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 $T$.

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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.4.14, 15$