$0\leq X^{2}\leq X$
HTML articles powered by AMS MathViewer
- by Ralph Gellar PDF
- Trans. Amer. Math. Soc. 173 (1972), 341-352 Request permission
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.References
- Taen-yu Dai, On some special classes of partially ordered linear algebras, J. Math. Anal. Appl. 40 (1972), 649β682. MR 316342, DOI 10.1016/0022-247X(72)90011-X
- Ralph DeMarr, On partially ordering operator algebras, Canadian J. Math. 19 (1967), 636β643. MR 212579, DOI 10.4153/CJM-1967-057-6
- Ralph Gellar, Spectrum of $X$ satisfying $0\leq X^{p}\leq X$, Proc. Amer. Math. Soc. 30 (1971), 32β36. MR 283542, DOI 10.1090/S0002-9939-1971-0283542-3 K. Knopp, Theory and applications of infinite series, Hafner, New York, 1963.
- HidegorΓ΄ Nakano, Modern Spectral Theory, Maruzen Co. Ltd., Tokyo, 1950. MR 0038564
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008 F. R. Gantmacher, The theory of matrices, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959. MR 16, 438; MR 21 #6372c.
- Ralph DeMarr, Partially ordered spaces and metric spaces, Amer. Math. Monthly 72 (1965), 628β631. MR 179760, DOI 10.2307/2313852 R. Gellar, Shift operators in Banach space, Dissertation, Columbia University, New York, 1968.
- Norbert Wiener, The Fourier integral and certain of its applications, Dover Publications, Inc., New York, 1959. MR 0100201
- Richard S. Varga, Matrix iterative analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0158502
- Ralph DeMarr, A generalization of the Perron-Frobenius theorem, Duke Math. J. 37 (1970), 113β120. MR 254592
Additional Information
- © Copyright 1972 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 173 (1972), 341-352
- MSC: Primary 46A40
- DOI: https://doi.org/10.1090/S0002-9947-1972-0318833-6
- MathSciNet review: 0318833