Finding an element $x$ of a Hilbert space with $\|Tx\|=\|T\|$

Question

Let $T:X\to Y$ be a bounded linear operator between Hilbert spaces $X$ and $Y$.

Is there always some $x \in X$ with $\|x\|_X=1$, such that $\|Tx\|_Y = \|T\|$?

I am guessing this is not always the case, but can't think of an example where this does not happen.

Answer 1

Let $T:\ell^2\to\ell^2$, with $$T(x_1,\dots x_n,\dots)=\left(\frac{x_1}{2},\frac{2x_2}{3},\frac{3x_3}{4},\dots\frac{nx_n}{n+1},\dots\right).$$ Then, $$\sum_{n=1}^{\infty}\left(\frac{nx_n}{n+1}\right)^2\leq\sum_{n=1}^{\infty}x_n^2,$$ therefore $\|T\|\leq 1$. On the other hand, $$\|Te_n\|=\left\|\frac{ne_n}{n+1}\right\|\to 1,$$ therefore $\|T\|=1$. However, there is no $x\in\ell^2$ with $\|x\|=1$ such that $\|Tx\|=1$.