De Branges spaces of entire functions closed under forming difference quotients

Abstract

With a de Branges spaceH(E) of entire functions a functionq, analytic in ℂ⁺ and satisfying there Imq(z)≥0, is associated. In this note we give necessary and sufficient conditions forH(E) to be closed under forming certain difference quotients in terms of the poles and zeros ofq. Moreover, we obtain a criterion whether a functionq possessing the above mentioned properties can be written as the quotient of the right upper and right lower entry of an entire matrix functionW (z) satisfying a certain kernel condition.