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# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## On spectral continuity of positive elements

### Tom 174 / 2006

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

#### Streszczenie

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.

#### Autorzy

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

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