I have noticed numerically that the spectral radius of $I-KH$, where $K$ is a Kalman gain, is less than or equal to 1. In other words, for some symmetric positive definite matrices $R$ and $C$, and another matrix $H$, is the spectral radius of $I - CH^T(HCH^T + R)^{-1}H$ less than or equal to 1?
In the scalar case, it is clearly true that $1 - ch^2/(h^2 c+r) \leq 1$ for $c>0$, $r>0$, and the inequality is strict when $h\neq 0$.
In general, the spectral radius of $I-KH$ is not guaranteed to be less or equal to 1. However, a spectral radius less than or equal to 1 implies the filter will be stable.
In other words, if the spectral radius is less than 1, the covariance matrix will not grow unbounded. This makes sense as the spectral radius is the maximum of the absolute values of the eigenvalues; thus, the components of $P$ along the eigenvectors of $I-KH$ should be decreased in magnitude.