Let $T$ be a compact self adjoint operator on a Hilbert space $H$. If $(e_n)_n$ is an orthonormal system of eigenvectors then setting $H_0= \{ z : \langle z , e_n \rangle=0$ forall $n$ $\}$, we have $T(H_0)=0$.
My attempt:
If $H_0=0$ then clearly we are done. So assume otherwise. If $z\in H_0$, $\langle Tz,e_n\rangle =0$. Hence, $T(H_0)\subseteq H_0$. So, $T|_{H_0}:H_0\rightarrow H_0$ is a compact, self adjoint operator. Note $H_0$ is closed in $H$.
Since $T|_{H_0}$ is compact and self adjoint, assume wlog that $||T|_{H_0}||$ is an eigenvalue, with corresponding eigen vector $w$.
Now, I'm trying to show that $||T|_{H_0}||=0$, but am unsure how to proceed.