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

Let $ \mathcal {L}(\mathscr {H}) $ L (H) denote the $ C^* $ C ∗ -algebra of adjointable operators on a Hilbert $ C^* $ C ∗ -module $ \mathscr {H} $ H . In this paper, we introduce the generalized Cauchy–Schwarz inequality for operators in $ \mathscr {L}(\mathscr {H}) $ L (H). More precisely, an operator $ A\in \mathscr {L}(\mathscr {H}) $ A ∈ L (H) is said to satisfy the generalized Cauchy–Schwarz inequality if there exists $ \nu \in (0, 1) $ ν ∈ (0 , 1) such that \[ \|\langle Ax, y\rangle\|\leq (\|Ax\|\|y\|)^{\nu}(\|Ay\|\|x\|)^{1 - \nu} \quad (x, y \in \mathscr{H}). \] ‖ ⟨ A x , y ⟩ ‖ ≤ (‖ A x ‖ ‖ y ‖) ν (‖ A y ‖ ‖ x ‖) 1 − ν (x , y ∈ 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 $ \left \|\langle Ax, y\rangle \right \| \leq \left \|{|A|}^{1/2}x\right \|\left \|{|A|}^{1/2}y\right \| $ ‖ ⟨ A x , y ⟩ ‖ ≤ ‖ | A | 1 / 2 x ‖ ‖ | A | 1 / 2 y ‖ for all $ x, y \in \mathscr {H} $ x , y ∈ H . [ABSTRACT FROM AUTHOR]