Let $H$ a hilbert space with an orthonormal basis $(e_n)_{n\in \mathbb{N}}$ and $F$ a linear operator, such that $\langle e_k,F e_n\rangle =:\phi(n,k)$. Find a good estimate for $\lVert F\lVert$ in terms of $\phi(n,k)$. Apply your estimate to the special case of $\phi(n,k)=\frac{1}{n+k}$.
I tried applying parsevals identity and hölder's inequality: \begin{align} \lVert F x\lVert^2&=\lVert \sum_{n=1}^{\infty}\langle x,e_n\rangle F e_n\lVert^2=\lVert\sum_{n=1}^{\infty}\langle x,e_n\rangle \sum_{k=1}^{\infty}\langle e_k,F e_n\rangle e_k \lVert^2 \\&=\sum_{k=1}^{\infty} \left(\sum_{n=1}^{\infty} \langle x,e_n\rangle \phi(n,k)\right)^2 \leq \sum_{k=1}^{\infty} \left [\left(\sum_{n=1}^{\infty} \langle x,e_n\rangle ^2\right)\left(\sum_{n=1}^{\infty}\phi(n,k)^2\right)\right]\\&= \lVert x\lVert^2 \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \phi(n,k)^2 \end{align}
Which implies $\lVert F\lVert \le \sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \phi(n,k)^2$. But applied to $\phi(n,k)=\frac{1}{n+k}$ this sum doesn't converge at all.