$\lim_{n\rightarrow\infty} ||T^n||$ where $T$ is left-shift operator on $l^2$

Question

Let $T$ be the left-shift operator on $l^2$. I want to calculate the two limits $\lim_{n\rightarrow\infty} \|T^nx\|_2\;\text{and}\; \lim_{n\rightarrow\infty} \|T^n\|$

I have no problems with the first one $$ \lim_{n\rightarrow\infty} \|T^nx\|_2 = \lim_{n\rightarrow\infty} \left(\sum_{i=n}^\infty|x_i|^2 \right)^\frac{1}{2} = 0.$$

But I'm not sure about the second one. I tried to write it out as $$ \|T^n\|= \inf\{c>0: \forall x_i \; \|T^nx\|_2 \leq c\|x\|_2\} $$

but I can't find a proper $c$ as $n\rightarrow \infty$.

Answer 1

$\|T^{n}x\|\leq \|x\|$, so $\|T^{n}\| \leq 1$. Also $T^{n}(e_{n+1})=e_{1}$ Hence $\|T^{n}\| \geq 1$. So $\|T^{n}\| = 1$ for all $n$.