Representation of shift-invariant operators on L ² by H ^∞ transfer functions: An elementary proof, a generalization to L ^p, and a counterexample for L ^∞

We give an elementary proof of the well-known fact that shift-invariant operators on L ²[0, ∞) are represented by transfer functions which are bounded and analytic on the right open half-plane. We prove a generalization to Banach space-valued L ^p -functions, where 1≤p<∞. We show that the result no longer holds for p=∞.