The operator distances from projections to an idempotent

The main purpose of this paper is to give a full characterization of the operator distances from projections to an idempotent, which includes the minimum value, the maximum value and the intermediate values. Let H be a Hilbert space and B(H) be the set of bounded linear operators on H. Given an arbitrary idempotent Q∈ B(H), it is proved that \|m(Q)-Q\|≤ \|P-Q\|≤ \|I-m(Q)-Q\| for every projection P on H, in which I is the identity operator on H and $m(Q)$ is a specific projection called the matched projection of Q. When Q(H) is a non-projection idempotent, it is proved that for every number α contained in the interval [\|m(Q)-Q\|,\|I-m(Q)-Q\|], there exists a projection P∈ B(H) such that \|P-Q\|=α. Two uniqueness problems concerning the projections that attain the minimum value or the maximum value are also dealt with.