Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 987–995 | Cite as

Outer Functions and Divergence in de Branges–Rovnyak Spaces

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Abstract

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\), in other words, \(\lim _{r\rightarrow 1^-}\Vert f_r-f\Vert =0\). We construct a de Branges–Rovnyak space \(\mathcal{H}(b)\) in which the polynomials are dense, and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). The essential feature of our construction lies in the fact that b is an outer function.

Keywords

De Branges–Rovnyak space Outer function Toeplitz operator 

Mathematics Subject Classification

46E22 47B32 30H15 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de mathématiques et de statistiqueUniversité LavalQuébecCanada

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