Every bounded self-ajoint operator is a real linear combination of $4$ orthoprojections

Abstract

We prove that every bounded self-adjoint operator in Hilbert space is a real linear combination of $4$ orthoprojections. Also we show that operators of the form identity minus compact positive operator can not be decomposed in a real linear combination of $3$ orthoprojections. Using ideas applied in infinite dimensional space, we find $n\times n$ matrices that are not real linear combinations of $3$ orthoprojections for every $n\ge 76$.