The backward shift on Dirichlet-type spaces

We study the backward shift operator on Hilbert spaces ${\mathcal{H}}_{\alpha}$ (for ${\alpha \geq 0}$) which are norm equivalent to the Dirichlet-type spaces $D_{\alpha}$. Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on ${\mathcal{H}}_{\alpha}$ in terms of a weighted superposition of backward shifts.