 Kernel of a sum of positive operators

For bounded positive operators $S,T$, membership $x\in \ker (S+T)$ gives $0=⟨Sx,x⟩+⟨Tx,x⟩$. 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.