Linear maps preserving product of involutions

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 $M_n(\mathbb{F})$.

v3, 25 pages. Minor revision following referee report. Improved exposition; arguments clarified. Revised Propositions 2.1, 2.2, and 3.2. Added proof of Lemma 2.4; corrected proof of Lemma 2.5