### Abstract

Every matrix function A ∈L∞_{n×n}(R) generates a Wiener-Hopf integral operator on L^{2}_{n}(R^{+}), the direct sum of n copies of L^{2}(R^{+}). The associated Wiener-Hopf integral operator is the operator W(Ã) where Ã(x) := A(-x). We discuss the connection between the Fredholm indices IndW(A) and Ind W(Ã). Our main result says that if A has at most a finite number d of discontinuities on R∪{∞} and both W(A) and W(Ã) are Fredholm, then |Ind W(A) + Ind W(Ã)| ≤d(n - 1); conversely, given integers K and v satisfying |κ+Ν| ≤ d(n- 1), there exist A ∈L∞n×n(R) with at most d discontinuities such that W(A) is Fredholm of index K and W(Ã) is Fredholm of index v.

Original language | English (US) |
---|---|

Pages (from-to) | 1-29 |

Number of pages | 29 |

Journal | Journal of Integral Equations and Applications |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2000 |

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### ASJC Scopus subject areas

- Numerical Analysis
- Applied Mathematics

### Cite this

*Journal of Integral Equations and Applications*,

*12*(1), 1-29. https://doi.org/10.1216/jiea/1020282131

**On the Fredholm indices of associated systems of Wiener-Hopf equations.** / Böttcher, A.; Grudsky, S. M.; Spitkovsky, Ilya.

Research output: Contribution to journal › Article

*Journal of Integral Equations and Applications*, vol. 12, no. 1, pp. 1-29. https://doi.org/10.1216/jiea/1020282131

}

TY - JOUR

T1 - On the Fredholm indices of associated systems of Wiener-Hopf equations

AU - Böttcher, A.

AU - Grudsky, S. M.

AU - Spitkovsky, Ilya

PY - 2000/12/1

Y1 - 2000/12/1

N2 - Every matrix function A ∈L∞n×n(R) generates a Wiener-Hopf integral operator on L2n(R+), the direct sum of n copies of L2(R+). The associated Wiener-Hopf integral operator is the operator W(Ã) where Ã(x) := A(-x). We discuss the connection between the Fredholm indices IndW(A) and Ind W(Ã). Our main result says that if A has at most a finite number d of discontinuities on R∪{∞} and both W(A) and W(Ã) are Fredholm, then |Ind W(A) + Ind W(Ã)| ≤d(n - 1); conversely, given integers K and v satisfying |κ+Ν| ≤ d(n- 1), there exist A ∈L∞n×n(R) with at most d discontinuities such that W(A) is Fredholm of index K and W(Ã) is Fredholm of index v.

AB - Every matrix function A ∈L∞n×n(R) generates a Wiener-Hopf integral operator on L2n(R+), the direct sum of n copies of L2(R+). The associated Wiener-Hopf integral operator is the operator W(Ã) where Ã(x) := A(-x). We discuss the connection between the Fredholm indices IndW(A) and Ind W(Ã). Our main result says that if A has at most a finite number d of discontinuities on R∪{∞} and both W(A) and W(Ã) are Fredholm, then |Ind W(A) + Ind W(Ã)| ≤d(n - 1); conversely, given integers K and v satisfying |κ+Ν| ≤ d(n- 1), there exist A ∈L∞n×n(R) with at most d discontinuities such that W(A) is Fredholm of index K and W(Ã) is Fredholm of index v.

UR - http://www.scopus.com/inward/record.url?scp=84875322701&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875322701&partnerID=8YFLogxK

U2 - 10.1216/jiea/1020282131

DO - 10.1216/jiea/1020282131

M3 - Article

AN - SCOPUS:84875322701

VL - 12

SP - 1

EP - 29

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

SN - 0897-3962

IS - 1

ER -