Linear shift-invariant input-output maps do not necessarily commute (Q2711558)

It is part of the engineering folklore that linear shift-invariant input-output operators that take a set of functions (closed under translation) into itself commute in the sense that \(H_1H_2= H_2H_1\) for any two such operators \(H_1\) and \(H_2\). The main purpose of this paper is to record theorems to the effect that, in a certain very reasonable discrete-space setting, it is not true that shift-invariant operators commute, even though \(H_1H_2= H_2H_1\) holds on certain interesting subsets of the set of inputs. A result showing the lack of commutativity for continuous-space systems is also given.