C*-module operators which satisfy the generalized Cauchy–Schwarz type inequality

Abstract

Let $\mathcal{L}(\mathcal{H})$ denote the $C^{\ast }$-algebra of adjointable operators on a Hilbert $C^{\ast }$-module $\mathcal{H}$. In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $\mathcal{L}(\mathcal{H})$. More precisely, an operator $A\in \mathcal{L}(\mathcal{H})$ is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $\nu \in (0,1)$ such that $\Vert ⟨Ax,y⟩\Vert \le (\Vert Ax\Vert \Vert y\Vert {)}^{\nu }(\Vert Ay\Vert \Vert x\Vert {)}^{1-\nu }(x,y\in \mathcal{H}).$We investigate various properties of operators which satisfy the generalized Cauchy–Schwarz inequality. In particular, we prove that if A satisfies the generalized Cauchy–Schwarz inequality such that A has the polar decomposition, then A is paranormal. In addition, we show that if for A the equality holds in the generalized Cauchy–Schwarz inequality, then A is cohyponormal. Among other things, when A has the polar decomposition, we prove that A is semi-hyponormal if and only if $\Vert ⟨Ax,y⟩\Vert \le \Vert {|A|}^{1/2}x\Vert \Vert {|A|}^{1/2}y\Vert$ for all $x,y\in \mathcal{H}$.

Keywords:

• Hilbert $C^{\ast }$-module
• hyponormal operators
• operator inequality
• Cauchy–Schwarz inequality

2010 Mathematics Subject Classifications:

• 46L08
• 47A63

Acknowledgements

The author would like to thank Prof. M.S. Moslehian, Prof. Q. Xu and Dr. R. Eskandari for their invaluable suggestions while writing this paper. He is also thankful to the referees for the useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).