On spectral continuity of positive elements

Volume 174 / 2006

S. Mouton Studia Mathematica 174 (2006), 75-84 MSC: 46H05, 47A10, 47B65, 06F25. DOI: 10.4064/sm174-1-6


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.


  • S. MoutonDepartment of Mathematics
    University of Stellenbosch
    Private Bag X1
    Matieland 7602, South Africa

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