Mathematica Bohemica, Vol. 143, No. 4, pp. 441-448, 2018
Subsets of nonempty joint spectrum in topological algebras
Received September 5, 2017. Published online June 12, 2018.
Abstract: We give a necessary and a sufficient condition for a subset $S$ of a locally convex Waelbroeck algebra $\A$ to have a non-void left joint spectrum $\sigma_l(S).$ In particular, for a Lie subalgebra $L\subset\A$ we have $\sigma_l(L)\neq\emptyset$ if and only if $[L,L]$ generates in $\A$ 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.
Keywords: joint spectrum; Waelbroeck algebra; commutator; spectral mapping formula