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 ≤ yx or yx ≤ xy. Furthermore, we show that the spectral radius is continuous at x, considered as an element of the set of all positive elements y ≥ x such that either xy ≤ yx or yx ≤ xy. We also show that the property ϱ(x + y) ≤ ϱ(x) + ϱ(y) of the spectral radius ϱ can be obtained for positive elements y which satisfy at least one of the above inequalities.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm174-1-6,
author = {S. Mouton},
title = {On spectral continuity of positive elements},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {75-84},
zbl = {1105.46031},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm174-1-6}
}
S. Mouton. On spectral continuity of positive elements. Studia Mathematica, Tome 173 (2006) pp. 75-84. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm174-1-6/