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Sums of Quadratic Endomorphisms of an Infinite-Dimensional Vector Space

Part of: Basic linear algebra General and miscellaneous

Published online by Cambridge University Press:  26 February 2018

Clément de Seguins Pazzis
Clément de Seguins Pazzis*
Affiliation:
Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, 45 avenue des États-Unis, 78035 Versailles cedex, France ([email protected])
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Abstract

We prove that every endomorphism of an infinite-dimensional vector space over a field splits into the sum of four idempotents and into the sum of four square-zero endomorphisms, a result that is optimal in general.

Keywords

infinite dimensiondecompositionsquare-zero endomorphismidempotentordinals

MSC classification

Primary: 15A24: Matrix equations and identities 16B50: Category-theoretic methods and results (except as in )
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 2 , May 2018 , pp. 437 - 447
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Ballantine, C. S., Products of positive definite matrices I, Pacific J. Math. 23(3) (1967), 427–433.Google Scholar
2.Ballantine, C. S., Products of positive definite matrices II, Pacific J. Math. 24(1) (1968), 7–17.Google Scholar
3.Ballantine, C. S., Products of positive definite matrices III, J. Algebra 10(2) (1968), 174–182.Google Scholar
4.Ballantine, C. S., Products of positive definite matrices IV, Linear Algebra Appl. 3(1) (1979), 79–114.CrossRefGoogle Scholar
5.Dawlings, 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.Google Scholar
6.Djoković, D. Ž., Product of two involutions, Arch. Math. 18(6) (1967), 582–584.Google Scholar
7.Erdos, J. A., On products of idempotent matrices, Glasg. Math. J. 8 (1967), 118–122.Google Scholar
8.Fillmore, P., On sums of projections, J. Funct. Anal. 4 (1969), 146–152.Google Scholar
9.Pearcy, C. and Topping, D., Sums of small numbers of idempotents, Michigan Math. J. 14(4) (1967), 453–465.Google Scholar
10.de Seguins Pazzis, C., On decomposing any matrix as a linear combination of three idempotents, Linear Algebra Appl. 433 (2010), 843–855.Google Scholar
11.Słowik, R., Sums of square-zero infinite matrices, Linear Multilinear Algebra 64 (2016), 1760–1768.CrossRefGoogle Scholar
12.Wang, J.-H. and Wu, P. Y., Sums of square-zero operators, Studia Math. 99 (1991), 115–127.CrossRefGoogle Scholar
13.Wu, P. Y., Additive combinations of special operators, Banach Center Publ. 30(1) (1994), 337–361.CrossRefGoogle Scholar