As a consequence, it is showed that if T is a completely non-normal     ( n , k )    -quasi-\u2217-paranormal operator for     k = 0 , 1     such that the defect operator     D T     is Hilbert-Schmidt class, then     T \u2208  C 10     .<\/jats:p>","DOI":"10.3390\/sym11010064","type":"journal-article","created":{"date-parts":[[2019,1,9]],"date-time":"2019-01-09T03:06:06Z","timestamp":1547003166000},"page":"64","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Asymmetric Putnam-Fuglede Theorem for (n,k)-Quasi-\u2217-Paranormal Operators"],"prefix":"10.3390","volume":"11","author":[{"given":"Ahmed","family":"Bachir","sequence":"first","affiliation":[{"name":"Department of Mathematics, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia"}]},{"given":"Abdelkader","family":"Segres","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Mascara University, Mascara 29000, Algeria"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,8]]},"reference":[{"key":"ref_1","first-page":"381","article-title":"On a class of operators","volume":"21","author":"Arora","year":"1986","journal-title":"Kyngpook Math. J."},{"key":"ref_2","first-page":"410","article-title":"On a class of operators","volume":"18","author":"Yoshino","year":"1966","journal-title":"T\u00f4hoku Math. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"357","DOI":"10.2307\/2372180","article-title":"On the normal operators on Hilbert space","volume":"73","author":"Putnam","year":"1951","journal-title":"Am. J. 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