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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\tau $$\end{document} defined on the de Branges space H(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal H}(E)$$\end{document} is bounded, then a suitably defined measure dμ(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\mu (z)$$\end{document} is a Carleson measure for the associated model space K(Θ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(\Theta )$$\end{document}. This relation allows us to state necessary conditions for the boundedness of the vertical translation Tτ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\tau $$\end{document}. Finally, similar results are also obtained for the horizontal translation Tσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_\sigma $$\end{document}.

Our interest for this problem has got a double origin. Firstly, we know that the vertical translation operator T τ is bounded in the Paley-Wiener spaces, that is, H(e −iaz ), and more generally, in the Bernstein spaces [8]. This fact makes us wonder if the vertical translation operator T τ is bounded also in other de Branges spaces as well.
Secondly, translation operator is closely related to the differentiation operator. The Bernstein inequality and the boundedness of the differentiation operator has been already widely studied by Dyakonov [6], and by Baranov [2][3][4] among the other papers.
We associate to the vertical translation operator the following measure defined in C + dμ(z) := n π δ t n +iτ (z) ϕ (t n ) where the values of {t n } will be specified later. Our first result is obtained in the second section, where we prove that dμ(z) is a Carleson measure for the associated model space K( ) if T τ is bounded. It means that the characterization of Carleson measures in model spaces is meaningful for the boundedness of T τ . About this topic, the first fundamental results were obtained by A.L. Vol'berg and S.R. Treil' in [15], where they gave a sufficient condition for a measure to be a Carleson measure in K( ) and a necessary and sufficient condition to be a Carleson measure in K( ) when is a one component inner function. Subsequently A.B. Aleksandrov gave a different proof of the same theorems in [1]. More recently, a complete characterization of Carleson measures in model spaces has been obtained [7,[9][10][11]. Thanks to these results, in the second section we find also the necessary and sufficient condition the measure dμ(z) needs to satisfy to be a Carleson measure in K( ) according to [9].
In the third section we find another necessary condition for the boundedness of T τ . We prove that if T τ is bounded in H(E), then the Hermite-Biehler function E satisfies the following condition: In order to do this, we link the measure dμ(z) to an infinite matrix , which defines a bounded operator in 2 (Z). Finally, in the fourth section we study the horizontal translation operator T σ ; it is defined in the same way: As already done for T τ , we associate a Carleson measure to T σ , which, this time, is supported on the real line. This fact allows us to give an easier necessary condition that the horizontal translation operator must satisfy in order to be bounded on H(E), (4.5).
The Condition (4.5) holds only when the derivative of the phase of the Hermite-Biehler function E is lower and upper bounded on R.
Let us first recall some definitions concerning the de Branges spaces H(E). For an extended description of H(E), one can study the book [5] and the papers [13,14]. An entire functions E belongs to the Hermite-Biehler class if it satisfies the inequality We define H(E) in the following way: where H 2 (C + ) is the Hardy space of the upper half plane.
Using the reproducing kernel of the Hardy space H 2 (C + ), we also define the reproducing kernel k w of the de Branges space H(E) [5,Problem 50], Moreover, we associate to the Hermite-Biehler function E the phase function ϕ, that is the C ∞ (R) function such that The following relationship between ϕ and k w holds: is the reproducing kernel of K ( ). We say that x ≡ π α if there is k ∈ Z such that x = α + kπ . A positive function f (x) 1 if there are two positive constants C, B such that 0 < C ≤ f (x) ≤ B < ∞, for every x ∈ R.

Vertical Translation, Carleson Measure and Two Weighted Hilbert Transform
In this section we would like to show that the problem of the boundedness of vertical translation operator T τ in the de Branges space H(E) can be reformulated as a problem concerning Carleson measures in the model space K( ). Next, we will show that this problem is equivalent to the boundedness of the two weighted Hilbert transform. The measures involved will be purely atomic.
Notation Given the de Branges space H(E), we can choose families of points t α n n∈Z ⊂ R with 0 ≤ α < π so that the set of reproducing kernel k t α n (z) is a basis of H(E) for every α, except at most one α [5].
We make the following computations:

Remark 2.3
In the proof of the above theorem, if we consider the points is still a Carleson measure for K( ).
Notation To simplify the notation, we would say that E( , μ α ) is the immersion operator from K ( ) to L 2 (dμ α ). In Theorem 2.1, we proved that the operator E( , μ α ) is continuous.
Before providing the necessary condition for the boundedness of T τ , in the rest of this section we want to prove the necessary and sufficient condition that the measure dμ α must satisfy in order to be a Carleson measure for K( ) [9]. From now on we omit the letter α as our calculations work for any 0 ≤ α < π.
First of all let us prove an equivalent description for K ( ).
The operator H 0 , defined as Proof This theorem is the Nyquist-Shannon sampling theorem for Model spaces [5,12]. Indeed for every f ∈ K( ) Indeed we choose t n so that (t n ) = 1.
The boundedness of E( , μ) is equivalent to the boundedness of the two weighted Hilbert transform H τ .

