Abstract: 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.
Keywords: ordered Banach algebra; eventually positive element; spectral property; Perron-Frobenius property
DOI: DOI 10.14712/1213-7243.2023.030
AMS Subject Classification: 46B40 46H05