Evaluation-functional-preserving maps

• Published in: Expositiones mathematicae Vol. 27; no. 3; pp. 211 - 226
• Main Authors: Jaoua, Nizar, Skhiri, Haïkel
• Format: Journal Article
• Language: English
• Published: Elsevier GmbH 2009
• Subjects: Composition operators; Evaluation functionals; Norm-contracting maps; Norm-preserving maps; Preservers; secondary; Norm-preserving maps; Composition operators; Preservers; Evaluation functionals; Norm-contracting maps; primary
• ISSN: 0723-0869; 1878-0792
• DOI: 10.1016/j.exmath.2008.06.003

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