The maximum operator distance from an idempotent to the set of projections

Abstract

For each pair of bounded linear operators A and B on a Hilbert space H, let ‖A−B‖ be their operator distance. Given an idempotent Q on H, consider the operator distances from Q to all of the projections on H. It is proved that there always exists a projection whose operator distance away from Q takes the maximum value. An example is constructed to show that such a projection may fail to be unique.