Compatible rewriting of noncommutative polynomials for proving operator identities

Abstract

The goal of this paper is to prove operator identities using equalities between noncommutative polynomials. In general, a polynomial expression is not valid in terms of operators, since it may not be compatible with domains and codomains of the corresponding operators. Recently, some of the authors introduced a framework based on labelled quivers to rigorously translate polynomial identities to operator identities. In the present paper, we extend and adapt the framework to the context of rewriting and polynomial reduction. We give a sufficient condition on the polynomials used for rewriting to ensure that standard polynomial reduction automatically respects domains and codomains of operators. Finally, we adapt the noncommutative Buchberger procedure to compute additional compatible polynomials for rewriting. In the package OperatorGB, we also provide an implementation of the concepts developed.

Algebraic proof methods for identities of matrices and operators: improvements of Hartwig's triple reverse order law

“…For more details, see Section 5.2 in [4]. Based on a refined version of the framework using rewriting, it is possible to obtain a similar theorem where polynomials in F are allowed to have a constant term, see Theorem 32 in [36].…”