A counterexample concerning the coefficient formula for de Branges-Rovnyak norms

Abstract

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $ϕ=b/a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $ϕ\in H^2$, then \[ f\in\mathcal{H}(b) \iff \Bigl(f\in H^2 \text{ and }\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr), \] in which case we have \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2. \] It was also shown that there exists a space $\mathcal{H}(b)$ for which $ϕ\in\cap_{p<2}H^p$, yet \[ f\in\mathcal{H}(b)\not\Rightarrow \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2<\infty. \] Here we complete this circle of ideas by showing that there exists a space $\mathcal{H}(b)$ for which $ϕ\in\cap_{p<1}H^p$, yet \[ \Bigl(f\in H^2\text{ and } \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hatϕ(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr) \not\Rightarrow f\in\mathcal{H}(b). \]