Let $E=l^p=\left\{x=\left(x_n\right): x_n\in\mathbb{R}, \sum_{1}^{\infty}\left|x_n\right|^p<\infty\right\}$ with $1\le p<\infty$, $\left\|x\right\|_E=\left(\sum_{1}^{\infty}\left|x_n\right|^p\right)^{1/p}.$ Let $\left(\lambda_n\right)$ is a bounded sequence in $\mathbb{R}$ and consider the operator $T$ defined by
$$T(x)=\left(\lambda_1x_x,\ldots,\lambda_nx_n,\ldots\right)$$
where $x\in l^p$.
Prove that $T$ is a compact operator from $E$ to $E$ iff $\lambda_n\rightarrow 0$.
Can anyone give me some hints to solve this problem? Thank you!