Operating functions on Fourier multipliers (Q1338405)

We consider the algebra of translation invariant operators of \(L^ p({\mathbf T})\) to \(L^ q({\mathbf T})\). If \(1\leq p< q\leq \infty\), the spectrum of such an operator coincides with the closure of the range of the corresponding multiplier. Furthermore, if \(p\leq 2< q\), the operating functions on the algebra of multipliers are characterized.