de branges spaces of entire functions closed under forming difference quotients

With a de Branges space H(E) of entire functions a function q, analytic in C+ and satisfying there Im q(z) greater than or equal to 0, is associated. In this note we give necessary and sufficient conditions for H(E) to be closed under forming certain difference quotients in terms of the poles and zeros of q. Moreover, we obtain a criterion whether a function q possessing the above mentioned properties can be written as the quotient of the right upper and right lower entry of an entire matrix function W(z) satisfying a certain kernel condition.