example of compact operator
Solution 1:
Hint for $\implies$: If $\lambda_n \not \to 0,$ then for some $\epsilon>0,|\lambda_{n_k}| > \epsilon$ along a subsequence $n_k.$ Letting $e_n$ denote the usual "basis" vector, consider the sequence $e_{n_k}$ in the unit ball of $l^p$ and its images under $T.$