Let $L$ be a normed space with norm $\|\cdot\|_1: L\to \mathbf{R}$. Let $T$ be a linear invertible operator on $L$ such that $\|T^n x\|_1 < c \|x\|_1$ for all $x\in L$ and $n\in \mathbf{N}$. Show that there is a norm $\|\cdot\|_2: L\to \mathbf{R}$ equivalent to norm $\|\cdot\|_1$ such that $\| Tx\|_2=\|x\|_2$ for all $x\in L$.
My thought is to consider $\|\cdot\|_2: L\to \mathbf{R}$ such that
$\|x\|_2=\|x\|_1 + \sup_{n\in \mathbf{N}} \|T^n x\|_1$
I have proven equivalence of the two norms but I am finding it uneasy proving that $\|T\|_2=1$.
Any help would be appreciated. Is my thought in track? Thanks
The inequality in the hypothesis can never hold when $x=0$ so I will change $x \in L$ to $x\in L\setminus \{0\}$.
The assertion is false. If $T=\frac 1 2 I$ then the hypothesis is satisfied with $c=\frac 3 4$ but there is no norm $\|.\|_2$ such that $\|Tx\|_2=\|x\|_2$ for all $x$.