I am looking for a reference, if any, to the following statement: "Let $X$ and $Y$ be Banach spaces, $A\in\mathcal{B}(X,X)$ be a linear bounded operator, and $B\in\mathcal{B}(X,Y)$ be surjective. Then, there exists $C\in\mathcal{B}(Y,Y)$ such that $BA=CB$ if and only if $\mathrm{Ker}(B)$ is $A$-invariant."

The proof is not difficult, it is an application of the Closed Graph Principle, and it must be known for a long time. It might be the case that it is just a folklore result.