Formal proofs of operator identities by a single formal computation

Computing elements of certain form in ideals to prove properties of operators

“…To end this section, we recall some of the notions related to the framework developed in [17] that are also relevant for the work presented here. For a short, more formal summary see also the appendix in [4].…”

“…Informally speaking, a polynomial is compatible with Q if and only if it can be translated into a valid operator in the setting encoded by Q. The main result of [17] then says the following. If the polynomials F ⊆ K X representing the assumptions of a statement about operators as well as the polynomial f ∈ K X corresponding to the claimed property are all compatible with the quiver Q encoding the domains and codomains, then the ideal membership f ∈ (F ) is equivalent to the correctness of the statement about operators.…”

“…The following corollary from [17,Theorem 16] ensures under very mild assumptions on the generators of I that, when working with ρ Q (I) instead of ρ(I), we can still form all compatible polynomials in I.…”

“…However, not every computation with noncommutative polynomials can be translated into a computation with operators, since addition and composition of operators are restricted by their domains and codomains. In [17], a general algebraic framework was developed that provides conditions on the generators of the ideal and the given polynomial to ensure that ideal membership of that polynomial is equivalent to the existence of a valid computation with actual operators proving the corresponding identity. More precisely, in this way a single computation with polynomials verifying ideal membership proves the statement about operators in an abstract setting in such a way that analogous statements in various concrete settings (e.g.…”

“…To keep the presentation self-contained, we briefly recall the most important notions about noncommutative polynomials and the algebraic proof framework from [17] in the following section.…”