Linear shift-invariant input-output maps do not necessarily commute

It is part of the engineering folklore that linear shift-invariant input-ou tput operators that take a set of functions (closed under translation) into itself commute in the sense that H1H2 = H2H1 for any two such operators H- 1 and H-2. The main purpose of this paper is to record theorems to the effe ct that, in a certain very reasonable discrete-space setting, it is not tru e that shift-invariant operators commute, even though H1H2 = H2H1 holds on certain interesting subsets of the set of inputs. A result showing the lack of commutativity for continuous-space systems is also given. Copyright (C) 2000 John Wiley & Sons, Ltd.