Let $X$ be a finite measure space and $D,M$ be bounded linear operators on a $(L^1(X;\mathbb C))^2$. $D$ is a diagonal operator matrix whose entries are multiplication operators by the invertible functions $d_1,d_2\in L^\infty(X;\mathbb C)$.
My question: Is it possible to characterize those $M$ with $M{\bf 1}=0$ and for which the equation $$ DM-MD=J K $$ is satisfied for some injective bounded linear operator $K$ and some diagonal block operator matrix whose entries are multiplication operators by $\Phi(d_1,d_2)$ for some function $\Phi:\mathbb C^2\to\mathbb C^2$ that only vanishes along the diagonal $\{(x,x):x\in \mathbb C\}$.
