Let H be Hilbert with an orthonormal basis $(e_n)_n$. Consider $A,B>0$ and two sequence $a_n>0, b_n>0$ such that $$\sum_{i=1}^\infty b_i(T e_i, e_j)\leq Aa_j \quad\text{and}\quad \sum_{j=1}^\infty a_j(T e_i, e_j)\leq Bb_i.$$ Show that $T$ extends to continuous linear operator and that $\|T\|\leq \sqrt{AB}$
Assume that $(Te_i, e_j)=\frac{1}{j+i-1}$ prove that $\|T\|\leq \pi.$
If $(Te_i, e_j)=\frac{1}{2^{j+i-1}}$ then compute $\|T\|.$
As explained in this question it is possible to get that,
$$\|T\|\leq \Big( \sum_{j=1}^\infty \sum_{i=1}^\infty (Te_i, e_j)^2\Big)^{1/2}. $$
How do I move from here?