Necessary Conditions for Boundedness of Translation Operator in de Branges Spaces

Abstract The translation operator is bounded in the Paley–Wiener spaces and, more generally, Bernstein spaces. goal of this paper to find some necessary conditions for boundedness de Branges spaces, which are special cases. Indeed, if vertical $$T_\tau $$ T τ defined on space $${\mathcal H}(E)$$ H ( E ) bounded, then a suitably measure $$d\mu (z)$$ d μ z Carleson associated model $$K(\Theta )$$ K Θ . This relation allows us state Finally, similar results also obtained horizontal $$T_\sigma σ