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: $\{0\}$