On spectral continuity of positive elements

S. Mouton¹

¹ Department of Mathematics University of Stellenbosch Private Bag X1 Matieland 7602, South Africa

Studia Mathematica, Tome 174 (2006) no. 1, pp. 75-84

Résumé

Let $x$ be a positive element of an ordered Banach algebra. We prove a relationship between the spectra of $x$ and of certain positive elements $y$ for which either $xy \leq yx$ or $yx \leq xy$. Furthermore, we show that the spectral radius is continuous at $x$, considered as an element of the set of all positive elements $y \geq x$ such that either $xy \leq yx$ or $yx \leq xy$. We also show that the property $\varrho (x+y) \leq \varrho (x) + \varrho (y)$ of the spectral radius $\varrho $ can be obtained for positive elements $y$ which satisfy at least one of the above inequalities.

Détail

DOI : 10.4064/sm174-1-6

Keywords: positive element ordered banach algebra prove relationship between spectra certain positive elements which either leq leq furthermore spectral radius continuous considered element set positive elements geq either leq leq property varrho leq varrho varrho spectral radius varrho obtained positive elements which satisfy least above inequalities