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Volume 40, Issue 3
Invariance of Conjugate Normality Under Similarity

Cun Wang, Meng Yu & Minyi Liang

DOI: 10.4208/cmr.2024-0002

Commun. Math. Res., 40 (2024), pp. 245-260.

Published online: 2024-09

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  • Abstract

An operator $T$ on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$ is called conjugate normal if $C|T|C = |T^∗|$ for some conjugate linear, isometric involution $C$ on $\mathcal{H}.$ This paper focuses on the invariance of conjugate normality under similarity. Given an operator $T,$ we prove that every operator $A$ similar to $T$ is conjugate normal if and only if there exist complex numbers $λ_1$, $λ_2$ such that $(T−λ_1)(T−λ_2)=0.$

  • Keywords

$C$-normal operators, complex symmetric operators, similarity.

  • AMS Subject Headings

Primary 47B99, 47A05, Secondary 47A10, 47A58

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-40-245, author = {Wang , CunYu , Meng and Liang , Minyi}, title = {Invariance of Conjugate Normality Under Similarity}, journal = {Communications in Mathematical Research }, year = {2024}, volume = {40}, number = {3}, pages = {245--260}, abstract = {

An operator $T$ on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$ is called conjugate normal if $C|T|C = |T^∗|$ for some conjugate linear, isometric involution $C$ on $\mathcal{H}.$ This paper focuses on the invariance of conjugate normality under similarity. Given an operator $T,$ we prove that every operator $A$ similar to $T$ is conjugate normal if and only if there exist complex numbers $λ_1$, $λ_2$ such that $(T−λ_1)(T−λ_2)=0.$

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2024-0002}, url = {http://global-sci.org/intro/article_detail/cmr/23411.html} }
TY - JOUR T1 - Invariance of Conjugate Normality Under Similarity AU - Wang , Cun AU - Yu , Meng AU - Liang , Minyi JO - Communications in Mathematical Research VL - 3 SP - 245 EP - 260 PY - 2024 DA - 2024/09 SN - 40 DO - http://doi.org/10.4208/cmr.2024-0002 UR - https://global-sci.org/intro/article_detail/cmr/23411.html KW - $C$-normal operators, complex symmetric operators, similarity. AB -

An operator $T$ on a separable, infinite dimensional, complex Hilbert space $\mathcal{H}$ is called conjugate normal if $C|T|C = |T^∗|$ for some conjugate linear, isometric involution $C$ on $\mathcal{H}.$ This paper focuses on the invariance of conjugate normality under similarity. Given an operator $T,$ we prove that every operator $A$ similar to $T$ is conjugate normal if and only if there exist complex numbers $λ_1$, $λ_2$ such that $(T−λ_1)(T−λ_2)=0.$

Wang , CunYu , Meng and Liang , Minyi. (2024). Invariance of Conjugate Normality Under Similarity. Communications in Mathematical Research . 40 (3). 245-260. doi:10.4208/cmr.2024-0002
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