Fredholmness and compactness of truncated Toeplitz and Hankel operators

We prove the spectral mapping theorem $\sigma_e(A_\phi) = \phi(\sigma_e(A_z))$fortheFredholmspectrumofatruncatedToeplitzoperator$A_\phi$withsymbol$\phi$intheSarasonalgebra$C+H^\infty$ 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 $S^p$.