For any real sequence $(a_n)_{n\in \mathbb N}$ define the linear operator $A:\mathbb R ^\mathbb N \to \mathbb R ^\mathbb N$ by $Ax=(a_nx_n)_{n\in \mathbb N}$. Let $1\leq p\leq \infty$.
I know $A$ is a bounded linear operator iff $(a_n)_{n\in \mathbb N}$ is bounded. I want to now characterize the $(a_n)_{n\in \mathbb N}$ for which $A:\ell^p \to \ell^p$ is compact. I know $A$ is compact iff $A$ is bounded and $A(B_1(0))$ is precompact in $\ell^p$. Any hints/ideas how to proceed? Thanks in advance!
