The L-Convolution Type Operators

The notion of the L-convolution operator is introduced by replacing the Fourier operator in the definition of the (regular) convolution operator to the operator intertwining the Sturm-Liouville operator L with the multiplicator operator. Analogously the L-Wiener-Hopf + L-Henkel operator is introduced. In the case of a piecewise continuous symbol, we obtain a criterion for the Fredholm property and a formula for the index of the L-Wiener-Hopf + L-Henkel operator. The invertibility of L-Wiener-Hopf operator is studied in the case of reflectionless potential.