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On operators satisfying the generalized Cauchy-Schwarz inequality


Authors: Hanna Choi, Yoenha Kim and Eungil Ko
Journal: Proc. Amer. Math. Soc. 145 (2017), 3447-3453
MSC (2010): Primary 47A63; Secondary 47B20
DOI: https://doi.org/10.1090/proc/13473
Published electronically: January 31, 2017
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Abstract: In this paper, we introduce the generalized Cauchy-Schwarz inequality for an operator $ T\in {\mathcal {L(H)}}$ and investigate various properties of operators which satisfy the generalized Cauchy-Schwarz inequality. In particular, every $ p$-hyponormal operator satisfies this inequality. We also prove that if $ T\in {\mathcal {L(H)}}$ satisfies the generalized Cauchy-Schwarz inequality, then $ T$ is paranormal. As an application, we show that if both $ T$ and $ T^{\ast }$ in $ {\mathcal {L(H)}}$ satisfy the generalized Cauchy-Schwarz inequality, then $ T$ is normal.


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Additional Information

Hanna Choi
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 03760 Korea
Email: rms5835@gmail.com

Yoenha Kim
Affiliation: Institute of Mathematical Sciences, Ewha Womans University, Seoul, 03760 Korea
Email: yoenha@ewhain.net

Eungil Ko
Affiliation: Department of Mathematics, Ewha Womans University, Seoul, 03760 Korea
Email: eiko@ewha.ac.kr

DOI: https://doi.org/10.1090/proc/13473
Received by editor(s): April 27, 2016
Received by editor(s) in revised form: September 9, 2016
Published electronically: January 31, 2017
Additional Notes: This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827). The second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT $&$ Future Planning (2015R1C1A1A02036456).
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2017 American Mathematical Society