Fredholm Characterization of Wiener–Hopf–Hankel Integral Operators with Piecewise Almost Periodic Symbols



This chapter is concerned with the Fredholm property of matrix Wiener–Hopf–Hankel operators (cf. [BoCa08], [BoCa], and [LMT92]) of the form
$$ W_\phi \pm H_\phi :\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N , $$
for N × N matrix-valued functions Φ with entries in the class of piecewise almost periodic elements (see [BoCa] or [BKS02]), and where WΦ and HΦ denote matrix Wiener–Hopf and Hankel operators defined by
$$ W_\phi = \,_{r + } \mathcal{F}^{ - 1} \Phi \cdot \mathcal{F}:\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N $$
$$ H_\phi = \,_{r + } \mathcal{F}^{ - 1} \Phi \cdot \mathcal{F}J:\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N $$
respectively. We are denoting by \( L^2 (\mathbb{R}) \) and \( L^2 (\mathbb{R}_{+}) \) the Banach spaces of complex-valued Lebesgue measurable functions ϕ, for which |ϕ|2 is integrable on \( \mathbb{R} \) and \( \mathbb{R} \) respectively. Moreover, in (7.1)–(7.3) \( L_ + ^2 (\mathbb{R}) \) denotes the subspace of \( L ^2 (\mathbb{R}) \) formed by all functions supported in the closure of \( \mathbb{R}_ + = (0, + \infty ) \) the operator + performs the restriction from \( L ^2 (\mathbb{R}) \) into \( L ^2 (\mathbb{R}_+) \) denotes the Fourier transformation, and J is the reflection operator given by the rule \( J\phi (x) = \tilde \phi (x) = \phi ( - x),x \in \mathbb{R} \).


Matrix Function Fredholm Operator Scalar Case Hankel Operator Fredholm Property 


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

Authors and Affiliations

  1. 1.Andrea Razmadze Mathematical InstituteAndrea RazmadzeGeorgia
  2. 2.University of AveiroAveiroPortugal

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