Opuscula Math. 32, no. 2 (2012), 227-234
http://dx.doi.org/10.7494/OpMath.2012.32.2.227

Opuscula Mathematica

# On the extended and Allan spectra and topological radii

Hugo Arizmendi-Peimbert
Angel Carrillo-Hoyo
Jairo Roa-Fajardo

Abstract. In this paper we prove that the extended spectrum $$\Sigma(x)$$, defined by W. Żelazko, of an element $$x$$ of a pseudo-complete locally convex unital complex algebra $$A$$ is a subset of the spectrum $$\sigma_A(x)$$, defined by G.R. Allan. Furthermore, we prove that they coincide when $$\Sigma(x)$$ is closed. We also establish some order relations between several topological radii of $$x$$, among which are the topological spectral radius $$R_t(x)$$ and the topological radius of boundedness $$\beta_t(x)$$.

Keywords: topological algebra, bounded element, spectrum, pseudocomplete algebra, topologically invertible element, extended spectral radius, topological spectral radius.

Mathematics Subject Classification: 46H05.

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