Pairs of mutually annihilating operators

TL;DR

This paper provides an explicit, constructive method to classify pairs of mutually annihilating operators over any field by reducing them to canonical matrices through similarity transformations.

Contribution

It introduces a constructive approach to obtain canonical forms of mutually annihilating operator pairs over any field, extending previous classifications.

Findings

Canonical matrices are explicitly constructed.

Reduction to canonical form is achieved via similarity transformations.

The method applies over any field, not just algebraically closed ones.

Abstract

Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)--> S^{-1}AS, S^{-1}BS).