Wiener-Hopf operators on $L_{\omega}^{2}(\mathbb{R}^{+})$

Abstract.

Let $L_{\omega}^{2}(\mathbb{R}^{+})$ be a weighted space with weight ω. In this paper we show that for every Wiener-Hopf operator T on $L_{\omega}^{2}(\mathbb{R}^{+})$ and for every a ∈I_ω, there exists a function $v_a \in L^\infty (\mathbb{R})$ such that

$$(Tf)_a = P^ + \mathcal{F}^{ - 1} (v_a \widehat{(f)_a }),$$

for all $f \in C_c^\infty (\mathbb{R}^ + ).$ Here (g) _a denotes the function x → g(x)e^ax for $g \in L_\omega ^2 (\mathbb{R}^ + ),P^ + f = \chi _{\mathbb{R}^{+}}f$ and $I_\omega = [\ln R_\omega ^ - ,\ln R_\omega ^ + ],$ where R ⁺_ω is the spectral radius of the shift S : f(x) → f(x−1) on $L_{\omega}^{2}(\mathbb{R}^{+}),$ while $\frac{1}{R_{\omega}^{-}}$ is the spectral radius of the backward shift S⁻¹ : f(x) → (P⁺f)(x+1) on $L_{\omega}^{2}(\mathbb{R}^{+}).$ Moreover, there exists a constant C_ω, depending on ω, such that $||v_a ||_\infty \leqq C_\omega ||T||$ for every a ∈I_ω. If R ⁻_ω < R ⁺_ω , we prove that there exists a bounded holomorphic function v on ${\mathop A\limits^{\circ}}_{\omega}:=\{ z \in \mathbb{C} | \,{\text{Im}}\, z \in {\mathop I\limits^{\circ}}_{\omega}\}$ such that for $a \in {\mathop I\limits^{\circ}}_{\omega} ,$ the function v _a is the restriction of v on the line $\{z \in \mathbb{C} | \,{\text{Im}}\, z = a\} .$