  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 as well as the numerical radius, respectively. Among other
things, the positivity of a $2\times 2$ block 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.
