On operators satisfying the generalized Cauchy-Schwarz inequality

Choi, Hanna; Kim, Yoenha; Ko, Eungil

doi:10.1090/proc/13473

On operators satisfying the generalized Cauchy-Schwarz inequality
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by Hanna Choi, Yoenha Kim and Eungil Ko PDF

Proc. Amer. Math. Soc. 145 (2017), 3447-3453 Request permission

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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