On conjugations concerning idempotents.

We introduce the C-decomposition property for reducible bounded linear operators on a Hilbert space, and prove that an arbitrary idempotent operator has the C-decomposition property with respect to a particular space decomposition, which is related to Halmos’ two projections theory. Using this, we obtain a general explicit description for all the conjugations C such that a given idempotent operator is a C-projection. We also present a characterization of the ranges of C-projections for any conjugation C. [ABSTRACT FROM AUTHOR]