Convex combinations of projections (Q2640670)

The paper contains the following results: On an n-dimensional inner- product space, every operator T that satisfies \(0\leq T\leq 1\) is a convex combination of as few as \([\log_ 2n]+2\) projections and this number is sharp. If \(0\leq T\leq 1\) and trace T is a rational number, then T is an average of projections. Further results are also got for the cases when the projections are needed to have the same rank / or to be commuting. In each case, the optimal number of projections is found out. The convex combination part presented in this paper is very interesting.