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Linear maps preserving product of involutions

Chi-Kwong Li, Tejbir Lohan, Sushil Singla

Abstract

An element of the algebra $M_n(\mathbb{F})$ of $n \times n$ matrices over a field $\mathbb{F}$ is called an involution if its square equals the identity matrix. Gustafson, Halmos, and Radjavi proved that any product of involutions in $M_n(\mathbb{F})$ can be expressed as a product of at most four involutions. In this article, we investigate the bijective linear preservers of the sets of products of two, three, or four involutions in $M_n(\mathbb{F})$.

Linear maps preserving product of involutions

Abstract

An element of the algebra of matrices over a field is called an involution if its square equals the identity matrix. Gustafson, Halmos, and Radjavi proved that any product of involutions in can be expressed as a product of at most four involutions. In this article, we investigate the bijective linear preservers of the sets of products of two, three, or four involutions in .

Paper Structure

This paper contains 7 sections, 22 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem
  4. Related results
  5. Non-unital preservers of ${\mathcal{C}}_n$
  6. The characteristic $2$ case
  7. Future directions

Key Result

Theorem 1.1

Let ${\mathbb F}$ be a field with characteristic not equal to $2$. Let $T: M_n({\mathbb F}) \rightarrow M_n({\mathbb F})$ be a bijective linear map. Then the following statements hold.

Theorems & Definitions (48)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 38 more