For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K, ν)denote the closure of Rat(K) in L2(ν) and let Sν denote the operator of multiplication by the independent variable z on R2(K,ν), that is, Sνf = zf for every f ∈ R2(K,ν).Suppose Ω is a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Let σ denote a harmonic measure for Ω. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2 (-Ω, σ). We show that for a given ν supported on -Ω, Sν is quasi-similar to Sσ if and only if ν|(a)Ω