Geodesic
The strengthened versions of the additive and multiplicative Weyl inequalities
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 39-59

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The strengthened versions of the classical additive and multiplicative Weyl inequalities for the singular values of $A+B$ and $AB^*$, where $A$ and $B$ are rectangular matrices, and for the eigenvalues of $A+B$ and $AB$, where $A$ and $B$ are Hermitian matrices, are established under certain assumptions on the subspaces spanned by some singular vectors or eigenvectors of $A$ and $B$. 
@article{ZNSL_2003_296_a3,
     author = {L. Yu. Kolotilina},
     title = {The strengthened versions of the additive and multiplicative {Weyl} inequalities},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {39--59},
     publisher = {mathdoc},
     volume = {296},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a3/}
}
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L. Yu. Kolotilina. The strengthened versions of the additive and multiplicative Weyl inequalities. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XVI, Tome 296 (2003), pp. 39-59. http://geodesic.mathdoc.fr/item/ZNSL_2003_296_a3/