Let's suppose that we have an operator defined as $$ \begin{array}{cccc} A: & \ell_2 & \to & \mathbb{C} \\ & (x_n)_{n=1}^\infty & \mapsto & \sum_{n=1}^\infty c_nx_n \end{array}\ , \quad\text{where } (c_n)_{n=1}^\infty \in \ell_2\ . $$
We could calculate the norm by using the Riesz theorem, which tells us that there should be a vector $a$ such that $Ax = \langle a,x\rangle$ and then $||A|| = ||a||$. In this case this is easy, because by inspection we can see that $$ a = \left(c_n^*\right)_{n=1}^\infty\ , $$ and then we could calculate the norm as $$ ||A|| = ||a|| = \left(\sum_{n=1}^\infty |c_n^*|^2\right)^{1/2}\ , $$ given that this summation converges to any finite number.
Now, in general we know that for any vector in $\ell_2$, we have that $||Ax|| \leq ||A||\,||x||$. So my question is, how could we determine all the vectors that satisfy the equality, i.e. the equation $$ \left|\sum_{n=1}^\infty c_nx_n\right| = ||A|| \left(\sum_{n=1}^\infty |x_n|^2\right)^{1/2}\ . $$