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

G. Bogveradze () and L. P. Castro ()
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G. Bogveradze: Andrea Razmadze Mathematical Institute
L. P. Castro: University of Aveiro

Chapter 7 in Integral Methods in Science and Engineering, Volume 1, 2010, pp 65-74 from Springer

Abstract: Abstract This chapter is concerned with the Fredholm property of matrix Wiener–Hopf–Hankel operators (cf. [BoCa08], [BoCa], and [LMT92]) of the form (7.1) $$ W_\phi \pm H_\phi :\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N , $$ 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 (7.2) $$ W_\phi = \,_{r + } \mathcal{F}^{ - 1} \Phi \cdot \mathcal{F}:\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N $$ (7.3) $$ H_\phi = \,_{r + } \mathcal{F}^{ - 1} \Phi \cdot \mathcal{F}J:\left[ {L_ + ^2 (\mathbb{R})} \right]^N \to \left[ {L^2 (\mathbb{R}_ + )} \right]^N $$ 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 $$ J\phi (x) = \tilde \phi (x) = \phi ( - x),x \in \mathbb{R} $$ .

Keywords: Matrix Function; Fredholm Operator; Scalar Case; Hankel Operator; Fredholm Property (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-0-8176-4899-2_7

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