On spectral continuity of positive elements
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.