In this paper, we study direct problems with a fractional Caputo differential operator using the Fourier variable separation method. The equation contains a linear transformation of the involution in the second derivative. As a consequence of the proved theorem on the equivalence of eigenfunction expansions, we prove the basicity in the space of ${{L}_{2}}\left( -1,1 \right)$ of the eigenfunctions of the spectral problem. The existence and uniqueness of the solution of the studied problems for a fractional differential operator with an involution and with a complex-valued coefficient is proved.