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Let $\mathcal{H}$ be a Hilbert space, and let $T\in K(\mathcal{H})$ be a compact operator. There exists a theorem in the following way: "$T(\mathcal{H})$ is closed in $\mathcal{H}$ if, and only if, $\dim(T(\mathcal{H}))<\infty$."

Can anybody give me a reference for this theorem, please?

niki
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2 Answers2

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This is an easy conseqeuence of Open Mapping Theorem. If $T(H)$ is closed we can think of $T$ as a surjective operator from $H$ to $T(H)$. By Open Mapping Theorem it is an open map. Hence the image of the closed unit ball contains some open ball around $0$ in $H$. Since $T$ is compact this open ball is relatively compact and this makes $T(H)$ finite dimensional.

Kavi Rama Murthy
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P.R. Halmos: Hilbert Space Problem Book, Problem 141:

Every closed subspace included in the range of a compact operator is finite- dimensional.

Fred
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  • Sorry, I checked this problem. Problem 141 is "The set of all invertible operators is connected." Would you please give me the correct number of the problem? – niki May 16 '19 at 12:32
  • In my edition it is problem 141. You will find the problem in the chapter "Compact operators". – Fred May 16 '19 at 13:43