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Spectral theorem for multipliers on Lw^2 (R)

We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces Lω(R). For operators M in the algebra generated by the convolutions with φ ∈ Cc(R) we show that μ(Ω) = σ(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 μ(Ω) is included in σ(M). A generalization of these results is given for the weighted spaces Lω(R) where the weight ω has a special form. Mathematics Subject Classification (2000). 47A10.