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

A. Böttcher, S. M. Grudsky, I. M. Spitkovsky

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalJournal of Integral Equations and Applications
Volume12
Issue number1
DOIs
StatePublished - 2000

ASJC Scopus subject areas

  • Numerical Analysis
  • Applied Mathematics

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