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So I'm stuck at the following result, about compact operators on Hilbert spaces (which I think it's called Fredholm's theorem) from Stein. It's exercise 29 from Chapter 4.

Let $T$ be a compact operator on a Hilbert space $\mathcal{H}$, and assume $\lambda \neq 0$.

a) Show that the range of $\lambda I - T$ is closed.

b) Show that this is not true for $\lambda = 0$.

c) Show that the range of $\lambda I - T$ is all of $\mathcal{H}$ if and only if the nullspace of $\overline{\lambda} I - T^{*}$ is trivial.

Sorry if it's already here on the forum! Thanks

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    This are standard fact which you can find in most functional analysis books2012-04-18
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    Conway's Functional Analysis is a good reference for this.2012-04-18
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    Thanks a lot; got it. What about $\lambda = 0$. Do you know any simple counterexample?2012-04-19
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    @Anna you can answer your own question.2012-04-19
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    http://meta.math.stackexchange.com/q/3286/82712012-04-19

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