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

Authors:

Violeta Petkova

Institution:

(1) Laboratoire Bordelais d rsquo

Analyse et Géométrie U.M.R. 5467, Université Bordeaux 1, 351, cours de la Libération, F-33405 Talence Cedex, France

Abstract:

Let $L_{\omega}^{2}(\mathbb{R}^{+})$ be a weighted space with weight ohgr

. In this paper we show that for every Wiener-Hopf operator T on $L_{\omega}^{2}(\mathbb{R}^{+})$ and for every a isin

I, 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 rarr

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⁺ is the spectral radius of the shift S : f(x) rarr

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) rarr

(P⁺f)(x+1) on $L_{\omega}^{2}(\mathbb{R}^{+}).$ Moreover, there exists a constant C, depending on ohgr

, such that $||v_a ||_\infty \leqq C_\omega ||T||$ for every a isin

I. If R^– < R⁺, 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 $\{z \in \mathbb{C} | \,{\text{Im}}\, z = a\} .$ Received: 18 May 2004

Keywords:

Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47B37 Secondary 47B35

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