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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.
MSC:
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
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