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Products of idempotent linear transformations

Published online by Cambridge University Press:  14 November 2011

M. A. Reynolds  and
R. P. Sullivan
M. A. Reynolds
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, Australia6009
R. P. Sullivan
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, Australia6009
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Synopsis

In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 1-2 , 1985 , pp. 123 - 138
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1. Bourbaki, N.. Eléments de mathétnatique, Livre V; Espaces vectoriels topologiques (Paris: Hermann, 1967).Google Scholar
2. Clifford, A. H. and Preston, G. B.. The Algebraic Theory of Semigroups (Math. Surveys, No. 7) (Providence, R.I.: Amer. Math. Soc, Vol. 1, 1961;Vol. 2, 1967).Google Scholar
3. Dawlings, R. J. H.. Products of idempotents in the semigroup of singular endomorphisms of a finite-dimensional vector space. Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 123–133.CrossRefGoogle Scholar
4Dawlings, R. J. H.. Sets of idempotents that generate the semigroup of singular endomorphisms of a finite-dimensional vector space. Proc. Edinburgh Math. Soc. 25 (1982), 133–139.CrossRefGoogle Scholar
5Dawlings, R. J. H.. The idempotent generated subsemigroup of the semigroup of continuous endomorphisms of a separable Hilbert space. Proc. Roy. Soc. Edinburgh Sect. A 94 (1983), 351–360.CrossRefGoogle Scholar
6Erdos, J. A.. On products of idempotent matrices. Glasgow Math. J. 8 (1967), 118–122.CrossRefGoogle Scholar
7Fuhrmann, P. A.. Linear Systems and Operators in Hilbert Space (NewYork: McGraw-Hill, 1981).Google Scholar
8Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
9Howie, J. M.. An introduction to semigroup theory (London: Academic, 1976).Google Scholar
10Howie, J. M.. Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–167.CrossRefGoogle Scholar
11Jacobson, N.. Lectures in Abstract Algebra, Vol. II (New York: VanNostrand, 1953).CrossRefGoogle Scholar
12Köthe, G.. Topological Vector Spaces. Vol. I (Berlin: Springer, 1969).Google Scholar
13Petrich, M.. Rings and Semigroups. Lecture Notes in Mathematics 380 (Berlin: Springer, 1974).Google Scholar
14Reynolds, M. A. and Sullivan, R. P.. The ideal structure of idempotent-generated transformation semigroups (submitted).Google Scholar
15Weidmann, J.. Linear operators on Hilbert spaces (New York: Springer, 1980).CrossRefGoogle Scholar