Spectra of convolution operators on L spaces

Let 1 < p < infinity with p not equal 2. Let G denote the n-torus or Euclidean n-space, and let Gamma be the dual group of G. We show the existence of a multiplier transformation T on L(p)(G) which satisfies the following properties: (a) the transform T of T is smooth and vanishes at infinity on Gamma; (b) the spectrum of T on L(p) properly contains T(Gamma) union or logical sum{0}.