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On the extended and Allan spectra and topological radii

Arizmendi-Peimbert H., Carrillo-Hoyo A., Roa-Fajardo J.

Opuscula Mathematica

2012

Vol. 32, no. 2

227-234

In this paper we prove that the extended spectrum ∑(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 σA(x), defined by G.R. Allan. Furthermore, we prove that they coincide when ∑ (x) is closed. We also establish some order relations between several topological radii of x, among which are the topological spectral radius Rt(x) and the topological radius of boundedness βt(x).