I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume $kerA\subset kerB$. Is it true that then we can find an operator $V$ so that $B=VA$?
It appears that an operator defined by $V(Au)=Bu$ is a properly defined one, but I'm not sure whether there isn't some kind of a continuity issue on the complement of the range of $A$ that eludes me. I couldn't find any kind of reference for such a theorem (the only one being for the case of $A,B$ linear functionals - so that then $B=\alpha A$ for a scalar $\alpha$) and I'm fairly sure something of the sort should exist. It is possible that a little bit stronger assumptions are needed - Douglas' lemma is a fairly similar theorem (with inclusions of ranges instead of kernels), but it's not exactly what I would need.
I would be grateful for any help and/or reference.
Regards
You can extend $V$ to the closure of $\mathrm{ran}A$ by the limits, and to $(\mathrm{ran}A)^\perp$ by constant $0$, then it will stay bounded.