Suppose $T$ is a compact self-adjoint operator and $f$ is a unit vector such that $\| (T -3)f \| \leq 1/2$. Denote by $p$ the orthogonal projection onto the direct sum of eigenspaces of $T$ with eigenvalue $2 \leq \lambda \leq 4$. I want to show that $$\| p f \| \geq \frac{\sqrt{3}}{2}.$$
I'm unsure of how to consider the norm of $p$ and how to relate it to the eigenvalues. Perhaps $$\| Tf \| = \lambda \| p \circ T f \|?$$
hint: $\Vert(T-3)f\Vert\geq\Vert(T-3)(1-p)f\Vert\geq\Vert(1-p)f\Vert$