De Branges spaces of entire functions closed under forming difference quotients (Q1580922)

Let \(E(z)\) be an entire function on the complex plane \(\mathbb{C}\) such that \(|E^{\#}(z)|<|E(x)|\) for \(z\) in the upper halfplane, where \(E^{\#}(z)= \overline{E(\overline z)}\). Let \(H(E)\) denote the de Branges-Hilbert space defined by \(E\). The condition that \(H(E)\) be closed under difference quotients reads thus: \((*)\) given \(F\in H(E)\) and \(z_0\in C\), the function \((F(z)- F(z_0))/(z- z_0)\) also belongs to \(H(E)\). A necessary and sufficient condition for \(H(E)\) to satisfy \((*)\) was given by \textit{L. de Branges} [``Hilbert spaces of entire functions'', London (1968; Zbl 0157.43301)] in terms of growth conditions on the function \(E(z)\). In the present paper, the author writes \(E(z)= A(z)- iB(z)\), where \(A= (E+ E^{\#})/2\), \(B= i(E- E^{\#})/2\), and derives a necessary and sufficient condition for \(H(E)\) to be isometric to some \(H(E_1)\) which satisfies \((*)\) in terms of the zeros of \(A(z)\) and \(B(z)\).