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Douglas' lemma gives a necessary and sufficient condition for two bounded operators $A$ and $B$ on a Hilbert space $H$ to get $\operatorname{Ran} A \subset \operatorname{Ran} B$ (wikipedia article, original article).

Does there exist a condition on two kernels $K_1$ and $K_2$ in order to get the inclusion of the ranges of the integral operators associated with? In order words, I am looking for a specific Douglas type lemma for integral operators which gives a condition on the kernels.

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Too long for a comment. Why don't use your third criterion: if $K_L$ is the kernel of the operator $L$, that gives you $$ K_A=K_B\circ K_C, $$ i.e. $ K_A(x,y)=\int K_B(x,z) K_C(z,y) dz. $

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    $\begingroup$ Well, but Douglas' result doesn't say that $C$ needs to be an integral operator, too. $\endgroup$
    – Jochen Glueck
    Commented Aug 2, 2021 at 6:05
  • $\begingroup$ Is there not a "Schwartz Kernel Theorem" in that context saying that all linear bounded operators must have a (distribution) kernel? $\endgroup$
    – Bazin
    Commented Aug 2, 2021 at 14:31

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