Let $T:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N})$ defined by $$(Tx)_n = a_nx_{n+1} $$ for $n\in\mathbb{N}$, with $a=(a_n)_n$ a given sequence in $\ell^{\infty}(\mathbb{N})$. Show that $T$ is a bounded linear operator and compute the operator norm $\|T\|$.
This is my humble attemp:
We know that the sequence space $\ell^{p}(\mathbb{N})$ consists of all infinite sequences $(x_n)_{n=1}^{+\infty}$ such that $\sum_{n=1}^{+\infty}|x_n|^p<+\infty$, with the $p$-norm $$\|x\|_p = \left(\sum_{n=1}^{+\infty}|x_n|^p\right)^{1/p}.$$ Then, in this case we have that $$\|x\|_2 =\left(\sum_{n=1}^{+\infty}|x_n|^2\right)^{1/2}.$$
Therefore, by the definition of the operator and the definition of a bounded linear operator we get $$\left(\|(Tx)_n\|_2\right)^2 = \sum_{n=1}^{+\infty}|a_nx_{n+1}|^2 = \sum_{n=1}^{+\infty}|a_n|^2|x_{n+1}|^2$$
But here is where I am stuck. I want to get an expression in the form $\left(\|(Tx)_n\|_2\right)^2\leq M \|x_{n+1}\|_2 ^2$, but I don't know how to deal with the $|a_n|^2$ terms.
In addition, for the computation of the operator norm, I want to use one of the equivalent definitions $\|T\| = \sup_{\|x\|=1} \|Tx\|$ to get $$\|T\| = \sup_{\|x\|=1} \|(Tx)_n\|_2 = \sup_{\|x\|=1}\sum_{n=1}^{+\infty}|a_n|^2|x_{n+1}|^2,$$ but again, I have troubles in dealing with the $|a_n|^2$ terms.