### Abstract

Let α and β be orientation-preserving diffeomorphism (shifts) of R_{+}=(0,∞) onto itself with the only fixed points 0 and ∞. We establish a Fredholm criterion and calculate the index of the weighted singular integral operator with shifts (aI−bU_{α})P_{γ} ^{+}+(cI−dU_{β})P_{γ} ^{−}, acting on the space L^{p}(R_{+}), where P_{γ} ^{±}=(I±S_{γ})/2 are the operators associated to the weighted Cauchy singular integral operator S_{γ} given by (S_{γ}f)(t)=1πi∫R_{+}(tτ)^{γ}f(τ)τ−tdτ with γ∈C satisfying 0<1/p+ℜγ<1, and U_{α},U_{β} are the isometric shift operators given by U_{α}f=(α^{′})^{1/p}(f∘α),U_{β}f=(β^{′})^{1/p}(f∘β), under the assumptions that the coefficients a,b,c,d and the derivatives α^{′},β^{′} of the shifts are bounded and continuous on R_{+} and admit discontinuities of slowly oscillating type at 0 and ∞.

Original language | English |
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Pages (from-to) | 606-630 |

Number of pages | 25 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 450 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jun 2017 |

### Keywords

- Fredholmness
- Index
- Orientation-preserving shift
- Semi-almost periodic function
- Slowly oscillating function
- Weighted Cauchy singular integral operator

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*450*(1), 606-630. https://doi.org/10.1016/j.jmaa.2017.01.052