Norm equalities in pre-Hilbert C*-modules

We discuss the connection between the equality case in the triangle inequality for two elements x and y in a pre-Hilbert module (E,〈·,·〉) over the C∗-algebra A and the equality case in the corresponding Cauchy–Schwarz inequality. We firstly show that the triangle “equality” associated to the “rank one” operators θx,x and θy,y holds true if and only if ∥〈x,y〉∥A=∥x∥∥y∥. The special situations when 〈x,y〉 is a perturbation, by a scalar α, of an idempotent (∥x+y∥=∥x∥+∥y∥ iff ) or it has positive real part (∥x+y∥=∥x∥+∥y∥ iff ∥R〈x,y〉∥A=∥x∥∥y∥) are also considered. In the last part, we characterize Pythagoras’ equality in pre-Hilbert C∗-modules. Our results extend or improve some theorems due to L. Arambašić and R. Rajić.