Inner multipliers of de Branges' spaces (Q916976)

The author considers the question of when an arbitrary \(H^{\infty}\) function m is a multiplier of the de Branges space \({\mathcal H}(b)\), where \(b\in unit\) ball of \(H^{\infty}\). Necessary and sufficient conditions that m be a multiplier of \({\mathcal H}(b)\) are obtained, and it is shown that there are no nonconstant inner multipliers of \({\mathcal H}(b)\), when b is a nonconstant extreme point of the unit ball of \(H^{\infty}\). The author provides a new proof of the known fact that \({\mathcal H}(b)\) is invariant under multiplication by z when b is not an extreme point of the unit ball of \(H^{\infty}\). Also, a new proof is given that an inner function m is a multiplier of \({\mathcal H}(b)\) for \(b(z)=(1+z)/2\) if and only if m belongs to the range of the Toeplitz operator \(T_{\overline{(1-z)/2}}\).