 Positive part of a Hermitian operator

For a Hermitian operator with spectral decomposition $X={\sum }_i{\lambda }_i{P}_i$, its positive part is $X_{+}={\sum }_i\max ({\lambda }_i,0)P_i=(|X|+X)/2$. It is a positive operator supported on the positive spectral subspace. Together with the negative part of a Hermitian operator it satisfies $X=X_{+}-X_{-}$ and $|X|=X_{+}+X_{-}$.