LEt $\mathbf{X} \in \mathbb{R}^{N \times N}$ be a positive definite matrix, and $\mathbf{Y} \in \mathbb{R}^{N \times M}$ be a rectangular matrix. Consider the following resolvent matrix: $$ \mathbf{G}(z) = (z \mathbf{I}_M - \mathbf{Y}^T \mathbf{X} \mathbf{Y} )^{-1} $$
Is there any relation between this resolvent and resolvents of $\mathbf{X}$ and/or $\mathbf{Y}^T \mathbf{Y}$?