Suppose that $A$ and $B$ are square matrices. We know that the spectral norm of $A$ is less than or equal to one. We also know that
$C = BCA^T + D$
where $C, D$ are square and positive definite matrices.
Can we say anything about the spectral norm of $AB$? For example, under what conditions spectral norm of $AB$ is less than or equal to one?