  We prove the spectral mapping theorem $\sigma_e(A_\phi) =
\phi(\sigma_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator
$A_\phi$ with symbol $\phi$ in the Sarason algebra $C+H^\infty$ acting on a
coinvariant subspace $K_\theta$ of the Hardy space $H^2$. Our second result
says that a truncated Hankel operator on the subspace $K_\theta$ generated by a
one-component inner function $\theta$ is compact if and only if it has a
continuous symbol. We also suppose a description of truncated Toeplitz and
Hankel operators in Schatten classes $S^p$.
