Outer Functions and Divergence in de Branges–Rovnyak Spaces

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, $\lim_{r\to1^-}\|f_r-f\|=0$. 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 $\lim_{r\to1^-}\|f_r\|_{{\mathcal H}(b)}=\infty$. The essential feature of our construction lies in the fact that $b$ is an outer function.