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Let $X$ be an infinite dimensional Banach space (over $\mathbb{C}$) and let $T\colon X\to X$ be a compact operator. Let $\sigma(T)$ denote the spectrum of $T$ and let $\sigma_{\text{p}}(T)$ denote the point spectrum (= eigenvalues) of $T$. Is it true that $$\sigma(T)=\sigma_{\text{p}}(T)\cup\{0\}?$$ I have seen some theorems, but none of them explicitly says that the above equality is true.

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    $\begingroup$ Rudin, Functional Analysis, Theorem 4.25 explicitly says that. $\endgroup$ – Daniel Fischer Nov 11 at 18:42

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