How to find the norm of the operator $(Ax)_n = \frac{1}{n} \sum_{k=1}^n \frac{x_k}{\sqrt{k}}$? How to find the norm of the following operator
$$
A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right)
$$
Any help is welcome.

Consider diagonal operator
$$
S:\ell_p\to\ell_p:(x_n)\mapsto(n^{-1/2}x_n)
$$
It is bounded and its norm is $\Vert S\Vert=\sup\{|n^{-1/2}|:n\in\mathbb{N}\} =1$
Consider Caesaro operator
$$
T:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^nx_k\right)
$$
As it was proved earlier its norm is is $\Vert T\Vert\leq p(p-1)^{-1}$. Since $A=T\circ S$, then
$$
\Vert A\Vert\leq\Vert T\Vert\Vert S\Vert=p(p-1)^{-1}
$$
But unfortunately I can''t say this is the best constant.
