Let $a = \{a_n\}_{n = 1}^{\infty}$ be inside $l_{\infty}$ where $l_{\infty}$ here is the sequences with supermum norm such that that norm is finite. Let $T : l_p \rightarrow l_p$ be defined as $Tx = \{x_na_n\}_{n = 1}^{\infty}$. I am wondering what is the norm here ?
I think it is $||a||_{\infty}$.
$\|T\| = \sup_{\|x\| = 1} \frac{\|Tx\|}{\|x\|} = \frac{\|x_na_n\|_p}{\|x_n\|_p}$. I think one has to use Cauchy-Schwarz here, but I am not sure how I can do it?
Let $n\in\mathbb N$ and let $e(n)\in\ell^p$ be the sequence such that$$e(n)_k=\begin{cases}1&\text{ if }k=n\\0&\text{ otherwise.}\end{cases}$$Then $\bigl\|e(n)\bigr\|_p=1$ and $\bigl|T\bigl(e(n)\bigr)\bigr|=|a_n|$. Therefore, $(\forall n\in\mathbb{n}):\|T\|\geqslant|a_n|$. Since this happens for each $n\in\mathbb N$, $\|T\|\geqslant\|a\|_\infty$.
On the other hand, if $\|x\|_p=1$, then$$\bigl|T(x)\bigr|=\sqrt[p]{\sum_{n=0}^\infty|a_nx_n|^p}\leqslant\|a\|_\infty\sqrt[p]{\sum_{n=0}^\infty|x_n|^p}=\|a\|_\infty\|x\|_p.$$So, $\|T\|=\|a\|_\infty$.