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

Bračič, Janko (avtor), Kandić, Marko (avtor)

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 $\mathrm{Col}(J_2 \oplus J_2)$.

• Jezik: Angleški jezik
• Ključne besede: invariant subspace, collineation, normalizer, reflexive cover
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: NTF - Naravoslovnotehniška fakulteta ; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2022
• Št. strani: Str. 207-230
• Številčenje: Vol. 653
• PID: 20.500.12556/RUL-139720
• UDK: 517.983:512.643
• ISSN pri članku: 0024-3795
• DOI: 10.1016/j.laa.2022.08.013
• COBISS.SI-ID: 119088643
• Datum objave v RUL: 06.09.2022

BRAČIČ, Janko in KANDIĆ, Marko, 2022, On the normalizer of the reflexive cover of a unital algebra of linear transformations. Linear algebra and its applications [na spletu]. 2022. Vol. 653, p. 207–230. [Dostopano 4 april 2025]. DOI 10.1016/j.laa.2022.08.013. Pridobljeno s: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=slv&id=139720

Gradivo je del revije

• Naslov: Linear algebra and its applications
• Skrajšan naslov: Linear algebra appl.
• Založnik: Elsevier
• ISSN: 0024-3795
• COBISS.SI-ID: 1119247

Licence

• Licenca: CC BY-NC-ND 4.0, Creative Commons Priznanje avtorstva-Nekomercialno-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nc-nd/4.0/deed.sl
• Opis: Najbolj omejujoča licenca Creative Commons. Uporabniki lahko prenesejo in delijo delo v nekomercialne namene in ga ne smejo uporabiti za nobene druge namene.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: invarianten podprostor, kolineacija, normalizator, refleksivno pokritje