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Let $H$ be a Hilbert space over $\mathbb{C}$. Let $T:H\rightarrow H$ be a compact operator. We know that if $\lambda\neq 0$ is an eigen value of of $T$, then the eigen space of $\lambda$ is finite dimensional. Can we say the same when $\lambda=0$, ie is it true that the dimension of the null space of $T$ is finite ?

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Definitely not. For example, the $0$ operator is compact and has null space all of $H$.

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