When is this $\ell^2(\mathbf C)\to\ell^2(\mathbf C)$ application continuous?

Asked Sep 14 '21 at 13:21

Active Sep 14 '21 at 13:21

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Let $a=(a_n)_{n\in\mathbf N}$ be a sequence of complex numbers and let us define the application $A\colon\ell^2(\mathbf C)\to\ell^2(\mathbf C)$ by $$A(x)\stackrel{\mathrm{def}}=(a_n\,x_n)_{n\in\mathbf N}$$ for all $x=(x_n)_{n\in\mathbf N}\in\ell^2(\mathbf C)$.

I am trying to find a necessary and sufficient condition on the sequence $a$ for the application $A$ to be continuous but I failed to go any further than the following: $$\text{$A$ is continuous}\iff\left(\forall x=(x_n)_{n\in\mathbf N}\in\ell^2(\mathbf C),\ \|x\|=1\implies\sum_{n=0}^{+\infty}| a_n|^2|x_n|^2<+\infty\right)$$

asked Sep 14 '21 at 13:21

$a_n$ in $l^2(C)$ – Kroki Sep 14 '21 at 13:33
Thanks for your answer, I can see why it is necessary (by taking $x=(1)_{n\in\mathbf N}$) but I don't get why it is sufficient – Sep 14 '21 at 13:41
The $x$ that you are taking is not in $l^2$. But for the sufficient part you can use Cauchy-Schwarz – Kroki Sep 14 '21 at 13:45
I will try that, thank you – Sep 14 '21 at 13:46
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@Youem Maybe you misread the question. What you said would be right if $Ax=\sum a_n x_n$, but here $Ax=(a_nx_n)$; that's totally different. – David C. Ullrich Sep 14 '21 at 13:49
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This is an example of a multiplication operator. A necessary and sufficient condition is that $a\in\ell_\infty$ – Mittens Sep 14 '21 at 13:49
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In fact tfae: (i) that formula defines a map from $\ell_2$ to itself, (ii) $A$ is continuous, (iii) $(a_n)\in\ell_\infty$. – David C. Ullrich Sep 14 '21 at 13:50

When is this $\ell^2(\mathbf C)\to\ell^2(\mathbf C)$ application continuous?

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