Show that $\sum_{k=1}^{\infty} \int_{0}^{\pi} \frac{\cos(kt)}{k}f(s)\cos(ks)ds$ is a compact operator

Question

Consider $X=L[0,\pi]$ and define $$T(f)=\sum_{k=1}^{\infty} \int_{0}^{\pi} \frac{\cos(kt)}{k}f(s)\cos(ks)ds.$$ Show that $T$ is a compact operator and find the eigenvalues and eigenvectors.

First I define $T_n(f)=\sum_{k=1}^{n} \int_{0}^{\pi} \frac{\cos(kt)}{k}f(s)\cos(ks)ds$ but does $T_n \rightarrow T$ in the operator norm? I am not sure about this. Is this the right way? I will be very grateful for any hint or suggestion.