Block Positivity and Optimal Mixed-Schwarz Inequalities on Hilbert $C^*$-Modules

We propose two interrelated advances in the theory of adjointable operators on Hilbert C*-modules. First, we give a set of equivalent, verifiable conditions characterizing positivity of general $n\times n$ block operator matrices acting on finite direct sums of Hilbert C*-modules. Our conditions generalize and remove several classical range-closedness and Moore-Penrose assumptions by expressing positivity in terms of a finite family of mixed inner-product inequalities and an explicit Gram-type factorization. Second, we investigate a parametric family of mixed-Schwarz inequalities for adjointable operators and determine optimal factor functions and constants which make these inequalities sharp; we characterize the extremal operators attaining equality in key cases. The two developments are tied together: the optimal mixed-Schwarz bounds are used to obtain sharp, computable tests in the $n\times n$ positivity criterion, and conversely the block-factorizations yield structural information used in the extremal analysis. We include applications to solvability of operator equations without Moore-Penrose inverses and spectral gap estimates for block operator generators.