A bounded linear operator T: H→ H is a C-normal operator if there exists a conjugation C on H such that [ CT, (CT) ∗] = 0 where [ R, S] : = RS- SR. In this paper we study properties of C-normal operators. In particular, we prove that T- λ is C-normal for all λ∈ C if and only if T is a complex symmetric operator with the conjugation C. Moreover, we show that if T is C-normal, then the following statements are equivalent; (i) T is normal, (ii) T is quasinormal, (iii) T is hyponormal, (iv) T is p-hyponormal for 0 < p≤ 1. Finally, we consider operator transforms of C-normal operators.
Bibliographical noteFunding Information:
The authors would like to thank the referee for his/her valuable comments which helped to improve the paper. These authors contributed equally to this work. The first author was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (2019R1F1A1058633) and the Ministry of Education (2019R1A6A1A11051177). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2019R1A2C1002653). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2019R1A6A1A11051177) and (2020R1I1A1A01064575).
© 2021, Tusi Mathematical Research Group (TMRG).
- C-normal operator
- Complex symmetric operator
- Operator transforms