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Zbl 1185.47026
Invertibility characterization of Wiener-Hopf plus Hankel operators via odd asymmetric factorizations.
(English)
[J] Banach J. Math. Anal. 3, No. 1, 1-18, electronic only (2009). ISSN 1735-8787/e

The paper is devoted to the invertibility and Fredholm criteria for the Wiener-Hopf plus Hankel operator $$WH_\varphi:=\chi_+{\cal F}^{-1}\varphi{\cal F}(I+J): L^2(\Bbb{R}_+)\to L^2(\Bbb{R}_+),$$ where $\chi_+$ is the characteristic function of $\Bbb{R}_+$, ${\cal F}$ is the Fourier transform, $\varphi\in L^\infty(\Bbb{R})$, $I$ is the identity operator, $J$ is the reflection operator, $(Jf)(x)=f(-x)$ for $x\in\Bbb{R}$, and the Lebesgue space $L^2(\Bbb{R}_+)$ is identified with the subspace $\chi_+L^2(\Bbb{R})$ of $L^2(\Bbb{R})$. A function $\varphi\in L^\infty(\Bbb{R})$ with $1/\varphi\in L^\infty(\Bbb{R})$ is said to admit an odd asymmetric factorization in $L^2(\Bbb{R})$ if it is represented in the form $$\varphi(x)=\varphi_-(x)\left(\frac{x-i}{x+i}\right)^m\varphi_0(x),\quad x\in\Bbb{R},$$ where $m\in\Bbb{Z}$ is the index of such a factorization, and {\parindent=8mm \item{(i)} $\frac{x}{(x-i)^2} \varphi_-\in H_-^2(\Bbb{R}),\quad \frac{1}{(x-i)^2} \varphi_-^{-1}\in H_-^2(\Bbb{R});$ \item{(ii)} $\frac{1}{x^2+1} \varphi_0\in L^2_{\text{odd}}(\Bbb{R}),\quad \frac{|x|}{x^2+1} \varphi_0^{-1}\in L^2_{\text{odd}}(\Bbb{R});$ \item{(iii)} the linear operator $W_{\varphi_0^{-1}}^0(I-J)\chi_+W_{\varphi_-^{-1}}^0:L^2(\Bbb{R})\to L^2_{\text{even}}(\Bbb{R})$ is bounded. \par} Here, $W_{\varphi}^0={\cal F}^{-1}\varphi{\cal F}:L^2(\Bbb{R})\to L^2(\Bbb{R})$ is a convolution operator, the space $H_-^2(\Bbb{R})= W_{\chi_+}^0 L^2(\Bbb{R})$ is identified with the corresponding Hardy space, $L^2_{\text{odd}}(\Bbb{R})$ and $L^2_{\text{even}}(\Bbb{R})$ are the spaces of odd and even functions in $L^2(\Bbb{R})$, respectively. Main result: If $\varphi$ is an invertible element in $L^\infty(\Bbb{R})$, then the operator $WH_\varphi$ is Fredholm on the space $L^2(\Bbb{R}_+)$ if and only if $\varphi$ admits an odd asymmetric factorization in $L^2(\Bbb{R})$. In that case, $$\dim\text{Ker}\,WH_\varphi=\max\{0,-m\},\quad \dim\text{Ker}\,WH^*_\varphi=\max\{0,m\},$$ where $m$ is the index of an odd asymmetric factorization of $\varphi$ in $L^2(\Bbb{R})$. In particular, the operator $WH_\varphi$ is invertible if it is Fredholm with zero index.
[Yuri I. Karlovich (Cuernavaca)]
MSC 2000:
*47B35 Toeplitz operators, etc.
47A68 Factorization theory of linear operators
47A53 (Semi-)Fredholm operators; index theories

Keywords: Wiener-Hopf plus Hankel operator; odd asymmetric factorisation; Fredholmness; invertibility

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