A note on \(k\)-potence preservers on matrix spaces over complex field
type: Item
instance of: scholarly article
title: A note on \(k\)-potence preservers on matrix spaces over complex field
zbMATH Open document ID: 1291.15069
DOI: 10.1155/2013/581683
author: Xiaofei Song
author: Baodong Zheng
author: Chongguang Cao
published in: Abstract and Applied Analysis
publication date: 2014-06-23T00:00:00Z
review text: Summary: Let \(\mathbb{C}\) be the field of all complex numbers, \(M_n\) the space of all \(n\times n\) matrices over \(\mathbb{C}\), and \(S_n\) the subspace of \(M_n\) consisting of all symmetric matrices. The map \(\phi:S_n\to M_n\) satisfies that \(A-\lambda B\) is \(k\)-potent in \(S_n\)? implying that \(\phi(A)-\lambda\phi(B)\) is \(k\)-potent in \(M_n\), where \(\lambda\in\mathbb{C}\) then there exist an invertible matrix \(P\in M_n\) and \(\epsilon\in\mathbb{C}\) with \(\epsilon^k=\epsilon\) such that \(\phi(X)=\epsilon P^{-1}(X)P\) for every \(X\in S_n\). Moreover, the inductive method used in this paper can be used to characterise similar maps from \(M_n\) to \(M_n\).
Mathematics Subject Classification ID: 15A86
zbMATH DE Number: 6307024
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/581683
OpenAlex ID: W2017051947
Wikidata QID: Q58917186
cites work: Linear Preserver Problems
cites work: Linear Transformations Preserving Potent Matrices
cites work: \(k\)-potence preserving maps without the linearity and surjectivity assumptions
cites work: A note on \(k\)-potence preserving maps
cites work: Maps on upper-triangular matrix algebras preserving \(k\)-potences
cites work: Q5062472
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description: scientific article; zbMATH DE number 6307024