Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result

Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections.

The author referred to a 1967 paper by Fillmore, Sums of operators of square zero. However, this paper is not online.

I wonder whether someone has a hint on how this could be true since there are all kinds of operators while projections have such a regular and restricted form.

Thanks!

I believe the answers you're looking for can be found in a paper by Pearcy and Topping, Sums of small numbers of idempotents., which is openly accessible.