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\(T_{\overline {{{(1 - z)} \mathord{\left/ {\vphantom {{(1 - z)} 2}} \right. \kern-\nulldelimiterspace} 2}} } \).