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 \(\beta_t(x)\).

Keywords

• topological algebra
• bounded element
• spectrum
• pseudocomplete algebra
• topologically invertible element
• extended spectral radius
• topological spectral radius