Remarks on spectra and 𝐿¹ multipliers for convolution operators

We prove that the spectrum of a convolution operator on a locally compact group G G by a self-adjoint L 1 L^1 -function f f is the same on L 1 ( G ) L^1(G) and L 2 ( G ) L^2(G) and consequently on all L p L^p spaces, 1 ≤ p > ∞ , 1\leq p>\infty , if and only if a Beurling algebra contains non-analytic functions on R \mathbb {R} operating on f f into L 1 L^1 .