On the Fredholm Property of a Class of Convolution-Type Operators

The notions of the $\mathscr L$-convolution operator and the $\mathscr L$-Wiener–Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm–Liouville operator on the axis $\mathscr L$. In the case of the zero potential, the introduced operators coincide with the convolution operator and the Wiener–Hopf integral operator, respectively. A connection between the $\mathscr L$-Wiener–Hopf operator and singular integral operators is revealed. In the case of a piecewise continuous symbol, a criterion for the Fredholm property and a formula for the index of the $\mathscr L$-Wiener–Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator $\mathscr L$ are obtained.