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I need to find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$ in $L_2[0,\pi]$.

I know that this operator is self-adjoint, so its residual spectrum is empty, and i tried to find its eigenvalues, but failed.

Solved:

$f(x) = \sum\limits_{m=1}^{\infty} a_m cos(mx)dx$

\begin{align*} Af(x) &= \sum\limits_{n=1}^\infty 5^{-n} \cos(nx) \sum\limits_{m=1}^\infty a_m \int\limits_0^\pi cos(nt)cos(mt)dt \\\ &= \sum\limits_{n=1}^\infty 5^{-n} \cos(nx) \frac{\pi}{2} a_n \end{align*}

eigenvalues: $\{ \frac{5^{-n} \pi}{2}\}_{n=1}^\infty$

continius spectrum: $\{0\}$

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Hint: Write $f(x)$ in terms of its cosine series $f(x) = \sum_n a_n\cos(nx)$. Then the cosine series of $Af(x)$ is readily determinable in terms of that of $f(x)$. –  Zarrax Nov 30 '11 at 16:36
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