Convergence properties of positive elements in Banach algebras
S Mouton - Mathematical Proceedings of the Royal Irish Academy, 2002 - muse.jhu.edu
Mathematical Proceedings of the Royal Irish Academy, 2002•muse.jhu.edu
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 …
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 …
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 () 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 . 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 and, for all n big enough, is a Riesz point of . 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 . Some of these results are applicable to certain types of operators.
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