On the extended and Allan spectra and topological radii

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 $\beta_t(x)$.