Symbole d'un multiplicateur sur <f>Lω2(R)</f>

Let <f>Lω2(R)</f> be a weighted space with weight <f>ω</f>. In this paper we show that for every bounded operator <f>T</f> on <f>Lω2(R)</f> which commutes with translations, and for every <f>a∈Iω</f>, there exists a function <f>νa∈L∞(R)</f> such that Here <f>(g)a</f> denotes the function <f>x↠g(x)eax</f> for <f>g∈Lω2(R)</f> and <f>Iω=[lnRω-,lnRω+]</f>, where <f>Rω+</f> is the spectral radius of the bilateral shift <f>S :f(x)↠f(x-1)</f> on <f>Lω2(R)</f>, while <f>1/Rω-</f> is the spectral radius of <f>S-1</f>. Moreover there exists a constant <f>Cω</f> depending on <f>ω</f> such that <f>ǁνaǁ∞⩽CωǁTǁ</f> for every <f>a∈Iω</f>. If <f>Rω-<Rω+</f>, there exists a bounded holomorphic function <f>ν</f> on <f>Åω:={z∈C∣Imz∈&Iring;ω}</f>, such that <f><ovl type="circ" STYLE="S">Tf</ovl>=ν&fcirc;</f>, <f>∀f∈D(R)</f> and <f>ǁνǁ∞⩽CωǁTǁ</f>, where <f><ovl type="circ" STYLE="S">Tf</ovl>(a+ix)=<ovl type="circ" STYLE="S">(Tf)a</ovl>(x)</f> for <f>a∈&Iring;ω</f>. [Copyright &y& Elsevier]