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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ 0\leq X\sp{2}\leq X$


Author: Ralph Gellar
Journal: Trans. Amer. Math. Soc. 173 (1972), 341-352
MSC: Primary 46A40
MathSciNet review: 0318833
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Abstract: This paper studies the structure of elements $ X$ satisfying $ 0 \leqslant {X^2} \leqslant X$ in a Dedekind $ \sigma $-complete partially ordered real linear algebra. The lollipop-shaped possible spectrum of $ X$ had been described previously. Three basic example types are described, each with possible spectrum a characteristic part of the lollipop and the possibility of splitting $ X$ into a sum of these types is considered. The matrix case is scrutinized. There are applications to operator theory. Contributions to the theory of convergence in partially ordered algebras are developed for technical purposes.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0318833-6
PII: S 0002-9947(1972)0318833-6
Keywords: Dedekind $ \sigma $-complete algebra, polynomial inequality, order unit, order convergence, formal series, nonnegative matrix, localization of spectra, spectral theory, polynomial bounded operator
Article copyright: © Copyright 1972 American Mathematical Society