Corollary: Bounded-defect implication

If ${\Lambda }_s(T)>0$, there is a uniformly bounded sequence $v_n\in M$ with spectral support shrinking to $s$ and ${\inf }_n\Vert T{v}_n\xi \Vert >0$.

Consequently, if

then ${\Lambda }_s(T)=0$.

Proof. For radii $1/n$, choose unit vectors whose quadratic-form values in (eq:averagedform) stay above a fixed positive fraction of ${\Lambda }_s(T)$. Apply Theorem thm:bounded. The second claim is its contrapositive.