Sums of orthogonal projections

In this paper, we consider the problem of characterizing Hilbert space operators which are expressible as a sum of (finitely many) orthogonal projections. We obtain a special operator matrix representation and some necessary/sufficient conditions for an infinite-dimensional operator to be expressible as a sum of projections. We prove that a positive operator with essential norm strictly greater than one is always a sum of projections, and if an injective operator of the form I+KI+K, where K   is compact, is a sum of projections, then either traceK+=traceK−=∞ or K is of trace class with trace K a nonnegative integer. We also consider sums of those projections which have a fixed rank. The closure of the set of sums of projections is also characterized.