Let $(x_k)_{k\geq 1}\in\ell_2$. Consider $\left(\sum\limits_{k=1}^\infty \dfrac{x_k}{j+k}\right)_{j\geq 1}$.
Now my question is that whether $\left(\sum\limits_{k=1}^{\infty}\frac{x_k}{j+k}\right)_{j\geq 1}$ belongs to $\ell_2$ or not.
The following is an idea, but I am not sure whether it's right.
Let $\varphi(t)=i(\pi-t)$ be a $2\pi$-period function. Then it's easy to see that $\widehat\varphi(n)=\frac{1}{n}$ for every nonzero $n\in\mathbb Z$ and $\widehat \varphi(0)=0$. Then we obtain $\sum\limits_{1\leq j,k\leq N}\frac{a_jb_k}{j+k}=\frac{1}{2\pi} \int_0^{2\pi}\left(\sum\limits_{j=1}^N a_j e^{-ijt}\right)\left(\sum\limits_{k=1}^Nb_ke^{-ikt}\right)\varphi(t)dt$ where $(a_1,\cdots,a_N),(b_1,\cdots,b_N)\in\mathbb C^N$.Then by Cauchy-Schwarz inequality, we have
$\left|\sum\limits_{1\leq j,k\leq N}\frac{a_jb_k}{j+k}\right|\leq\|\varphi\|_{\infty}\left|\left(\sum\limits_{j=1}^N|a_j|^2\right)^{1/2}\left(\sum\limits_{k=1}^N|b_k|^2\right)^{1/2}\right|.$ Now for any $x,y\in \ell_2$, we have $\left|\langle u(x),y\rangle\right|\leq \|\varphi\|_{\infty}\|x\|_2\|y\|_2$ where $u(x)=(\sum\limits_{k=1}^{\infty}\frac{x_k}{j+k})_{j\geq 1}$. Note that $\langle u(x),y\rangle$ exists, since $\lim\limits_{N\to\infty}\sum\limits_{1\leq j,k\leq N}\frac{|x_j||y_j|}{j+k}$ exists. Then $\langle u(x),\cdot\rangle$ is a linear functional on $\ell_2$ for fixed $x\in\ell_2$. By Riesz representation theorem, there exists $z\in \ell_2$ such that $\langle u(x),y\rangle=\langle z,y\rangle$ for all $y\in\ell_2$. Hence $z=u(x)$, i.e. $u(x)=z\in\ell_2$.
Maybe I have made some mistakes in the proof. And anyone know some other proofs?
Thank you for you help.