On irreducibility for a class of parameterized multiplication operators by polynomials

We study the reducibility of multiplication operators defined by quasi-homogeneous polynomials on the weighted Hardy space H²(ω, δ). The study of these operators essentially reduces to that of the parameterized multiplication operators Mz+α w for α ∈ (0, +∞). By employing determinant techniques and algebraic methods, we prove that the operator Mz+α w is reducible for at most a countable subset of α. Furthermore, we construct examples to show that such a set of α can be empty or a singleton. Our results also provide support for a conjecture due to Guo and Wang on the reducibility of Mz+w.