Fredholm Index Formula for a Class of Matrix Wiener–Hopf Plus and Minus Hankel Operators with Symmetry



The main goal of this chapter is to obtain a Fredholm index formula for a class of Wiener.Hopf plus and minus Hankel operators which contain a certain symmetry between their Fourier symbols. It is relevant to mention that Wiener. Hopf plus and minus Hankel operators (with and without symmetries) appear in several different kinds of applications [CST04]; therefore, further knowledge about their Fredholm property and index is relevant for both theoretical and applied reasons. In view of this, several works concerning these classes of operators have appeared recently [BoCa06, BoCa, CaSi09, NoCa07]. The Fourier matrix symbols considered in this chapter belong to the C.*algebra of piecewise almost periodic functions. Besides the Fredholm index formula, conditions that ensure the Fredholm property of the operators under study will also be obtained.


Fredholm Operator Hankel Operator Piecewise Continuous Function Fredholm Property Extension Relation 


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  1. [BoCa06]
    Bogveradze, G., Castro, L.P.: Wiener–Hopf plus Hankel operators on the real line with unitary and sectorial symbols. Contemp. Math., 414, 77–85 (2006).MathSciNetGoogle Scholar
  2. [BoCa]
    Bogveradze, G., Castro, L.P.: On the Fredholm property and index of Wiener–Hopf plus/minus Hankel operators with piecewise almost periodic symbols. Appl. Math. Inform. Mech., (to appear).Google Scholar
  3. [BKS02]
    Böttcher, A.,Karlovich , Yu.I., Spitkovsky, I.M.: Convolution Operators and Factorization of Almost Periodic Matrix Functions, Birkhäuser, Basel (2002).MATHGoogle Scholar
  4. [CST04]
    Castro, L.P., Speck, F.-O., Teixeira, F.S.: On a class of wedge diffraction problems posted by Erhard Meister. Operator Theory Adv. Appl., 147, 211–238 (2004).MathSciNetGoogle Scholar
  5. [CaSi09]
    Castro, L.P., Silva, A.S.: Invertibility of matrix Wiener–Hopf plus Hankel operators with symbols producing a positive numerical range. Z. Anal. Anwendungen, 28, 119–127 (2009).MATHMathSciNetGoogle Scholar
  6. [NoCa07]
    Nolasco, A.P., Castro, L.P.: A Duduchava-Saginashvili’s type theory for Wiener–Hopf plus Hankel operators. J. Math. Anal. Appl., 331, 329–341 (2007).MATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Universidade de AveiroAveiroPortugal

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