Abstract
We study the spectrum σ(M) of the multipliers M which commute with the translations on weighted spaces \({L_{\omega}^{2}(\mathbb{R})}\) For operators M in the algebra generated by the convolutions with \({\phi \in {C_c(\mathbb {R})}}\) we show that \({\overline{\mu(\Omega)} = \sigma(M)}\), 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 \({\overline{\mu(\Omega)}}\) is included in σ(M). A generalization of these results is given for the weighted spaces \({L^2_{\omega}(\mathbb {R}^{k})}\) where the weight ω has a special form.
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Petkova, V. Spectral theorem for multipliers on \({L_{\omega}^{2}(\mathbb{R})}\) . Arch. Math. 93, 357–368 (2009). https://doi.org/10.1007/s00013-009-0043-2
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DOI: https://doi.org/10.1007/s00013-009-0043-2