= Positive-negative decomposition of a Hermitian operator
= Positive-negative decomposition
{synonym}
For a bounded <Hermitian operator> $A$, the <continuous functional calculus> defines $|A|=(A^2)^{1/2}$ and
$$
A_+=\frac{|A|+A}{2},\qquad A_-=\frac{|A|-A}{2},\qquad A=A_+-A_-.
$$
Both parts are <positive operators>, commute, and satisfy $A_+A_-=0$, so their <supports> are orthogonal. They apply the scalar functions $\max(t,0)$ and $\max(-t,0)$ to the <spectrum>. The <spectral theorem> gives the same definitions even with continuous spectrum; a finite eigenvalue sum is only a special case.
If $A$ is additionally a <trace-class operator>, $\|A\|_1=\operatorname{Tr}A_++\operatorname{Tr}A_-$. Consequently $\operatorname{Tr}A=0$ implies $\operatorname{Tr}A_+=\operatorname{Tr}A_-=\|A\|_1/2$. The trace-class assumption is essential: a general bounded Hermitian operator need not have a finite trace or trace norm. For $A=\operatorname{diag}(2,-1)$, the two parts are $\operatorname{diag}(2,0)$ and $\operatorname{diag}(0,1)$.
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