The matched projections of idempotents on Hilbert C∗-modules

The aim of this paper is to give new characterizations of some fundamental issues about idempotents. In the general setting of adjointable operators on Hilbert C∗-modules, a new notion of quasi-projection pair is introduced. For each idempotent Q, a projection m(Q), called the matched projection of Q, is constructed. It is shown that Q and m(Q) as idempotents are homotopic, and (m(Q),Q) is a quasi-projection pair. Some formulas for m(Q) are derived. Based on these formulas, representations and norm estimations associated with m(Q) are dealt with.