The Sylvester equation is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension $m\times n.$ It is a well known fact that the equation has an unique solution if and only if the matrices $A$ and $B$ have disjoint spectrum. If they do not have disjoint spectrum, then the result in general depends on $C.$

While determining perturbation of eigenvalues in certain context I was naturally drawn to the the problem of determining the minimum, $min_{X}||AX-XB-C||,$ where $||.||$ is the Frobenius norm. Clearly, if the spectrum of $A$ and $B$ is disjoint then there is a choice of $X$ for which the norm is zero. Otherwise, we need to resort to certain optimization techniques. One approach could be to vectorize the matrices using Kronecker products and determine the minimum of a linear system.

The problem is: "What is the choice of $X$ for which the norm $||AX-XB-C||$ attains minimum (if it exists) when the spectra of $A$ and $B$ are not disjoint?"

I have not found any literature on discussion about similar problems. I would be very thankful for any references or suggestions in this direction.