I have been thinking about the following problem on the upper bound of the spectral norm of the product: Consider $||\cdot||$ as the spectral norm, by the definition of matrix norm we have $$||AB||\leq ||A||||B|| $$ for any n by n matrices $A$ and $B$.
If we have the information that the spectral radius of the non-symmetric matrix $ A$ is precisely 1, under what condition can we have a sharper upper bound that $$||AB||\leq ||B|| $$ by replacing the $||A||$ (max modulus of singular values) with the spectral radius 1?
Without extra conditions, it seems hard to hold because the spectral radius is upper-bounded by any matrix norm.