Geodesic
Subsets of nonempty joint spectrum in topological algebras
Mathematica Bohemica, Tome 143 (2018) no. 4, pp. 441-448
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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. \endgraf 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.
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. \endgraf 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.
DOI : 10.21136/MB.2018.0098-17
Classification : 46H30, 47A13, 47A60
Keywords: joint spectrum; Waelbroeck algebra; commutator; spectral mapping formula 
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Wawrzyńczyk, Antoni. Subsets of nonempty joint spectrum in topological algebras. Mathematica Bohemica, Tome 143 (2018) no. 4, pp. 441-448. doi: 10.21136/MB.2018.0098-17

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