Groups and semigroups generated by a single unitary orbit
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Groups and semigroups generated by a single unitary orbit Heydar Radjavi1 · Ahmed Ramzi Sourour2 Received: 11 December 2019 / Accepted: 12 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given invertible matrix A. In particular, we give necessary and sufficient conditions for such a semigroup to be the special linear group or the general linear group or other classical matrix groups. Furthermore, we use the above to determine the subgroups of the general linear group that are normalized by the unitary group. Keywords Unitary orbit · Matrix semigroups · Unitary groups · Linear groups
1 Introduction Let A be an n × n matrix over the field ℂ of complex numbers and let U(A) stand for the unitary orbit of A, i.e., the set of all matrices of the form U ∗ AU , where U is an n × n unitary matrix. The (multiplicative) semigroup generated by U(A) will be denoted by S(A) and the group generated by U(A) will be denoted by G(A). We investigate the structure of S(A) and G(A). In particular we determine when S(A) is a group. If A is itself a unitary, then U(A) is a conjugacy class and G(A) is a normal subgroup of the unitary group. When A is not unitary, the situation is quite different and the semigroup S(A) is more complicated.
Communicated by Jan Okniński. Research of both authors is partially supported by NSERC discovery grants. * Ahmed Ramzi Sourour [email protected] Heydar Radjavi [email protected] 1
Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
2
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3P4, Canada
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H. Radjavi, A. R. Sourour
In the general linear group, a conjugacy class is a similarity orbit {T −1 AT ∶ T invertible} and the semigroup and the group generated by one such conjugacy class over an arbitrary field were identified in [5]. The semigroup generated by the unitary orbit of a singular complex matrix were characterized in [10] and the semigroup generated by the similarity orbit of a singular matrix over an arbitrary field were identified in [5]. For operators on infinite-dimensional Hilbert space, the structure of groups and semigroups generated by a unitary orbit or a similarity orbit were investigated in [3]. One motivation of the current investigation is the observation that in several groups the semigroup generated by a certain conjugacy class is the full group. For example, every permutation is a product of transpositions; every matrix of determinant 1 is a product of transvections [1]; every unitary operator on Hilbert space is a product of symmetries [6]. We now fix some notation and terminology. We denote the set of all n × n matrices over the complex field by Mn or Mn (ℂ) . We denote the general linear group and special linear group over a field 𝔽 by 𝐆𝐋n (𝔽 ) and 𝐒𝐋n (𝔽 ) respectively while 𝐔n (ℂ) and 𝐒𝐔n (ℂ) denote the unit
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