Let $X$ be a Banach space with a norm $\|\cdot\|_1$ and $A$ be a linear operator on $X$ such that
$\|A\|_1\leq 1$;
$\|A^m\|_1<1$ for some $m\in \mathbb N$.
Is that true that there is an equivalent norm $\|\cdot\|_2$ on $X$ such that $\|A\|_2<1$? If there exists such a norm, how can it be constructed?
Here for operator we use associated (induced norm): given a norm $\|\cdot\|$ on $X,$ $ \|B\| :=\sup\limits_{\|x\|=1}\|Bx\| $ for any linear operator $B$.