linear algebra - subspace separation and M-matrices

The separation between two square matrices $A$ and $B$, often used as a measure of the sensitivity of invariant subspace problems, is defined as
$$operatorname{sep}(A,B)=min_{Xneq 0}frac{leftVert AX-XBrightVert_F}{leftVert XrightVert_F}$$
(see e.g. Golub and Van Loan, Section 7.2.4).

I would like to express the separation between two M-matrices in terms of their Perron vector and values $Au=lambda u$, $Bv=mu v$. All I can do is estimating $operatorname{sep}(A,B)geq lambda+mu$. Is there any better bound, maybe from above? The question looks like an "eigenvalues vs. singular values" bound, so I am not sure that the answer is positive.

Alternatively, do you know any "handier" way to deal with $operatorname{sep}(A,B)$, other than using its definition and its alternative formulation as the smallest singular value of a Kronecker sum $sigma_{min}(B otimes I + I otimes A^T)$? I always find it unwieldy to use this definition, it is not easy to squeeze something simple to compute/estimate out of it. For instance, if $A$, $A'$, $B$ are M-matrices and $A'geq A$, does $operatorname{sep}(A',B)geq operatorname{sep}(A,B)$ hold?

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