Theorem 2.5 The immersion operator E( , μ) is bounded if and only if the two weighted Hilbert transform H
For any g ∈ K( ), let g = H 0 f according to (2.2); the above inequality says that Thus, we have that H τ 2 ≤ C 2 . Since the argument is completely reversible, we can say that H τ = C which proves the theorem.
Using the conclusive result of Lacey-Sawyer-Shen-Uriarte-Tuero-Wick [10] we can characterize when the two weights Hilbert transform H τ is bounded.
We would like to apply the general theorem of [9] to our particular situation (2.3), (2.4) and (2.5). The first condition becomes (2.6) where we used Lagrange theorem The second condition becomes where we used again Lagrange theorem. The third condition becomes .
Finally the fourth condition becomes (2.9) Therefore we can state the following theorem.

Boundedness of T and the Matrix 0
In this section we prove a necessary condition the Hemite-Biehler E must satisfy if the vertical translation operator T τ is bounded in H(E). Indeed if T τ < C, then, due to Theorem 2.1, This implies that ∀n ∈ Z and for every 0 ≤ α < π The above relation assure that there exists a family of uniformly bounded operators (α) in 2 (Z).

Proposition 3.1 For every 0 ≤ α < π, there exists an infinite matrix (α)
which acts in 2 (Z), such that for every 0 ≤ α < π Proof First of all, thanks to Shannon sampling theorem for de Branges spaces [5], we know that for every 0 ≤ α < π, wherek w is the normalized reproducing kernel of H(E). We can now identify in the canonical way the space Span n∈Z k t α n H with 2 (Z) as k t n ,k t m H = δ n,m . Therefore we define the operator Thanks to estimate (3.1), for every 0 ≤ α < π the operator T (α) is bounded. Consequently we can define the family of infinite matrices (α) as in (3.2) such that (α) ≤ C.

Remark 3.2 We can explicitly described the matrix (α). Indeed
Of course, the diagonal elements are bounded; Indeed, for any 0 ≤ α < π We are now ready to state the necessary condition the Hermite-Biehler function E must satisfy when the vertical translation operator T τ is bounded in H(E).

Theorem 3.3 Let us assume that the vertical translation operator T τ is bounded in H(E). Then the Hermite-Biehler function E satisfies
Proof As already explained in the above proposition, relation (3.5) holds. We explicitly compute (3.5) in order to obtain (3.6). Indeed .
Remark 3. 4 We point also that the estimate (3.6) can be rewritten as ≤C.

Horizontal Translation and Carleson Measure
In this section we prove a necessary condition for the boundedness of the horizontal translation T σ . First of all we have the analogue of Theorem 2.1 with almost the same proof.

Theorem 4.1 If the operator T σ is bounded in H(E), the measure
is a Carleson measure for K( ).
Therefore if T σ is bounded the measure dν α (z) is a Carleson measure for K( ) and it is supported in R.
The following theorem, proved in [2], will give us a necessary and sufficient condition the measure dν α (z) of (4.1) must satisfy in order to be a Carleson measure for K( ).
and let μ be a Borel measure on the line R. Then (1) implies (2): (1) There are constants L, C L > 0 such that μ(I ) < C L L for any interval I such that Under the additional condition |ϕ (t)| > 0, the condition (2) implies (1).
We can apply the above result to the measure dν α (z) of (4.1).
We can finally state the necessary condition the Hermite-Biehler function E has to satisfy so that T σ is bounded.

Theorem 4.4 Let ϕ (x)
1. If the horizontal translation operator T σ is bounded, then The Condition (4.5) comes directly from (4.4) by varying α ∈ [0, π) and n ∈ Z. Indeed, for every x ∈ R, x = α + nπ for some 0 ≤ α < π and n ∈ Z. Consequently, since C L in (4.4) does not depend on α, Funding Open access funding provided by Universitá degli Studi di Milano within the CRUI-CARE Agreement.
Data Availability All data generated or analysed during this study are included in this published article (and its supplementary information files).

Conflict of interest
The author declares that they no conflict of interest.
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