Products of involutions of an infinite-dimensional vector space

TL;DR

This paper proves that any automorphism of an infinite-dimensional vector space can be expressed as a product of four involutions, and characterizes those that are products of three involutions, with detailed polynomial conditions.

Contribution

It establishes the minimal number of involutions needed to factorize automorphisms in infinite-dimensional spaces and characterizes the cases with three involutions.

Findings

Every automorphism is a product of four involutions.

Characterization of automorphisms as products of three involutions.

Analysis of decompositions with specific polynomial conditions.

Abstract

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$.