Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-25T16:22:16.115Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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