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 $$T_\tau $$ 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 .

Subspaces of de Branges Spaces Generated by Majorants

For a given de Branges space ℋ (E ) we investigate de Branges subspaces defined in terms of majorants on the real axis. If ω is a nonnegative function on ℝ, we consider the subspaceWe show that ℛω (E ) is a de Branges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorants.

Admissible Majorants for Model Subspaces of H2, Part II: Fast Winding of the Generating Inner Function

This paper is a continuation of [6]. We consider the model subspaces Kϴ = H2 ϴ ϴH2 of the Hardy space H2 generated by an inner function ϴ in the upper half plane. Our main object is the class of admissible majorants for Kϴ, denoted by Adm ϴ and consisting of all functions ω defined on ℝ such that there exists an f ≠ 0, f ∈ Kϴ satisfying |f(x)| ≤ ω(x) almost everywhere on ℝ. Firstly, using some simple Hilbert transformtechniques, we obtain a general multiplier theorem applicable to any Kϴ generated by a meromorphic inner function. In contrast with [6], we consider the generating functions ϴ such that the unit vector ϴ(x) winds up fast as x grows from –∞ to ∞. In particular, we consider ϴ = B where B is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from ℝ. It is shown, among other things, that for any such B, any even ω decreasing on (0,∞) with a finite logarithmic integral is in Adm B (unlike the “vertical” case treated in [6]), thus generalizing (with a new proof) a classical result related to Adm exp(iσz), σ > 0. Some oscillating ω's in Adm B are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm exp(iσz), σ > 0, and to de Branges’ space ℋ(E).