  It is well known that on the Hardy space $H^2(\mathbb{D})$ or weighted
Bergman space $A^2_{\alpha}(\mathbb{D})$ over the unit disk, the adjoint of a
linear fractional composition operator equals the product of a composition
operator and two Toeplitz operators. On $S^2(\mathbb{D})$, the space of
analytic functions on the disk whose first derivatives belong to
$H^2(\mathbb{D})$, Heller showed that a similar formula holds modulo the ideal
of compact operators. In this paper we investigate what the situation is like
on other weighted Hardy spaces.
