Recently in a post (link) upper bounds on the singular values $\sigma_j(X)$ of a matrix $X$ have been considered. To restate the central observation, it says that if $AX−XB=F$ for $A$ and $B$ normal matrices, then we have that $$σ_{1+νk}(X)≤Z_k(σ(A),σ(B)) \sigma_1, \;\;\;\;\;\; ν=rank(F),$$ where $σ(A)$ and $σ(B)$ are the spectra of $A$ and $B$, respectively, and $Z_k(E,F)$ the Zolotarev number. This nice result is from here. In the reference, as far as I can see the examples are throughly for disjoint $E$ and $F$.

Are there any result when $E$ and $F$ are not disjoint?



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