Two-way operator norm?

Question

Suppose I have a matrix $A$ and a symmetric positive definite matrix $W$ and a vector $f$. I am interested in the magnitude $||AWf||$. I understand that this is bounded above by $||A||||W||||f||$ where for the matrices the norm is the operator norm. What I am wondering is whether there is some norm $|.|$ so that $||AWf||\leq |W|||Af||$, that is whether there is some kind of two-way operator norm and if so what this is called and/or its properties? Any help greatly appreciated.

Answer 1

You can easily construct examples where $AWf\ne0$ but $Af=0$. That would contradict your inequality for any norm.

For instance, $$ A=\begin{bmatrix} 1&-1\\-1&1\end{bmatrix},\ \ W=\begin{bmatrix} 2&1\\1&1\end{bmatrix},\ \ f=\begin{bmatrix} 1\\1\end{bmatrix} . $$