The original problem I'm looking at is: given a bound on the operator norm of $\Lambda A \Lambda,$ where $\Lambda, A$ are positive definite matrices and $\Lambda$ is diagonal, what is the tightest bound on the operator norm of $A \Lambda^2.$

My starting point is the fact that these two matrices have the same eigenvalues, so the operator norm of $\Lambda A \Lambda$ upper bounds the spectral radius of $A \Lambda^2.$

For normal matrices, the numerical radius is the same as the spectral radius and the operator norm. While $A \Lambda^2$ is not normal, one might hope that it is nice enough that there is still some nontrivial connection between its numerical and spectral radii ( a bound on the former is a bound on the operator norm, up to a constant). Is this the case, or am I barking up the wrong tree?