The effects of integer powers on a linear operator

Question

Let $H$ be a Hilbert space, and let $T : H \to H$ be a bounded self-adjoint linear operator, with $T \ne 0$

How can we prove that $T$ to some integer power is not equal to $0$, i.e that $T^{2^k}\ne0$ for every $k \in N$

Is this proof different when considering the case in which $T^n\ne0$ for every $n \in N$?

Answer 1

$T^{2n}=0$ implies $0=(x,T^{2n}x)=(T^nx,T^nx)=\|T^nx\|^2$ by self-adjointness. Therefore $T^{2n}=0$ implies $T^n=0$.

Contrapositively, $T^n\ne0$ implies $T^{2n}\ne0$. If $T\ne0$ then $T^{2^k}\ne0$ for all $k$. Then for any $n$, $T^nT^{2^k-n}\ne0$ for large enough $k$, and so $T^n\ne0$.