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Subsets of nonempty joint spectrum in topological algebras. (English) Zbl 06997377
Summary: We give a necessary and a sufficient condition for a subset $$S$$ of a locally convex Waelbroeck algebra $$\mathcal{A}$$ to have a non-void left joint spectrum $$\sigma_l(S).$$ In particular, for a Lie subalgebra $$L\subset\mathcal{A}$$ we have $$\sigma_l(L)\neq\emptyset$$ if and only if $$[L,L]$$ generates in $$\mathcal{A}$$ a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.
##### MSC:
 47A13 Several-variable operator theory (spectral, Fredholm, etc.) 47A60 Functional calculus for linear operators 46H30 Functional calculus in topological algebras
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##### References:
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