Evaluation-functional-preserving maps

Given any space of holomorphic functions in the open unit disc DD, satisfying certain conditions, we characterize the self-mappings of its algebraic dual space which preserve the set of all evaluation functionals δzδz. Among these maps, we give a description of those which contract the norm and those which preserve it. In the case where the norm ∥δz∥∥δz∥ depends strictly increasingly on |z||z|, we show that the first ones arise exactly from the self-maps of DD vanishing at 0. When this dependence is only injective, we prove that the second ones are precisely induced by the rotations of DD. We provide a nice generalization of those results in the case where ∥δz∥∥δz∥ grows with |θ(z)||θ(z)|, for a given automorphism θθ of DD.