  Contractive selfadjoint extensions of a Hermitian contraction $B$ in a
Hilbert space ${\mathfrak H}$ with an exit in some larger Hilbert space
${\mathfrak H}\oplus{\mathcal H}$ are investigated. This leads to a new
geometric approach for characterizing analytic properties of holomorphic
operator-valued functions of Kre\u{i}n-Ovcharenko type, a class of functions
whose study has been recently initiated by the authors. Compressed resolvents
of such exit space extensions are also investigated leading to some new
connections to transfer functions of passive discrete-time systems and related
classes of holomorphic operator-valued functions.
