Condition for Nonzero Eigenvalue of Compact Operator

Theorem
Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$.

Let $T \in \map {B_0} H$ be a compact operator.

Let $\lambda \in \Bbb F, \lambda \ne 0$ be a nonzero scalar.

Suppose that the following holds:


 * $\inf \set {\norm {\paren {T - \lambda I} h}_H: \norm h_H = 1} = 0$

Then $\lambda \in \map {\sigma_p} T$, that is, $\lambda$ is an eigenvalue for $T$.