Classification of linear operators satisfying $(Au,v)=(u,A^rv)$ or $(Au,A^rv)=(u,v)$ on a vector space with indefinite scalar product

TL;DR

This paper classifies linear operators on vector spaces with indefinite scalar products that satisfy specific symmetric or Hermitian-like relations involving their adjoints and powers, covering real, complex, and quaternionic cases.

Contribution

It provides a comprehensive classification of operators satisfying these relations across various types of scalar products and fields, extending previous results to more general settings.

Findings

Complete classification of operators satisfying $(Au,v)=(u,A^rv)$

Complete classification of operators satisfying $(Au,A^rv)=(u,v)$

Results applicable to real, complex, and quaternion vector spaces

Abstract

We classify all linear operators $A:V\to V$ satisfying $(Au,v)=(u,A^rv)$ and all linear operators satisfying $(Au,A^rv)=(u,v)$ with $r=2,3,\dots$ on a complex, real, or quaternion vector space with scalar product given by a nonsingular symmetric, skew-symmetric, Hermitian, or skew-Hermitian form.