  Given two multiplicity-free operators $T_1$ and $T_2$ of class $C_0$ having
the same finite Blaschke product as minimal function, the operator algebras
$H^\infty(T_1)$ and $H^\infty(T_2)$ are isomorphic and $T_1$ is similar to
$T_2$. We find conditions under which the norm of the similarity between the
operators can be controlled by the norm of the algebra isomorphism. As an
application, we improve upon earlier work and obtain results regarding
similarity when the minimal function is an infinite product of finite Blaschke
products satisfying the generalized Carleson condition.
