(See title.) I think that I know how to prove $\implies$: If $(x_n)$ is a bounded sequence in $\ell^p$, then there is a $c\in\mathbb{R}$ such that $\|x_n\|\leq c$ for all $n$. So $$\|T_a x_n\|=\|a_n x_n\|\leq c\|a_n\|\to0,$$ from which it follows that any subsequence of $(T_a x_n)$ in $\ell^p$ converges (to $0$). (Is this correct?) However, for the converse, I have no idea. Any suggestions are greatly appreciated!
2026-03-26 19:02:50.1774551770
Let $(a_n)\subset\mathbb{C}$ be bounded. Then $T_a\colon\ell^p\to\ell^p, \ (T_a x)_n:=a_n x_n$ compact iff $a_n\to0$.
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