Isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form

TL;DR

This paper classifies isometric, selfadjoint, and skewadjoint operators on vector spaces with diagonalizable forms over various fields, providing canonical matrices based on known classifications of forms.

Contribution

It extends the classification of such operators to fields beyond the complex numbers, including fields with characteristic not equal to 2, using known form classifications.

Findings

Canonical matrices for isometric and selfadjoint operators over complex fields.

Canonical matrices for isometric, selfadjoint, and skewadjoint operators over fields with characteristic not 2.

Connection to classification of symmetric and Hermitian forms over finite extensions.

Abstract

Let $V$ be a vector space over a field $\mathbb F$ with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis. If $\mathbb F=\mathbb C$, then we give canonical matrices of isometric and selfadjoint operators on $V$ using known classifications of isometric and selfadjoint operators on a complex vector space with nondegenerate Hermitian form. If $\mathbb F$ is a field of characteristic different from $2$, then we give canonical matrices of isometric, selfadjoint, and skewadjoint operators on $V$ up to classification of symmetric and Hermitian forms over finite extensions of $\mathbb F$.