Invertible operators and norms

Question

Let $A$ be an invertible opeartor, $B$ being another operator, is |ABA^{-1}| \leq |B|?

Answer 1

When $A = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, we have $ABA^{-1} = \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$. So, $\|ABA^{-1}\| = 2 \not\leq 1 = \|B\|$.