Let φ be a holomorphic self-map of the open unit disk U, and let F be a holomorphic function on U that is absolutely convergent in the same domain. This study introduces and analyzes a new class of composition operators in the Hardy space H², known as F-composition operators, which are induced by the pair (F, φ). The main objective of this study was to investigate their analytical structure and functional properties. We define the F-composition operator and explore its adjoint representation, emphasizing its relationship with several families of bounded linear operators, including the self-adjoint, normal, unitary, and isometric operators. Furthermore, we establish the necessary and sufficient conditions for the invertibility of these operators, which depend on the analytic behavior of both F and φ. The results contribute to a broader understanding of composition-type operators and provide a new framework that generalizes the classical operator theory on Hardy spaces by incorporating an analytic weight function, F.
Hussein et al. (Wed,) studied this question.