C*-module operators which satisfy in 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 $\mathcal{H}$. We introduce the generalized Cauchy-Schwarz inequality for operators in $\mathcal{L}(\mathcal{H})$ and investigate various properties of operators which satisfy the generalized Cauchy--Schwarz inequality. In particular, we prove that if an operator $A\in \mathcal{L}(\mathcal{H})$ 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 $\Vert ⟨Ax,y⟩\Vert \le \Vert {|A|}^{1/2}x\Vert \Vert {|A|}^{1/2}y\Vert$ for all $x,y\in \mathcal{H}$.

10 pages, submitted to a Journal