Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T17:17:48.293Z Has data issue: false hasContentIssue false
  • English
  • Français

Products of Involutions of an Infinite-dimensional Vector Space

Part of: Basic linear algebra

Published online by Cambridge University Press:  15 November 2019

Clément de Seguins Pazzis
Clément de Seguins Pazzis*
Affiliation:
Université de Versailles Saint-Quentin-en-Yvelines, Laboratoire de Mathématiques de Versailles, 45 avenue des Etats-Unis, 78035Versailles cedex, France Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.

Keywords

infinite dimensiondecompositioninvolutionquadratic automorphismordinalbilateral shift

MSC classification

Primary: 15A23: Factorization of matrices
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 1 , February 2021 , pp. 195 - 220
Check for updates
Copyright
© Canadian Mathematical Society 2019

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

Ballantine, C. S., Products of involutory matrices. I. Linear Multilinear Algebra 5(1977), 53–62. https://doi.org/10.1080/03081087708817174CrossRefGoogle Scholar
Breaz, S., Endomorphisms of free modules as sums of four quadratic endomorphisms. Linear Multilinear Algebra 66(2018), 2215–2217. https://doi.org/10.1080/03081087.2017.1389853CrossRefGoogle Scholar
Djoković, D.Ž., Product of two involutions. Arch. Math. (Basel) 18(1967), 582–584. https://doi.org/10.1007/BF01898863CrossRefGoogle Scholar
Gustafson, W. H., Halmos, P. R., and Radjavi, H., Products of involutions. Linear Algebra Appl. 13(1976), 157–162. https://doi.org/10.1016/0024-3795(76)90054-9CrossRefGoogle Scholar
Halmos, P. R. and Kakutani, S., Products of symmetries. Bull. Amer. Math. Soc. 64(1958), 77–78. https://doi.org/10.1090/S0002-9904-1958-10156-1CrossRefGoogle Scholar
Hoffman, F. and Paige, E. C., Products of two involutions in the general linear group. Indiana Univ. Math. J. 20(1971), 1017–1020. https://doi.org/10.1512/iumj.1971.20.20096CrossRefGoogle Scholar
Liu, K.-M., Decomposition of matrices into three involutions. Linear Algebra Appl. 111(1988), 1–24. https://doi.org/10.1016/0024-3795(88)90047-XCrossRefGoogle Scholar
Radjavi, H., The group generated by involutions. Proc. Roy. Irish Acad. Sect. A 81A(1981), 9–12.Google Scholar
de Seguins Pazzis, C., Products of involutions in the stable general linear group. J. Algebra 530(2019), 235–289. https://doi.org/10.1016/j.jalgebra.2019.04.009CrossRefGoogle Scholar
de Seguins Pazzis, C., Sums of quadratic endomorphisms of an infinite-dimensional vector space. Proc. Edinburgh Math. Soc. 61(2018), 437–447. https://doi.org/10.1017/s0013091517000323CrossRefGoogle Scholar
de Seguins Pazzis, C., Sums of three quadratic endomorphisms of an infinite-dimensional vector space. Acta Sci. Math. (Szeged) 83(2017), 83–111.CrossRefGoogle Scholar
de Seguins Pazzis, C., The sum and the product of two quadratic matrices. arxiv:1703.01109.Google Scholar
Shitov, Y., Sums of square-zero endomorphisms of a free module. Linear Algebra Appl. 507(2016), 191–197. https://doi.org/10.1016/j.laa.2016.06.020CrossRefGoogle Scholar
Wonenburger, M. J., Transformations which are products of two involutions. J. Math. Mech. 65(1966), 327–338. https://doi.org/10.1512/iumj.1967.16.16023Google Scholar