Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$.

Let $T,S\in\mathcal{B}(F)$. Assume that there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. I want to establish a necessary and sufficient condition under which ($TS=\lambda ST\Longrightarrow [T,S]=0$).

I guess that

($TS=\lambda ST\Longrightarrow [T,S]=0$) if and only if ($TS\geq 0$ and $ST\geq 0$)

If $TS\geq 0$ and $ST\geq 0$, then $\lambda\geq 0$. Moreover since $\|TS\|=\|ST\|$, then $\lambda=1$.