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Let $E = \ell^p$ with $1 \le p \le \infty$. Let $(\lambda_n)$ be a bounded sequence in $\mathbb{R}$ and consider the operator $T \in \mathcal{L}(E)$ defined by$$T(x) = (\lambda_1x_1, \lambda_2x_2, \dots, \lambda_nx_n, \dots),$$where$$x = (x_1, x_2, \dots, x_n, \dots).$$How do I see that $T$ is a compact operator from $E$ into $E$ if and only if $\lambda_n \to 0$?

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    $\begingroup$ See this for some ideas. $\endgroup$ – David Mitra Dec 28 '15 at 8:09