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Wiener-Hopf operators onL_{omega}^{2}(mathbb{R}^{+})

Authors:

Violeta Petkova

Affiliation:

(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 ${mathop Alimits^{circ}}_{omega}:={ z in mathbb{C} | ,{text{Im}}, z in {mathop Ilimits^{circ}}_{omega}}$ such that for $a in {mathop Ilimits^{circ}}_{omega} ,$ the function v_a is the restriction of v on the line ${z in mathbb{C} | ,{text{Im}}, z = a} .$

${z in mathbb{C} | ,{text{Im}}, z = a} .$

Received: 18 May 2004

Keywords:

KeywordHeading" >Mathematics Subject Classification (2000). Primary 47B37 Secondary 47B35

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