This paper is devoted to the introduction and systematic study of (P,Q)-normal operators in the context of semi-Hilbertian spaces, where P and Q are non-constant complex polynomials in one variable. This class generalizes the well-known notion of polynomially normal operators and offers a natural setting to study their structural properties in spaces endowed with a semi-inner product induced by a positive operator. We establish fundamental properties of (P,Q)-normal operators, including conditions for commutativity with respect to the A-adjoint and relations to other classes of A-operators. Several examples are provided to illustrate the theory and demonstrate how (P,Q)-normality extends classical concepts in operator theory.
Mahmoud et al. (Sun,) studied this question.