Necessary Conditions for Boundedness of Translation Operator in de Branges Spaces

The translation operator is bounded in the Paley–Wiener spaces and, more generally, in the Bernstein spaces. The goal of this paper is to find some necessary conditions for the boundedness of the translation operator in the de Branges spaces, of which the Paley–Wiener spaces are special cases. Indeed, if the vertical translation operator Tτ defined on the de Branges space H(E) is bounded, then a suitably defined measure dμ(z) is a Carleson measure for the associated model space K(Θ). This relation allows us to state necessary conditions for the boundedness of the vertical translation Tτ. Finally, similar results are also obtained for the horizontal translation Tσ.