Semi-harmonious and harmonious quasi-projection pairs on Hilbert C^*-modules

For each adjointable idempotent Q on a Hilbert C^*-module H, a specific projection $m(Q)$ called the matched projection of Q was introduced recently due to the characterization of the minimum value among all the distances from projections to Q. Inspired by the relationship between $m(Q)$ and Q, another term called the quasi-projection pair $(P,Q)$ was also introduced recently, where P is a projection on H satisfying Q^*=(2P-I)Q(2P-I), in which Q^* is the adjoint operator of the idempotent Q and I is the identity operator on H. Some fundamental issues on quasi-projection pairs, such as the block matrix representations for quasi-projection pairs and the C^*-morphisms associated with quasi-projection pairs, are worthwhile to be investigated. This paper aims to make some detailed preparations. Two objects called the semi-harmonious quasi-projection pair and the harmonious quasi-projection pair are introduced and are systematically studied in the general setting of the adjointable operators on Hilbert C^*-modules. Some applications concerning the common similarity of operators and a norm equation associated with the Friedrichs angle are also dealt with. Furthermore, many examples are provided to illustrate the non-triviality of the associated characterizations.