### 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) |
---|---|

Pages (from-to) | 404-416 |

Number of pages | 13 |

Journal | Mathematical Notes |

Volume | 104 |

Issue number | 3-4 |

DOIs | |

State | Published - Sep 1 2018 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Notes*,

*104*(3-4), 404-416. https://doi.org/10.1134/S0001434618090080

**On the Fredholm Property of a Class of Convolution-Type Operators.** / Kamalyan, A. G.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Mathematical Notes*, vol. 104, no. 3-4, pp. 404-416. https://doi.org/10.1134/S0001434618090080

}

TY - JOUR

T1 - On the Fredholm Property of a Class of Convolution-Type Operators

AU - Kamalyan, A. G.

AU - Spitkovsky, Ilya

PY - 2018/9/1

Y1 - 2018/9/1

N2 - 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.

AB - 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.

KW - Fredholm property

KW - singular integral operator

KW - the operator L-Wiener–Hopf

UR - http://www.scopus.com/inward/record.url?scp=85056739330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85056739330&partnerID=8YFLogxK

U2 - 10.1134/S0001434618090080

DO - 10.1134/S0001434618090080

M3 - Article

VL - 104

SP - 404

EP - 416

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 3-4

ER -