Concrete Examples of $$\varvec{\mathscr {H}(b)}$$ H ( b
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Concrete Examples of H (b) Spaces Emmanuel Fricain1 · Andreas Hartmann2 · William T. Ross3
Received: 10 April 2015 / Revised: 13 August 2015 / Accepted: 21 August 2015 / Published online: 19 October 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract In this paper, we give an explicit description of de Branges–Rovnyak spaces H (b) when b is of the form q r , where q is a rational outer function in the closed unit ball of H ∞ and r is a positive number Keywords de Branges–Rovnyak spaces · Non-extreme points · Kernel functions · Corona pairs Mathematics Subject Classification
30J05 · 30H10 · 46E22
Communicated by Dmitry Khavinson. This work was initiated while the E. Fricain and A. Hartmann were staying at the University of Richmond. E. Fricain and A. Hartmann would like to thank that institution for the great hospitality. Work supported by Labex CEMPI (ANR-11-LABX-0007-01).
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William T. Ross [email protected] Emmanuel Fricain [email protected] Andreas Hartmann [email protected]
1
Laboratoire Paul Painlevé, Université Lille 1, 59 655 Villeneuve d’Ascq Cedex, France
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Institut de Mathématiques de Bordeaux, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France
3
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173, USA
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1 Introduction The purpose of this paper is to explicitly describe the elements of the de Branges– Rovnyak space H (b) for certain b ∈ b(H ∞ ). Here H ∞ denotes the space of bounded analytic functions on the open unit disk D normed by f ∞ := supz∈D | f (z)|, and b(H ∞ ) := {g ∈ H ∞ : g∞ 1} is the closed unit ball in H ∞ and, for b ∈ b(H ∞ ), the de Branges–Rovnyak space H (b) is the reproducing kernel Hilbert space of analytic functions on D whose kernel is kλb (z) :=
1 − b(λ)b(z) 1 − λz
, λ, z ∈ D.
Besides possessing a fascinating internal structure [11], H (b) spaces play an important role in several aspects of function theory and operator theory, most importantly, in the model theory for many types of contraction operators [3,4]. Despite the important role H (b) spaces play in operator theory, the exact contents of H (b) often remain mysterious. What functions belong to H (b)? Certainly all of the kernel functions kλb do (and have dense linear span). What else? In this paper, we give a precise description of the elements of H (b) for certain relatively simple b, namely positive powers of rational outer functions. Our description needs the following setup. If b ∈ b(H ∞ ) is a non-extreme point of b(H ∞ ), equivalently, log(1 − |b|) ∈ L 1 (T, m) (where T := {ζ ∈ C : |ζ | = 1} and m is Lebesgue measure on T normalized so that m(T) = 1), then there exists a unique outer function a ∈ b(H ∞ ), called the Pythagorean mate for b, such that a(0) > 0 and |a|2 +|b|2 = 1 almost everywhere on T. The pair (a, b) is said to be a Pythagorean pair. Our first observation says that in certain situations, H (br ) does not depend on r > 0. Theorem 1.1 (1) For b ∈ b(H
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