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I know of a theorem that tells me, that every compact linear operator on an infinitedimensional Hilbert space has to have the eigenvalue $0$. On the other hand I have the operator \begin{eqnarray*} & T:\ell^{2}\rightarrow\ell^{2}\\ & \left(x_{1},x_{2},\ldots\right)\mapsto\left(\lambda_{1}x_{1},\lambda_{2}x_{2},\ldots\right), \end{eqnarray*} where $\left(\lambda_{n}\right)_{n}$ is a sequence of real nonnegative numbers, tending to $0$. Then this mapping can't have $0$ as an eigenvalue, since if that were the case, there had to be a $\left(y_{1},y_{2},\ldots\right)\in\ell^{2}$ with not all $y_{n}$'s being zero, such that $\lambda_{n}y_{n}=0$ for all $n\in\mathbb{N}$. Since $\lambda_{n}\neq0$, that would imply that all $y_{n}$'s are there.

Where is my error ? The operator $T$ is compact and $\ell^{2}$ is infinitedimensional, so this should be a counterexample to the theorem above.

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The theorem does not say that $0$ is an eigenvalue, only that the spectrum contains $0$. See Cocopuff's answer. – Erick Wong Jul 28 '12 at 5:36
Unlike in finite dimensions, an operator can have a residual and continuous spectrum. The operator may be injective but not surjective. – copper.hat Jul 28 '12 at 5:54
up vote 5 down vote accepted

$0$ being in the spectrum means that $T$ isn't invertible, which in infinite-dimensional space no longer means that it's not injective. You should be able to show that $T$ isn't surjective.

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You might want to replace the word bijective by the word invertible. – Rasmus Jul 28 '12 at 9:28
@Rasmus Alright. They should be the same in Banach spaces, though. – Cocopuffs Jul 28 '12 at 9:50
Right, this theorem helps us out. – Rasmus Jul 28 '12 at 12:02
@Cocopuffs So just to gmake sure I understood this properly: $0$ is in the spectrum, since $T$ isn't surjectiv, but since $0$, as I have shown, isn't an eigenvalue, $T$ has to be injective ? – pink_pyjamas Jul 28 '12 at 12:45
@user36675: Related info: Compact operators on infinite dimensional spaces are never surjective (although they can be injective as in your example). All nonzero elements of the spectrum of a compact operator are eigenvalues (e.g. see the Fredholm alternative). – Jonas Meyer Jul 29 '12 at 2:47

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