Certifying operator identities via noncommutative Gröbner bases

Abstract

Matrices or linear operators and their identities can be modelled algebraically by noncommutative polynomials in the free algebra. For proving new identities of matrices or operators from given ones, computations are done formally with noncommutative polynomials. Computations in the free algebra, however, are not necessarily compatible with formats of matrices resp. with domains and codomains of operators. For ensuring validity of such computations in terms of operators, in principle, one would have to inspect every step of the computation. In [9], an algebraic framework is developed that allows to rigorously justify such computations without restricting the computation to compatible expressions. The main result of that paper reduces the proof of an operator identity to verifying membership of the corresponding polynomial in the ideal generated by the polynomials corresponding to the assumptions and verifying compatibility of this polynomial and of the generators of the ideal.