Some inequalities for adjointable operators in Hilbert C^*-modules

In this article, we discuss some problems related to adjointable operators between two Hilbert modules over a C^*-algebra A. In particular, we discuss the positivity of block matrices in this class, where equivalent conditions are provided. Then, we generalize the mixed Schwarz inequality for the operators mentioned above. This generalization's significance arises because the mixed Schwarz inequality for bounded linear operators on Hilbert spaces does not apply for adjointable operators between Hilbert A-modules. Then, some applications that involve the numerical radius of Hilbert space operators will be provided, where the positivity of certain block matrices is essential. Numerical examples to support the validity and advantage of the obtained results are given in the end.