Author:
Ralph Gellar

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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper studies the structure of elements satisfying in a Dedekind -complete partially ordered real linear algebra. The lollipop-shaped 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.

**[1]**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), 636-643. MR**35**#3450. MR**0212579 (35:3450)****[3]**R. Gellar,*Spectrum of satisfying*, Proc. Amer. Math. Soc.**29**(1971), 32-36. MR**0283542 (44:773)****[4]**K. Knopp,*Theory and applications of infinite series*, Hafner, New York, 1963.**[5]**H. Nakano,*Modern spectral theory*, Maruzen, Tokyo, 1950. MR**12**, 419. MR**0038564 (12:419f)****[6]**K. Hoffman,*Banach spaces of analytic functions*, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**24**#A2844. MR**0133008 (24:A2844)****[7]**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), 628-631. 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)****[11]**R. S. Varga,*Matrix iterative analysis*, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR**28**#1725. MR**0158502 (28:1725)****[12]**R. DeMarr,*A generalization of the Perron-Frobenius theorem*, Duke Math. J.**37**(1970), 113-120. MR**40**#7800. MR**0254592 (40:7800)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
46A40

Retrieve articles in all journals with MSC: 46A40

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0318833-6

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