Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to be $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. What is a necessary and sufficient condition on operators $S$ and $T$ such that $(T,S)$ is $\lambda$-commute implies they commute ?

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. I am looking for necessary and sufficient conditions on the operators $S$ and $T$ such that $(T,S)$ $\lambda$-commute, then they already commute.

I know already that $S,T \geq 0$ or $S,T \leq 0$ is a sufficient condition for $S$ and $T$ to commute if they $\lambda$-commute, but this is far from necessary.

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to be $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. What is a necessary and sufficient condition on operators $S$ and $T$ such that $(T,S)$ is $\lambda$-commute iff they commute ?

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to be $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. What is a necessary and sufficient condition on operators $S$ and $T$ such that $(T,S)$ is $\lambda$-commute implies they commute ?

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

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to be $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. What is a necessary and sufficient condition on operators $S$ and $T$ such that $(T,S)$ is $\lambda$-commute iff they commute ?

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert $F$.

Let $T,S\in\mathcal{B}(F)$. The pair $(T,S)$ is said to be $\lambda$-commute if there exists $\lambda\in \mathbb{C}^*$ such that $TS=\lambda ST$. What is a necessary and sufficient condition on operators $S$ and $T$ such that $(T,S)$ is $\lambda$-commute iff they commute ?