Let $A$ be a bounded operator on $l^2(\mathbb C)$ and let $\{e_i\}$ be some orthonormal basis. Is there a sharp characterisation of the operator norm $\|A\|_{\mathcal L(l^2(\mathbb C))}$ in terms of the matrix elements $A_{ij}:=\langle e_i,A e_j\rangle$?
It is straightforward to show that $$\sup_k\sum_i|A_{ik}|^2\leq \|A\|_{\mathcal L(l^2(\mathbb C))}^2 \leq \sum_{i,k}|A_{ik}|^2, $$
but simple counterexamples show that neither of the two inequalities can be sharp.
There isn't, even in finite dimension. For instance, let us consider $4\times 4$ matrices where four entries are $1$ and the rest are zero. Let $$ A_1=\begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}, \ \ A_2=\begin{bmatrix} 1&1&0&0\\ 0&0&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix}, $$ $$ A_3=\begin{bmatrix} 1&1&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&0\\ \end{bmatrix}, \ \ A_4=\begin{bmatrix} 1&1&0&0\\ 1&1&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{bmatrix}. $$ Then, though the sets of entries of the four matrices is the same, we have the four operator norms $$ \|A_1\|=1,\ \ \ \|A_2\|=\sqrt2,\ \ \ \|A_3\|=\sqrt{\frac{3+\sqrt5}{2}},\ \ \|A_4\|=2. $$