The main purpose of this paper is, in the general setting of the adjointable operators on Hilbert \(C^*\) -modules, to develop two new tools that can be applied to deal with the positive solutions of certain operator equations, the operator norm and related quantities. Among other things, the positivity of a \(2\times 2\) operator matrix is clarified without any preconditions on its entries, and a generalized version of the mixed Schwarz inequality with a parameter is derived. Numerical examples are provided to illustrate the non-triviality of this newly obtained inequality.
