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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!

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    The number 257 was improved to 10 by Matsumoto 30 years ago. See this paper by Laurent Marcoux: http://www.math.uwaterloo.ca/~lwmarcou/Preprints/Mar2009Projections.pdf2012-06-24

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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.