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 contained in the interval \|m (Q) -Q\|, \|I-m (Q) -Q\|, there exists a projection P B (H) such that \|P-Q\|=. Two uniqueness problems concerning the projections that attain the minimum value or the maximum value are also dealt with.