# Spectrum of compact operator on an infinite dimensional Banach space

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.

• Rudin, Functional Analysis, Theorem 4.25 explicitly says that. – Daniel Fischer Nov 11 at 18:42