Let $H$ be a Hilbert space, and let $T : H \rightarrow H$ be a bounded self-adjoint linear operator, with $T \neq 0.$
I need to show that $T^{2^k} \neq 0$ $\forall k \in \mathbb{N}$. Here's what I've done so far:
$T^2x = T(Tx)$ and so $T^{2^k}x = T(T^{2k-1}x)$. Hence, as T is self-adjoint, $<Tx,y> = <x,T^*y>$ and so $<T^{2k}x,y> = <x,(T^{2k})^*y>$. However, I'm struggling to go from here, any help is appreciated.