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There is a theorem in functional analysis, that says that for a selfadjoint compact operator $T:H\rightarrow H$, either $\lVert T\rVert $ or $-\lVert T\rVert$ is an eigenvalue. For finite dimensional $H$ it is easy to construct examples of operators such that $\lVert T \rVert$ and $-\lVert T\rVert$ are eigenvalues.

But I couldn't find yet an example for infinite dimensional spaces of such operators. Could someone provide me with one or give me a reference ?

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up vote 2 down vote accepted

You can build yourself useful toy examples in $\ell^2$ space. Take a sequence $(a_n)$ of real numbers such that $a_n \to 0$ and define $T\mathbf{x}=(a_1x_1, a_2x_2, a_3x_3 \ldots)$. This operator is compact, selfadjoint, and its eigenvalues are precisely $a_1, a_2, a_3 \ldots$.

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