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 $$T_\tau $$ T τ defined on the de Branges space $${\mathcal H}(E)$$ H ( E ) is bounded, then a suitably defined measure $$d\mu (z)$$ d μ ( z ) is a Carleson measure for the associated model space $$K(\Theta )$$ K ( Θ ) . This relation allows us to state necessary conditions for the boundedness of the vertical translation $$T_\tau $$ T τ . Finally, similar results are also obtained for the horizontal translation $$T_\sigma $$ T σ .

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Three different characterizations of one-component bounded analytic functions are provided. The first one is related to the the inner-outer factorization, the second one is in terms of the size of the reproducing kernels in the corresponding de Branges-Rovnyak spaces and the last one concerns the associated Clark measure.