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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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