Outer functions and divergence in de Branges-Rovnyak spaces

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, ${\lim }_{r\to 1^{-}}\mathrm{\parallel }{f}_r-f\mathrm{\parallel }=0$. 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 ${\lim }_{r\to 1^{-}}\mathrm{\parallel }{f}_r{\mathrm{\parallel }}_{\mathcal{H}(b)}=\mathrm{\infty }$. The essential feature of our construction lies in the fact that $b$ is an outer function.