×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Harte, R. E., Spectral mapping theorem, Proc. R. Ir. Acad., Sect. A 72 (1972), 89-107 · Zbl 0206.13301
[2] Janas, J., Note on the joint spectrum of the Wiener-Hopf operators, Proc. Am. Math. Soc. 50 (1975), 303-308 · Zbl 0337.47017
[3] Müller, V.; Sołtysiak, A., Spectrum of generators of a noncommutative Banach algebra, Studia Math. 93 (1989), 87-95 · Zbl 0704.46027
[4] Nuñez, J. R., A Joint Spectrum Associated to an Ideal, Tesis de Maestría, Universidad Autónoma Metropolitana, Ciudad de México (2017)
[5] Pryde, A. J.; Sołtysiak, A., On joint spectra of non-commuting normal operators, Bull. Aust. Math. Soc. 48 (1993), 163-170 · Zbl 0810.47003
[6] Wawrzyńczyk, A., Joint spectra in Waelbroeck algebras, Bol. Soc. Mat. Mex., III. Ser. 13 (2007), 321-343 · Zbl 1178.46047
[7] Wawrzyńczyk, A., Schur lemma and the spectral mapping formula, Bull. Pol. Acad. Sci., Math. 55 (2007), 63-69 · Zbl 1118.46045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.