Author:
Ralph Gellar
Journal:
Trans. Amer. Math. Soc. 173 (1972), 341352
MSC:
Primary 46A40
MathSciNet review:
0318833
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Abstract: This paper studies the structure of elements satisfying in a Dedekind complete partially ordered real linear algebra. The lollipopshaped possible spectrum of 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 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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 T. Y. Dai, On some special classes of partially ordered linear algebras, J. Math. Anal. Appl. (to appear). MR 0316342 (47:4890)
 [2]
 R. DeMarr, On partially ordering operator algebras, Canad. J. Math. 19 (1967), 636643. MR 35 #3450. MR 0212579 (35:3450)
 [3]
 R. Gellar, Spectrum of satisfying , Proc. Amer. Math. Soc. 29 (1971), 3236. MR 0283542 (44:773)
 [4]
 K. Knopp, Theory and applications of infinite series, Hafner, New York, 1963.
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 H. Nakano, Modern spectral theory, Maruzen, Tokyo, 1950. MR 12, 419. MR 0038564 (12:419f)
 [6]
 K. Hoffman, Banach spaces of analytic functions, PrenticeHall Series in Modern Analysis, PrenticeHall, Englewood Cliffs, N. J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
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 F. R. Gantmacher, The theory of matrices, GITTL, Moscow, 1953; English transl., Chelsea, New York, 1959. MR 16, 438; MR 21 #6372c.
 [8]
 R. DeMarr, Partially ordered spaces and metric spaces, Amer. Math. Monthly 72 (1965), 628631. MR 31 #4003. MR 0179760 (31:4003)
 [9]
 R. Gellar, Shift operators in Banach space, Dissertation, Columbia University, New York, 1968.
 [10]
 N. Wiener, Fourier integral and certain of its applications, Cambridge Univ. Press, Cambridge, 1933; reprint, Dover, New York, 1959. MR 20 #6634. MR 0100201 (20:6634)
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 R. S. Varga, Matrix iterative analysis, PrenticeHall, Englewood Cliffs, N. J., 1962. MR 28 #1725. MR 0158502 (28:1725)
 [12]
 R. DeMarr, A generalization of the PerronFrobenius theorem, Duke Math. J. 37 (1970), 113120. MR 40 #7800. MR 0254592 (40:7800)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197203188336
PII:
S 00029947(1972)03188336
Keywords:
Dedekind 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
