# Spectral theory of compact, self-adjoint operators.

Let $T$ be a compact, self-adjoint operator on a separable Hilbert space H. Suppose that $f\in H$, $||f|| =1$ and $||(T-3)f||\leq 1/2$. Let P be the orthogonal projection onto the direct sum of all the eigenspace of T with eigenvalue $\lambda \in [2,4]$. Show that

$||Pf||\geq \frac{\sqrt{3}}{4}$.

I think what I need to do is use spectral theory to show that there exists an orthonormal basis consisting of the eigenvectors. Then I need to evaluate $P$ with regards to this basis. Am I on the right track?

• Apply standard theory on compact, normal operators ($T=T^*\Rightarrow TT^*=T^*T$) – Piquito Jan 12 '16 at 12:57