Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space.
By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ \left\|x^*\right\|\leq 1} \sum_{k=1}^{\infty}\left\langle h_k, T^*x^*\right\rangle^2 .$$ where $(h_k)$ is an orthonormal basis of $H$.
I was wondering if $$x_N: = \sup_{ \left\|x^*\right\|\leq 1} \sum_{k\geq N}\left\langle h_k, T^*x^*\right\rangle^2 \to 0.$$
I guess it's converging to a number $x$ with $0 \leq x \leq \left\|T^*\right\|^2$. I was hoping that it converges to $0$, but I know we can't simply interchange sum and supremum. If it fails to be true, can anyone come up with a simple counter example?