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Spectral theorem for multipliers on {L_{omega}^{2}(mathbb{R})}
Authors:Violeta Petkova
Affiliation:1. LMAM, Université de Metz UMR 7122, Ile du Saulcy, 57045, Metz Cedex 1, France
Abstract:We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces ${L_{omega}^{2}(mathbb{R})}We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lw2(mathbbR){L_{omega}^{2}(mathbb{R})} For operators M in the algebra generated by the convolutions with f ? Cc(mathbb R){phi in {C_c(mathbb {R})}} we show that [`(m(W))] = s(M){overline{mu(Omega)} = sigma(M)}, where the set Ω is determined by the spectrum of the shift S and μ is the symbol of M. For the general multipliers M we establish that [`(m(W))]{overline{mu(Omega)}} is included in σ(M). A generalization of these results is given for the weighted spaces L2w(mathbb Rk){L^2_{omega}(mathbb {R}^{k})} where the weight ω has a special form.
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