Kernel of a sum of positive operators
= Kernel of a sum of positive operators
{title2=$\ker(S+T)=\ker S\cap\ker T$}
For bounded <positive operators> $S,T$, membership $x\in\ker(S+T)$ gives $0=\langle Sx,x\rangle+\langle Tx,x\rangle$. Both terms are nonnegative, so both vanish. The <Cauchy-Schwarz inequality for a positive operator> makes $Sx=Tx=0$. The reverse inclusion is immediate. Commutativity is not required.