# Spectral theory of compact operators

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Let $X$ be a complex Banach space and $T$ a compact operator on $X$. Then $\sigma ( T )$, the spectrum of $T$, is countable and has no cluster points except, possibly, $0$. Every $0 \neq \lambda \in \sigma ( T )$ is an eigenvalue, and a pole of the resolvent function $\lambda \mapsto ( T - \lambda I ) ^ { - 1 }$. Let $\nu ( \lambda )$ be the order of the pole $\lambda$. For each $n \in \mathbf N$, $( T - \lambda I ) ^ { n } X$ is closed, and this range is constant for $n \geq \nu ( \lambda )$. The null space $N ( ( T - \lambda I ) ^ { n } )$ is finite dimensional and constant for $n \geq \nu ( \lambda )$. The spectral projection $E ( \lambda )$ (the Riesz projector, see Riesz decomposition theorem) has non-zero finite-dimensional range, equal to $N ( ( T - \lambda I ) ^ { \nu ( \lambda ) } )$, and its null space is $( T - \lambda l ) ^ { \nu ( \lambda ) } X$. Finally, $\operatorname { dim } ( E ( \lambda ) X ) \geq \nu ( \lambda ) \geq 1$.
The respective integers $\nu ( \lambda )$ and $\operatorname { dim } ( E ( \lambda ) X )$ are called the index and the algebraic multiplicity of the eigenvalue $\lambda \neq 0$.