Positive imaginary part of a Stieltjes matrix resolvent ($\mathrm{Im}{G}_X(z)=\eta G_X(z{)}^{\ast }{G}_X(z)(\eta =\mathrm{Im}z>0)$)

By an orthonormal eigenbasis, the imaginary part of the Stieltjes matrix resolvent has eigenvalues $\eta /|\lambda -z{|}^2>0$. Thus its normalized trace has positive imaginary part, and its quadratic form at any vector has nonnegative imaginary part. This yields $|z+g|\ge \eta$ and $|z+q|\ge \eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.