Spectral theorem for multipliers on ${L_{\omega}^{2}(\mathbb{R})}$

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.