The strengthened versions of the additive and multiplicative Weyl inequalities

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 $B$.