Condition for Nonzero Eigenvalue of Compact Operator

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

Let $T \in B_0 \left({H}\right)$ be a compact operator.

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

Suppose that the following holds:


 * $\inf \left\{{\left\Vert{\left({T - \lambda I}\right)h}\right\Vert_H: \left\Vert{h}\right\Vert_H = 1}\right\} = 0$

Then $\lambda \in \sigma_p \left({T}\right)$, i.e., $\lambda$ is an eigenvalue for $T$.

Corollary
Suppose $\lambda \notin \sigma_p \left({T}\right)$ and $\bar \lambda \notin \sigma_p \left({T^*}\right)$.

Then $T - \lambda I$ is invertible.

Furthermore, $\left({T - \lambda I}\right)^{-1}$ is a bounded linear operator.