## Abstract

The notions of the L-convolution operator and the 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 ℒ. In the case of the zero potential, the introduced operators coincide with the convolution operator and theWiener–Hopf integral operator, respectively. A connection between the 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 ℒ-Wiener–Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator ℒ are obtained.

Original language | English (US) |
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Pages (from-to) | 404-416 |

Number of pages | 13 |

Journal | Mathematical Notes |

Volume | 104 |

Issue number | 3-4 |

DOIs | |

State | Published - Sep 1 2018 |

## Keywords

- Fredholm property
- singular integral operator
- the operator L-Wiener–Hopf

## ASJC Scopus subject areas

- Mathematics(all)