Inner multipliers of de Branges's spaces

Abstract

For a given functionb in the unit ball ofH ^∞ and an arbitraryH ^∞ functionm, the question of whenm is a multiplier of the de Branges space$\mathcal{H}(b)$ (that is, when$\mathcal{H}(b)$ is invariant under multiplication bym) is examined. Some necessary and sufficient conditions thatm be a multiplier of$\mathcal{H}(b)$ are found and it is shown that there are no nonconstant inner multipliers of$\mathcal{H}(b)$ whenb is a nonconstant extreme point of the unit ball ofH ^∞. A new proof is given of the known fact that$\mathcal{H}(b)$ is invariant under multiplication byz whenb is not an extreme point of the unit ball ofH ^∞. Finally, we give a new proof of the known fact that an inner functionm is a multiplier of$\mathcal{H}(b)$ forb(z)=(1+z)/2 if and only ifm belongs to the range of$T_{\overline {{{(1 - z)} \mathord{\left/ {\vphantom {{(1 - z)} 2}} \right. \kern-\nulldelimiterspace} 2}} }$.