Condition for Nonzero Eigenvalue of Compact Operator

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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 $T$.


Corollary

Suppose $\lambda \notin \map {\sigma_p} T$ and $\bar \lambda \notin \map {\sigma_p} {T^*}$.


Then $T - \lambda I$ is invertible.

Furthermore, $\paren {T - \lambda I}^{-1}$ is a bounded linear operator.


Proof


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