Negative part of a Hermitian operator ($X_{-}=(|X|-X)/2$)

The negative part of a Hermitian operator is $X_{-}={\sum }_i\max (-{\lambda }_i,0)P_i=(|X|-X)/2$. In this convention $X_{-}$ is itself a positive operator, and $X=X_{+}-X_{-}$. Its support is orthogonal to that of the positive part of a Hermitian operator.