Cauchy–Schwarz inequality in semi-inner product -modules via polar decomposition

By virtue of the operator geometric mean and the polar decomposition, we present a new Cauchy–Schwarz inequality in the framework of semi-inner product C ∗ -modules over unital C ∗ –algebras and discuss the equality case. We also give several additive and multiplicative type reverses of it. As an application, we present a Kantorovich type inequality on a Hilbert C ∗ -module.