Norm of this Linear Operator

Question

Let $H$ be an infinite dimensional Hilbert space and let $e_1,e_2\dots,$ be an orthonormal basis. Let $A$ be the operator defined by

$$ Au = \sum_{j=1}^\infty\frac{1}{j} \langle u,e_j\rangle e_j$$.

It is easy to show that $||A|| \leq \pi^2/6$ and I suspect that this is the norm of $A$, but I'm having difficulty showing it.

Answer 1

$\|Au\|^{2}=\sum\limits_{k=1}^{\infty} (\frac 1 k)^{2} |\langle u , e_k \rangle|^{2} \leq \sum\limits_{k=1}^{\infty} |\langle u , e_k \rangle|^{2} =\|u\|^{2}$, so $||A\|\leq 1$. Since $Ae_1=e_1$ and $e_1$ is a unit vector we get $\|A\| \geq \|Ae_1\|=1$ so $\|A\|=1$.