On Factoring an Operator Using Elements of its Kernel

TL;DR

This paper generalizes a classical factorization theorem for differential operators to a broader class of operators defined by an endomorphism on an algebra, providing explicit criteria and examples.

Contribution

It extends the factorization result to operators formed from an endomorphism on an algebra, with explicit construction of the generator operator.

Findings

Operators with specified kernel elements form a left ideal generated by a specific operator.

The generalized theorem applies to various contexts, including automorphisms of finite order.

Explicit examples demonstrate the theorem's applicability in different algebraic settings.

Abstract

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation $\partial$ acting on some ring of functions, this paper considers the more general situation of an endomorphism $\mathfrak{D}$ acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of $\mathfrak{D}$ followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which $\mathfrak{D}$ is an automorphism of finite order.