On the normalizer of the reflexive cover of a unital algebra of linear transformations

Bračič, Janko (author), Kandić, Marko (author)

Given a unital algebra ${\mathcal A}$ of linear transformations on a finite-dimensional complex vector space $V$, in this paper we study the set $\mathrm{Col}({\mathcal A})$ consisting of those invertible linear transformations $S$ on $V$ which map every subspace $M\in Lat({\mathcal A})$ to a subspace $SM\in \mathrm{Lat}({\mathcal A})$. We show that $Col({\mathcal A})$ is the normalizer of the group of invertible linear transformations in the reflexive cover of ${\mathcal A}$. For the unital algebra $(A)$ which is generated by a linear transformation $A$, we give the complete description of $\mathrm{Col}(A)$. By using the primary decomposition of $A$, we first reduce the problem of characterizing $\mathrm{Col}(A)$ to the problem of characterizing the group $\mathrm{Col}(N)$ of a given nilpotent linear transformation $N$. While $\mathrm{Col}(N)$ always contains all invertible linear transformations of the commutant of $(N)'$ of $N$, it is always contained in the reflexive cover of $(N)'$. We prove that $\mathrm{Col}(N)$ is a proper subgroup of $\mathrm{(AlgLat}(N)')^{-1}$ if and only if at least two Jordan blocks in the Jordan decomposition of $N$ are of dimension 2 or more. We also determine the group $\mathrm{Col}(J_2 \oplus J_2)$.

• Language: English
• Keywords: invariant subspace, collineation, normalizer, reflexive cover
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: NTF - Faculty of Natural Sciences and Engineering; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2022
• Number of pages: Str. 207-230
• Numbering: Vol. 653
• PID: 20.500.12556/RUL-139720
• UDC: 517.983:512.643
• ISSN on article: 0024-3795
• DOI: 10.1016/j.laa.2022.08.013
• COBISS.SI-ID: 119088643
• Publication date in RUL: 06.09.2022

Record is a part of a journal

• Title: Linear algebra and its applications
• Shortened title: Linear algebra appl.
• Publisher: North Holland
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licences

• License: CC BY-NC-ND 4.0, Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International
• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Description: The most restrictive Creative Commons license. This only allows people to download and share the work for no commercial gain and for no other purposes.

Secondary language

• Language: Slovenian
• Keywords: invarianten podprostor, kolineacija, normalizator, refleksivno pokritje