Mathematical Proceedings of the Royal Irish Academy

Abstract:

We recall the definition and properties of an algebra cone in an ordered Banach algebra and continue to develop spectral theory for the positive elements. If ($a_{n}$) is a sequence of positive elements converging to a, then an interesting question is that of which properties of the spectral radius r(a) of a are 'inherited' by $r(a_{n})$. We show that under suitable circumstances if r(a) is a Riesz point of the spectrum σ(a) of a (relative to some inessential ideal), then $r(a_{n})\rightarrow r(a)$ and, for all n big enough, $r(a_{n})$ is a Riesz point of $\sigma (a_{n})$. If the Laurent series of the corresponding resolvents are then investigated, some conclusions can be drawn regarding the convergence of the spectral idempotents, as well as the positive eigenvectors associated with $a_{n}$. Some of these results are applicable to certain types of operators.