The Twofold Ellis–Gohberg Inverse Problem in an Abstract Setting and Applications

  • S. ter HorstEmail author
  • M. A. Kaashoek
  • F. van Schagen
Part of the Operator Theory: Advances and Applications book series (OT, volume 272)


In this paper we consider a twofold Ellis–Gohberg type inverse problem in an abstract *-algebraic setting. Under natural assumptions, necessary and sufficient conditions for the existence of a solution are obtained, and it is shown that in case a solution exists, it is unique. The main result relies strongly on an inversion formula for a 2 × 2 block operator matrix whose off diagonal entries are Hankel operators while the diagonal entries are identity operators. Various special cases are presented, including the cases of matrixvalued L1-functions on the real line and matrix-valued Wiener functions on the unit circle of the complex plane. For the latter case, it is shown how the results obtained in an earlier publication by the authors can be recovered.


Inverse problem operator inversion Wiener functions abstract Toeplitz and Hankel operators integral operators 

Mathematics Subject Classification (2010)

Primary: 47A56 Secondary: 15A29 47B35 47G10 


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • S. ter Horst
    • 1
    Email author
  • M. A. Kaashoek
    • 2
  • F. van Schagen
    • 2
  1. 1.Department of Mathematics, Unit for BMINorth West UniversityPotchefstroomSouth Africa
  2. 2.Department of MathematicsVrije Universiteit AmsterdamAmsterdamThe Netherlands

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