Extending Polynomially Normal Operators to (P, Q)-Normal Operators in Semi-Hilbertian Spaces

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 $\mathbf{A}$-adjoint and relations to other classes of $\mathbf{A}$-operators. Several examples are provided to illustrate the theory and demonstrate how $(P,Q)$-normality extends classical concepts in operator theory.