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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.

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    $\begingroup$ It is a standard exercise in functional analysis courses that an operator $A$ on $X$ induces an operator on $X/Z$ iff $AZ\subset Z$. Since your statement is an immediate consequence of this via the open mapping theorem, it is not that surprising that I have not seen the statement in books. OTOH, I like your formulation better because it forces students to make the connection to the open mapping theorem. $\endgroup$
    – Bill Johnson
    Commented Oct 24, 2015 at 15:00
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    $\begingroup$ Thanks. I also assign it as an exercise when I teach functional analysis but now I review a manuscript and since the authors prove this as a theorem I want to indicate in a more consistent way that it is something well known. Probably I have just to say that it's a folklore statement and nobody can claim originality for it. $\endgroup$
    – Aurelian Gheondea
    Commented Oct 24, 2015 at 15:12

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