zbMATH — the first resource for mathematics

How to make nontrivial the spectrum of a translation invariant space of smooth functions. (English) Zbl 0852.46025
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.
46E10 Topological linear spaces of continuous, differentiable or analytic functions
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47B38 Linear operators on function spaces (general)
46F12 Integral transforms in distribution spaces
44A35 Convolution as an integral transform
47A10 Spectrum, resolvent
Full Text: DOI