Let $T:\ell^2 \to \ell^2$ with $T(x_n)_n=(\dfrac{x_2}{2}, \dfrac{x_3}{3},...,\dfrac{x_n}{n},...)$ and $\ell^2 = \{(x_n)_n \in \mathbb{C} | \sum_{n=1}^{\infty} |x_n|^2 < \infty \}$
I already proved that $T$ is linear and bounded, i.e. $||T(x_n)_n|| \leq \dfrac{1}{2} ||(x_n)_n||$.
This result gives me only an overestimation of the norm of $T$ in the space of all linear amd bounded operators ($\mathcal{L}(\ell^2)$).
Now, I know that the actual value of the operatorial norm I'm looking for is exactly $\frac12$, but to prove this I need an underestimation of it.
I tried to use the sup definition of the operatorial norm for sequences of norm $1$, but couldn't find the desired inequality.
Any ideas?
