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

Let $\mathcal{L}(\mathscr{H})$ denote the $C^*$-algebra of adjointable operators on a Hilbert $C^*$-module $\mathscr{H}$. We introduce the generalized Cauchy-Schwarz inequality for operators in $\mathcal{L}(\mathscr{H})$ and investigate various properties of operators which satisfy the generalized Cauchy--Schwarz inequality. In particular, we prove that if an operator $A\in\mathcal{L}(\mathscr{H})$ 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 $\big\|\langle Ax, y\rangle\big\| \leq \big\|{|A|}^{1/2}x\big\|\big\|{|A|}^{1/2}y\big\|$ for all $x, y \in\mathscr{H}$.