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Given a positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $D\in \mathbb{C}^{m\times m}$, we characterize under which conditions there exists a strictly contractive matrix $K\in \mathbb{C}^{n\times m}$ such that the non-Hermitian block-matrix \[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \] has a positive definite Schur complement with respect to its submatrix~$A$.
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J.I. Giribet, M. Langer, F. Martínez Pería, F. Philipp and C. Trunk, Spectral enclosures for a class of block operator matrices, submitted
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