Theoretical analysis demonstrates Perron–Frobenius spectral properties in ordered Banach algebras, highlighting asymptotic positivity under relaxed structural assumptions.
In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral radius is the only element in the peripheral spectrum; there are positive (approximate) eigenvectors for the spectral radius. Recently such types of results have been shown for operators on Banach lattices. Our results can be viewed as a complement, since our structural assumptions on the ordered Banach algebra are much weaker.
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Herzog et al. (2024) studied this question.
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