I was wondering if there is some way to solve the matrix equation $$XX^*-X^*X=A,$$ where $A$ is a given (necessarily Hermitian and vanishing trace) matrix and we are to solve for unknown square matrix $X$. We can assume further that $X$ is non-singular, non-normal.
If this is too difficult, can we find an $X$ when we know only the eigenvalues of $XX^*-X^*X$?
Here $X^*$ denotes "Hermitian conjugate" (i.e. complex conjugate+ transpose) of $X$.