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

TL;DR

This paper demonstrates that any bounded self-adjoint operator can be expressed as a real linear combination of four orthoprojections, and explores limitations on such decompositions in finite and infinite dimensions.

Contribution

It proves that four orthoprojections suffice for all bounded self-adjoint operators and identifies classes of operators that cannot be decomposed into three orthoprojections.

Findings

Every bounded self-adjoint operator is a linear combination of 4 orthoprojections.

Operators of the form identity minus a compact positive operator cannot be decomposed into 3 orthoprojections.

Existence of n×n matrices not expressible as a linear combination of 3 orthoprojections for n ≥ 76.

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 $n\ge 76$.