Detailed and rigorous study is made of operators of the ghost number $Q_c$ and ghost conjugation $U_c$ which are operators in Krein spaces arising in the BRST quantization formalism for constrained dynamical systems. A number of conditions are obtained which guarantee that $Q_c$ is well-defined and $J$-symmetric. It is shown that properties of $Q_c$ are related to the following geometrical problem: to find conditions under which a pair of lineals in the Krein space can be made neutral by the appropriate choice of $J$-metrics. The complete solution of this problem is given. Whole series of examples is constructed which demonstrate the connections between properties of $Q_c$ and geometry of its spectral subspaces.