Differentiation in Star-Invariant Subspaces I. Boundedness and Compactness

Given a discrete set Λ⊂C\R, let RΛp denote the subspace of Lp(R) generated by rational functions with poles in Λ. We determine for which Λʹs the differentiation operator d/dx:RΛp→Lp(R) is bounded or compact.