Subsets of nonempty joint spectrum in topological algebras. (English) Zbl 1463.47027

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.


47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A60 Functional calculus for linear operators
46H30 Functional calculus in topological algebras
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