How to make nontrivial the spectrum of a translation invariant space of smooth functions (Q1910111)

Let \(E(\mathbb{R}^n)\) denote the space of smooth functions on \(\mathbb{R}^n\) for \(n> 1\), and let \(E'(\mathbb{R}^n)\) denote its dual space. If \(V\) is a closed linear subspace of \(E\), and \(T\) belongs to \(E'\), then we define the linear operator \(A_T(f)= T*f\) for each \(f\in V\) in terms of the convolution operation. We say that \(V\) is translation invariant in case \(A_T(f)\in V\) for all \(f\in V\) and all \(T\in E'\). The author defines and studies the spectrum of the convolution operator \(A_T\) in some compactification of the complex plane. He then proceeds to define and study the joint spectrum of elements \(T_1,\dots, T_k\in E'\) and constructs a global resolvent on the set of regular points. He studies at length the case where \(k= n\) and \(T_j= \sqrt{- 1} \partial/\partial x_j\) and calls its spectrum the extended spectrum of \(V\). His main result is that the extended spectrum is not empty. Finally, he proposes a second definition of spectrum, proves some desirable properties of it, and conjectures that it coincides with his first definition.