Continuity of linear operators commuting with shifts

It is shown that linear operators on a wide class of function and measure spaces on [0, ∞) which commute with the right shift operators are necessarily continuous. Indeed, commutativity with a single nonidentity shift is often sufficient, and the results are valid for more general cones. As a consequence characterizations of perfect operators and other multipliers are obtained, and applied to prove an extension of a result of Diamond for derivations on measure algebras.