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

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 $d{m}_f(\zeta )=\mathrm{\mid }f(\zeta ){\mathrm{\mid }}^{2}dm(\zeta )$. Our first result provides a characterization of functions in $D(m_f)$, when $f$ satisfies the condition $\mathrm{\mid }{f}^{\mathrm{\prime }}(z){\mathrm{\mid }}^2=O(\frac{1}{(1-\mathrm{\mid }z\mathrm{\mid }{)}^{1-\epsilon }})$ for some $\epsilon >0$. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when $f^{\mathrm{\prime }}$ 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 $f^{\mathrm{\prime }},g^{\mathrm{\prime }}$ 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 $M_z{\mathrm{\mid }}_{\mathcal{M}}$ 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.