Relationship between $\left\Vert A-B \right\Vert$ and $\left\Vert A_s- B_s \right\Vert$ where $A$ is similar to $A_s$ and $B$ is similar to $B_s$?

Question

I'm looking for a bound that relates $\left\Vert A-B \right\Vert_{op}$ and $\left\Vert A_s- B_s \right\Vert_{op}$ where $A$ and $A_s$ are similar matrices and $B$ and $B_s$ are similar matrices. If it helps one can assume that all four matrices are real and that $A$ and $B$ are symmetric (but $A_s$ and $B_s$ may not be). I define the matrix norm here as the operator norm induced by the Euclidian norm.

It is easy to check these quantities are not equal. Is it possible to construct a bound of the form $$\left\Vert A-B \right\Vert_{op} \leq f\left\Vert A_s- B_s \right\Vert_{op}$$ where $f$ is a variable that depends on the matrices in question?

Answer 1

Let $A=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and $B=B_s=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$. Then $A$ is similar to $$A_s:=B=\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix}^{-1}A\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix}$$

As $\|A-B\|=\|\begin{pmatrix}-1&1\\1&1\end{pmatrix}\|=\sqrt2$ but $A_s-B_s=0$, there is no number $c$ such that $\|A-B\|\le c\|A_s-B_s\|$.