Self-adjoint Hilbert Space operators

Question

Let $H$ be a Hilbert space and $T$ is a self adjoint continuous operator in $\mathcal B(H)$. Show that $\|T^{2^{k}}\| = \|T\|^{2^{k}}$. Does this equality hold for all operators?

Now it is clear that this holds for all self adjoint operators but I simply cannot find a counter-example for a non-self adjoint operator. So, one must ask oneself does this hold for all operators?

Any help will be greatly appreciated.

Answer 1

Take $H=\mathbb R^2$, then $\mathcal B(H)=\mathbb R^{2\times 2}$, and $$ T=\left(\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right), \quad T^2=0 $$ Then $\|T\|=1$, $\|T^2\|=0$.

Answer 2

No. For example consider nilpotent operators on a finite dimensional Hilbert space.