Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces

RESUMO

We consider the harmonically weighted Dirichlet spaces $D(\mu)$ induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure $m$ on the unit circle $\mathbb T$. Let $H^2$ and $D$ denote the Hardy space and the Dirichlet space on the unit disc $\mathbb D$, respectively. For any $f\in H^2$, let $m_f$ denote the measure defined by $dm_f(\zeta)=|f(\zeta)|^2 dm(\zeta)$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $|f'(z)|^2=O(\frac{1}{(1-|z|)^{1-\epsilon}})$ for some $\epsilon>0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f'$ is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of $D(m_f)$ in terms of the radial zero set of $f$. As an application, we prove that if $f',g'$ are bounded and the radial zero sets of $f$ and $g$ are different, then the operators $(M_z, D(m_f))$ and $(M_z, D(m_g))$ are not similar. This generalizes a result of Richter showing that the operators $(M_z, D(m_{z-1}))$ and $(M_z, D)$ are not similar. Then, we classify all invariant subspaces $\mathcal M$ of $(M_z, D(\mu))$ for which $M_z|_{\mathcal M}$ is similar to $(M_z, D(\mu))$, where $\mu$ is any measure mutually absolutely continuous with respect to $m$. Finally, we study a special case of the problem of finding all functions $g\in D$ for which $gD(m_g)$ is a closed invariant subspace of $(M_z, D)$. A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.